. _. g r, . ._ _, 7.‘ _ y _ h. x V , a. . , , . . S ‘ m m , V M o m... w 7 H . x N , . , W M U M _ . ,2. , ‘ a Mmmw _ .m m 3 m I. an m W . f . mm is. w..." m ‘ $ m flu. mm M . . ,: _ .... arr: LIBRARY iéichigzn Sula University "mFSI. This is to certify that the thesis entitled NUMERICAL SOLUTIONS OF JEFFREY-HAMEL FLOW AT FIXED FLOW RATES presented by » Floyd Ernest LeCureux has been accepted towards fulfillment of the requirements for Ph.D. degree in Mechanical Engineering (4)133 Li/% 5 9‘1"“ Major professor Date ”VI/b1 H1 /772 0-169 ABSTRACT NUMERICAL SOLUTIONS OF JEFFREY-HAMEL FLOW AT FIXED FLOW RATES By Floyd Ernest LeCureux The radial flow of viscous incompressible fluid between non- parallel plane walls governed by the Navier-Stokes equations has been previously investigated. However, the known solutions include trans- cendental equations containing elliptic functions. Therefore, even though some solutions are known, there are no explicit equations or methods available for determining the shape of the velocity profile for specified boundary conditions that do not require an iterative or graphical method of solution. Further it is not readily apparent which profiles are possible for Specified boundary conditions. The major objective of this work is to develop a procedure for determining the shape of the velocity profile that may be assumed by the fluid for specified values of flow rate and angle of inclina- tion of the walls. Further, the limits on each type of flow profile are investigated and comparisons are made between the various profiles. The basic types of flow considered are symmetrical diverging where the centerline velocity is outward, symmetrical converging where the centerline velocity is inward, nonsymmetrical with one interior zero, and non-symmetrical with three interior zeroes. For each case the transcendental equations resulting from application of the boundary conditions are presented in the form of a flow parameter graph. Each graph has coordinates of flow rate and angle of Floyd Ernest LeCureux inclination and has two intersecting families of curves representing areas where solutions are possible. In each case the boundaries are defined and relations between different profiles are noted. These flow parameter graphs are then used to plot several examples of velo- city profiles by reading the flow parameters for use in the velocity distribution function from the graph for particular values of flow rate and angle of inclination. A minor objective of this work is the verification of a modified perturbation technique as applied to this problem. The modification of the normal perturbation procedure allows the deter- mination of velocity distributions for small flow rates at a critical angle of inclination where standard perturbation equations are not defined. The velocity profiles obtained by use of the flow para- meter graphs are used to verify the modified perturbation method and to analyze the accuracy of this method for various boundary conditions. NUMERICAL SOLUTIONS OF JEFFREY-HAMEL FLOW AT FIXED FLOW RATES By Floyd Ernest LeCureux A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mechanical Engineering 1972 To Jeanne - for her patience and understanding ii ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to his guidance committee chairman, Professor David H. Y. Yen, for his valuable assistance in conducting this research. He also wishes to thank Professors John F. Foss, J. Sutherland Frame, Donald J. Montgomery, and Merle C. Potter for serving on his guidance committee. Thankful acknowledgment is also extended to Mrs. Thelma Liszewski and Miss Annette Doyle for their excellent typing. iii TABLE OF CONTENTS Page LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . vi I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Procedure . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Order of Presentation . . . . . . . . . . . . . . . . 6 II. FLOW MODEL . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . 8 2.2 Type of Fluid Flow and Coordinate System . . . . . . . 9 2.3 Fundamental Equations . . . . . . . . . . . . . . . . 12 2.4 Dimensionless Velocity Function . . . . , . . , . . , 14 2.4.1 Dimensionless Velocity Function for SD Profiles 17 2.4.2 Dimensionless Velocity Function for SC Profiles 21 2.4.3 Dimensionless Velocity Function for Non-Symmetrical Profiles . . . . . . . . . . 24 2.4.4 Dimensionless Velocity Function for SDCO Flow . 24 2.5 Application of Boundary Conditions . . . . . . . . . . 29 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . 32 III. FLOW PARAMETER GRAPHS . . . . . . . . . . . . . . . . . . . . 34 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . 34 3.2 SD Flow . . . . . . . . . . . . . . . . . . . . . . . 35 3.2.1 Solution where C(F) has Three Real Roots . . . 36 3.2.2 Solution for SDCO Flow . . . . . . . . . . . . 54 3.3 SC Flow . . . . . . . . . . . . . . . . . . . . . . . 62 3.4 Non-Symmetrical Flow . . . . . . . . . . . . . . . . . 77 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . 90 IV. VERIFICATION OF A MODIFIED PERTURBATION TECHNIQUE . . . . . . 95 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . 95 4.2 Standard Perturbation Method . . . . . . . . . . . . . 97 4.3 Modified Perturbation Method . . . . . . . . . . . . 100 4.4 Comparison with Exact Solution . . . . . . . . . . . 106 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . 114 V. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . 116 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . 119 iv TABLE OF CONTENTS Page APPENDICES APPENDIX A: SUBROUTINE DIVERGE . . . . . . . . . . . . . . . 121 SUBROUTINECOMPLEX...............13O SUBROUTINECONVERG...............136 SUBROUTINE NONSYMI . . . . . . . . . . . . . . . 146 SUBROUTINENONSYM3..............,152 SUBROUTINES GRAPH, ELLI, & SN . . . . . . . . . 158 APPENDIX B: SUBROUTINE PRO SD . . . . . . . . . . . . . . . 163 SUBROUTINE PRO SDCC . . . . . . . . . . . . . . 169 SUBROUTINE PRO CON . . . . . . . . . . . . . . . 173 SUBROUTINE PRO N1 . . . . . . . . . . . . . . . 178 APPENDIX C: SUBROUTINE PRO CON . . . . . . . . . . . . . . . 183 SUBROUTINE PERT . . . . . . . . . . . . . . . . 191 LIST OF FIGURES Figure Page 2.1 Polar Coordinate System . . . . . . . . . . . . . . . . . . 10 2.2 Schematic Diagram of Inward and Outward Flow . . . . . . . 10 2.3 Velocity Profile Notation . . . . . . . . . . . . . . . . . 11 2.4 Possible Solutions of Equation (2.17) . . . . . . . . . . . 16 2.5 Plot Of F, SD Flow, for m1 = .85, k = .55 . . . . . . . . . 16 2.6 Plot of F for SD Profiles for k = .55, m1 = Various . . . . 20 2.7 Plot of F for SD Profiles, with m = .55, k = Various . . . 20 2.8 Plot of F for SC Profiles; Various k and m3 . . . . . . . . 23 2.9 Plot of F for SD Profiles, m1 = .85, k = .55 . . . . . . . 23 2.10 Various Non-symmetrical Velocity Profiles . . . . . . . . . 25 2.11 Plot of F for SDCO Flow, for Various m2 and k . . . . . . . 28 3.1 Symbolic Flow Chart for Constant k Curves, SDO Flow . . . . 38 3.2 Constant k Curves, SDO Flow . . . . . . . . . . . . . . . . 39 3.3 Constant k Curves, SDO & SD2 Flow . . . . . . . . . . . . . 42 3.4 Symbolic Flow Chart for Constant m Curves, SDO Flow . . . . 46 3.5 Constant m Curves, SDO Flow . . . . . . . . . . . . . . . . 47 3.6 Constant m Curves, SDO and SD2 . . . . . . . . . . . . . . 48 3.7 Flow Parameter Graph, SDO and SD2 Flows . . . . . . . . . . 50 3.8 Flow Parameter Graph SDO and SD2 Flows (Primary Solutions). 51 3.9 Typical Velocity Profiles, SD Flow . . . . . . . . . . . . 53 3.10 Symbolic Flow Chart for Constant k Curves, SDCO Flow . . . 56 3.11 Symbolic Flow Chart for Constant m Curves, SDCO Flow . . . 57 vi Figure Page 3.12 Flow Parameter Graph-SDCO Flow . . . . . . . . . . . . . . 58 3.13 Typical Velocity Profiles-SDCO Flow . . . . . . . . . . . 60 3.14 Flow Parameter Graph for Diverging Flow . . . . . . . . . 61 3.15 Symbolic Flow Chart for Constant k Curves, SC Flow . . . . 64 3.16 Symbolic Flow Chart for Constant m Curves, SC Flow . . . . 65 3.17 Flow Parameter Graph, SCO Flow . . . . . . . . . . . . . . 66 3.18 Plot of m3 = .95, 800 Flow . . . . . . . . . . . . . . . . 69 3.19 Flow Parameter Graph, SC Flow . . . . . . . . . . . . . . 71 3.20 Plot of m3 = .95, SC Flow . . . . . . . . . . . . . . . . 74 3.21 Flow Parameter Graph, SC Flow (Primary Solutions) . . . . 75 3.22 Typical Velocity Profiles, SC Flow . . . . . . . . . . . . 76 3.23 Symbolic Flow Chart for Constant k Curves, N1 Flow . . . . 80 3.24 Symbolic Flow Chart for Constant m Curves, N1 Flow . . . . 81 3.25 Flow Parameter Graph, N1 Flow . . . . . . . . . . . . . . 83 3.26 Flow Parameter Graph, N3 Flow . . . . . . . . . . . . . . 84 3.27 Sketch of F for N1 Flows Showing Limiting Cases . . . . . 88 3.28 Typical Velocity Profiles, N1 and N3 Flow . . . . . . . . 91 3.29 Flow Parameter Graph, SD and SC Flow (Primary Solutions) , 92 3.30 Chart of the Range of Possible Velocity Profiles . . . . . 94 4.1 Symbolic Flow Chart for Modified Perturbation Method . . . 107 4.2 Velocity Distributions Obtained by the Modified Perturbation Me thOd O O O O C O I O O O O . O O O O O O 109 4.3 Velocity Distributions, SCZ Flow . . . . . . . . . . . . . 110 4.4 The Dimensionless Distribution Function for Comparison of 8C2 Flow and Modified Perturbation Method . . . . . . . 112 4.5 Relation Between a and a0 Near ac from Modified Perturbation Me thOd O O O O O 0 O O . O O O O O O O O O 11-3 vii I . INTROD UCTION 1.1 Background One of the classic problems of incompressible, viscous fluid dynamics is that of flow between parallel plane walls. For this problem the non-linear Navier-Stokes equations may be reduced to a relatively simple form which can be integrated to yield an exact solu- tion for the familiar parabolic velocity distribution. A slightly less known but equally interesting problem is that of flow between non-parallel plane walls. For this case, although the non-linear terms in the Navier-Stokes equations do not vanish, an exact solution is still possible in terms of elliptic functions as first derived independently by Hamel [5] and Jeffery {7]. Since these first papers a number of authors have further investigated specialized aspects of the problem [3, ll, 12, 13]. Two of the more comprehensive studies are by Rosenblatt [12] and Rosenhead [13]. Rosenhead's work, eSpecially, contains a thorough study of the possible solutions, the effect of increasing Reynolds Number, and the definition of areas where particular solutions may or may not be mathematically possible. However, as stated in a more recent paper by Millsaps and Pohlhausen [9], the unavailability of an extensive table of elliptic functions appears to be one of the prin- cipal reasons why numerically calculated velocity profiles for assigned Reynolds Numbers have been published for only a few selected values. Millsaps and Pohlhausen develOped several solutions in their study of thermal distributions between non-parallel plane walls. 1 Their solutions are in terms of the more familiar Jacobian elliptic functions rather than the Weierstrass elliptic functions as used by some previous authors. Their results, however, still contain a trans- cendental equation which requires either a graphical solution or some iterative procedure. Thus the basic solutions for the problem of purely radial two-dimensional flow of viscous fluid between non-parallel walls are well known, and a few velocity profiles, dependent on the solution of a transcendental equation, have been calculated; and, these works have in fact been referred to in several textbooks, for example, [3, 8, and 11]. On the other hand, it should be mentioned that there is no. method' -that can . easily be - used a to determine the shape of the mathematically possible velocity distributions for a fixed flow rate and a particular angle of inclination of the walls.‘ More specifically, the following questions remain unanswered. First, will a particular set of values of flow rate and angle of inclination allow purely inward converging flow or purely outward diverging flow, or flow. with areas of both converging and diverging flows? These various possible solutions may all exist for a given flow rate and angle of inclination. Second, what is the exact shape of the velocity profile for this particular set of conditions? Several authors,[1l] for example, have obtained expressions describing the conditions necessary for purely outward flow. Rosenhead [13] has included a series of graphs which define the limits of the various types of flow. However, the determination of the shape of the velocity profile for a particular set of boundary conditions is left to the fairly complex procedure of solving a system of transcendental equations. It is 3 difficult as well as time consuming to use the "apparently available" exact solution for comparison with other investigations or for an indication of how the profiles change between different regions. 1.2 Objectives The present availability of computerized techniques of solu- tions not only make the handling of elliptic functions more amenable to calculations but also allow more straightforward investigation of the limitations of the various possible solutions. The major objec- tive of this work is to apply computer solutions to aid in the theo- retical analysis of the problem of two-dimensional, viscous flow between non-parallel walls to determine "flow parameter graphs." These graphs can then be used to readily determine which particular velocity profile is possible for a certain Specification of flow rate and angle of inclination, what the flow parameters are for this pro- file to allow easy determination of the velocity distribution function, and what the limits are for each type of velocity profile. As indicated by Rosenhead [13], for every specification of flow rate and angle of inclination the number of mathematically pos- sible velocity profiles of purely radial motion is infinite. For example, a particular set of conditions might have, mathematically speaking, possible symmetrical profiles with no interior zeroes, two interior zeroes, four interior zeroes, etc., as well as non-symmetri- cal profiles with 1,3,5,--- etc. interior zeroes.‘ In this investiga- tion only a few of these profiles are considered - in particular, sym- metrical flows with no interior zeroes and two interior zeroes, and non-symmetrical flows with one interior zero and three interior zeroes. Other solutions are indicated and one case of symmetrical flow with four interior zeroes will be noted. The determination of which of the "mathematically possible" flows would actually be assumed by the fluid when a flow rate and angle of inclination are specified can only be made after investigating stability considerations of the flow. Only the relatively simple profiles are to be considered in this study. A further minor objective of this investigation is the application of the flow parameter graphs to the verification of a modified perturbation technique. This perturbation procedure is used to predict velocity distributions for small flow rates near a critical angle of inclination where standard perturbation techniques are not applicable. 1.3 Procedure The governing equations of mass and momentum conservation reduce to a single equation which, for each of the different velocity profiles can be solved in terms of elliptic functions. For this inves- tigation each solution for the velocity distribution function is expressed in terms of one or more of the Jacobian elliptic functions. The velocity distribution so expressed is dependent on two "flow para- meters," referred to as k and m. These flow parameters are then determined from the boundary conditions by Specifying the flow rate and the angle of inclination. The resulting two equations for the 5 flow parameters k and m are transcendental equations in terms of the flow rate and the angle of inclination. Explicit solutions for k and m from these equations in terms of flow rate and angle of incli- nation are difficult. An alternative procedure used by past authors is to solve graphically or by iterative“ techniques for one 'of the variables and then solve explicitly for the other. The method used in this investi- gation is to relate the Jacobian elliptic functions to standard trigonometric functions. Then explicit solutions can be found for k and the flow rate or for m and the flow rate in terms of the trigo- nometric parameter and m, or k, respectively. Finally the angle of inclination that has been used, but not explicitly specified, can be determined in terms of the inverse Jacobian elliptic function. As a result graphs of constant m and constant k can be plotted on the flow parameter graph with flow rate and angle of inclination as coordi- nates and this graph can then be used with reasonable accuracy to determine the values of k and m for any point within the family of graphs. The above procedure is used and corresponding flow parameter graphs are plotted for each type of flow. The limits of each region are shown graphically and in many cases explicit expressions for the boundaries are derived. Several typical velocity profiles are then plotted from the values read from the flow parameter graphs. One of the values chosen for plotting a velocity distribu- tion is for small negative flow rate and an angle of inclination of about 128 degrees. For this particular angle the standard perturba- tion technique for determining the velocity profile breaks down as the unperturbed or linear solution approaches infinity. A general 6 modified perturbation technique developed by Yen and Tang [15 and 16] can be applied to this case. In order to assess the accuracy of the new perturbation technique, exact solutions for the velocity profile determined by use of the flow parameter graph at this critical angle of about 1280 can now be used. 1.4 Order of Presentation Chapter II contains the mathematical develOpment of the general problem. The basic equations and assumptions are presented first. Then the basic equations are reduced to a single second order non-linear differential equation. This equation is then solved in a manner similar to previous work to obtain general solutions for the three basic types of flow: symmetrical flow with outward flow on the centerline, symmetrical flow with inward flow on the centerline, and non-symmetrical flow. The resulting velocity profile functions are discussed and the boundary conditions are applied. The resulting sets of transcendental equations then serve as the basis for the flow para- meter charts to be presented in Chapter III. The sets of transcendental equations derived in Chapter II are used in Chapter FLIto obtain explicit equations for calculating constant k and constant m curves for the three types of veldcity profiles - symmetrical profile with outward flow on the centerline, symmetrical profile with inward flow on the centerline, and non-symme- trical profile. In each case the characteristics of the flow para- meter graphs and the limits of each type of flow are discussed. Also, several velocity profiles from various regions of each flow parameter graph are shown. . 7 In Chapter IV an example of the application of the flow para- meter graphs develOped in Chapter III is presented. The example pro- vides the verification of a modified perturbation technique. The stan- dard perturbation method is reviewed and its limitations of applica- tion near certain critical values are discussed. Then a modified method deve10ped by Yen and Tang and applied to other similar types of problems [15 and 16] is applied to this problem of velocity profile for radial flow between non-parallel planes. This modified method gives perturbation solutions for small flow rates near the critical value of the angle of inclination. The velocity distribution func- tions and flow parameter graphs develOped in Chapter III are then used to assess the accuracy of these results. A summary of the results of this study and suggested alter- natives for further study are presented in Chapter V. II . FLOW MODEL 2.1 Introduction The governing equations for viscous flows are the familiar Navier-Stokes' equations discussed at length in Schlichting [14]. These equations, even simplified for the conditions Specified for this study, still have relatively few exact solutions. However, purely radial flow between non-parallel plates is one case where an exact solution has been deve10ped by Hamel [5], and Jeffery [7]. The solu- tion, expressed in different forms by various authors [5,7,9, and 13], is still very unwieldy Since it involves transcendental equations con- taining various forms of elliptic functions. This difficulty seems to be the primary reason why even though the basic solution is well known the presentation of the solution in a form that would be readily appli- cable to a Specific problem has not been available. In this study the basic derivation of the equations given by Millsaps and Pohlhausen [9] is followed. Then by relating the elliptic functions to standard trigonometric functions the transcendental equations take on a more explicit form. Finally, the application of computer solutions pro- vides flow parameter graphs that can be used to easily determine the exact solutions based on specified values of flow rate and angle of inclination. This Chapter describes the various velocity profiles to be considered, gives the basic assumptions and governing equations, derives the dimensionless velocity profiles for each type of flow, and applies the boundary conditions to obtain Specific solutions. 8 2.2 Type of Fluid Flow and Coordinate System The following assumptions are made for this investigation. 1. The fluid is incompressible. 2. The flow is independent of time. 3. Gravitational and other body forces are negligible. 4. The flow is two-dimensional. 5. The fluid is linearly viscous, isotropic, and homogeneous. 6. The reference frame is inertial. A polar coordinate system (r,9) is used, as shown in Figure 2.1 where the walls are set at 9 “.1‘3- The velocity compo- nents in the (r,9) coordinates will be denoted by (u,v). To effect the flow conditions, necessary sources or sinks are assumed at the apex. Examples of pure inward flow towards the apex and pure outward flow away from the apex are shown in Figure 2.2. The dotted lines indicate.that r is not allowed to approach zero. The terms converging and diverging flows were not used in Figure 2.2 as they can be misleading. It is possible to have both con- verging flow and diverging flow in the same velocity profile, and the difference between purely converging, purely diverging, or part con- verging and part diverging flows can be a small change in the boundary conditions. To avoid confusion and to abbreviate the names of fre- quently used descriptive titles the system of identification shown in Figure 2.3 will be used. The first letter, N or S, describes the velocity profile as non-symmetrical or symmetrical, respectively. For symmetrical profiles, a second letter, C or D, denotes that flow on the center line is either converging or diverging, reSpectively. Also, a number, 0, l, 2, 3, ---, is used in each name to Specify the number of interior zeroes. 10 e = 0° FIGURE 2.1 POLAR COORDINATE SYSTEM / /' ’i/‘V: / .f'l’fy /,./ \FIN / ,1: .14 _ / rd __,__. iii-r" “I" __ Ego-cw... , h, . 23\ Le SINK \ “I \\ \ft. I in \7\ INWARD FLOW OUTWARD FLOW FIGURE 2.2 SCHEMATIC DIAGRAM OF INWARD AND OUTWARD FLOW OHM OHM ZQHHdHOZ OHM MAMMOMM VHHOOMM> m.N MMDOHM Nam 12 2.3 Fundamental Equations In the polar coordinate system for the flows described in Section 2.2.the governing equations are presented by Schlichting [14]. Conservation of Mass (Continuity) Ogru) éy. = 5r +-ae O (2.1) Conservation of Momentum ' 2 §g_ y_5u _ y_’ _ _ l_5 2 2 BV' u u r+r§5 r '- p§%+v(Vu-:§55.:§) (2.2) 5v v 6v uv __ l §£ 2 _2__5u _v_ 2 2 5.2 15 15‘ where V = —- + - —.- + —— —- and p =. density p = pressure and 2 . 2 2 ’ s a: r Br 59 v = kinematic viscosity. The reduction of these equations to a single equation is well-known (for example, see [9]). Since the flow is purely radial v50 and equation (2.2.1) then implies 9%? = o (2.4) Thus, put u = vF(9)/r where F is a function of 9 only and is the dimensionless velocity profile. Equations (2.2) and (2.3) can now be written as: 22 2 VF 1gp VF" .- ' = ‘— + (2.5) r3 par r3 l 5 2v2F' O = uSE-gg-F—r3 (2'6) where the prime denotes differentiation with respect to e. l3 3 Multiplying (2.5) by 55-2 gives L92: 2 .. Br F + F (2.7) Integrating (2.6) with respect to 9 gives 2 2v OF p = -—7f-‘+ S(r) (2.8) r where S(r) is a function of r. Substituting (2.8) into (2.7) gives 3 r 2 CV 2 = 4F + F + F" (2-9) 0.0. NU) The left hand Side of equation (2.9) is a function of r only and the right hand Side is a function of s only. Therefore, each must be equal to a constant. Let this constant be 'J. Then 2 4F + F + F" = -J (2.10) and S(r) may be chosen as 2 2 S(r) = -f 21—iig = EZ_£. (2.11) r3 2r2 Therefore, the conservation of mass and the conservation of momentum equations lead to the single second order non-linear ordinary differ- ential equation for F in (2.10). The boundary conditions are Speci- fied by the fact that velocity at the wall is zero, and that the total flow between the planes remains constant and can be expressed as an integral of the velocity function, i.e., 21: l. u(jg) = O, or since u = r' , Mia) = 0 (2.12) 14 (I 2. f u(9)rd9 = constant E Q where Q denotes the net volume ‘(1 of fluid passing between the plates per unit of depth in unit time. Expressing the integral in terms of F, a VI Name = Q -a or (2.13) O.’ f F(e)ds = 9; E e .a * where e is a dimensionless flow rate. Equations (2.10) and boundary conditions (2.12) and (2.13) describe completely the mathematical problem. 2.4 Dimensionless Velocity Function The solutions of equation (2.10) for the dimensionless velo- city F are described in [9] and the prOperties of the various solu- tions are well illustrated in [11] and [13]. The following is based on these works. Equation (2.10) may be integrated once after multiplying by 6F'. This results in 3 2 2F +12F2+3(F') +6JF- 2H = o (2.14) where H is a constant of integration. Solving for F' which is 3% gives a—F; _ + 2 % 2 3% 59 - __('§ ) (H - 3JF - 6F - F ) (2.15) * Most previous publications have defined flow rate in terms of Rey- nolds' Number. For example [9] defines R0 = F0 = uOr/v where “0 is the centerline velocity. However, when considering flow with interior zeroes “0 may not be a representative velocity which requires a re-definition of R0. Therefore, all calculations in this study are made with respect to e as defined above. 15 It then follows that lFZ e = i(%)§f__.._dF 1 (2.16) F1 [C(F)]? where C(F) H - 3JF - 6F2 - F3 (2.17) "I The choice of the limits of integration for (2.16) depends on the boundary conditions. Clearly 9 is given by an elliptic integral. From (2.12) if F = 0 when G = i<1 then C(F) in equation (2.16) must be real when F = 0. Therefore C(O) 3.0 (2.18) An approximate plot of equation (2.17) can be obtained since C(O) 3.0, C(F) < 0 for large values of F, and C(F) > 0 for large negative values of F. Therefore, there must be at least one positive root for C(F) = O and there may be three. However, assuming three roots for (2.17), say e1, e2, and 63, then C(F) H-3JF-6F2-F3 = (el-F)(e2-F)(e3-F) (2.19) 2 = e 1e 2e 3+F(-e‘1e2-e1e 3-e2e3)+F (e 1-+e2-ie 3) -F3 (2. 20) and equating powers of F it shows that H = e1e2e3 (2.21) 3J = elez + e1e3 + e2e3 (2.22) -6 = el + e2 + e3 (2.23) Therefore, from (2.23) the three roots must add to -6, so there can only be one real positive root, say e1. The other roots are both negative or complex conjugates. The possible solutions are sketched in Figure 2.4. The solid curve represents 3 real roots - one positive 16 . 1 Real Root \ .1... w\\ ,f‘ 2 Complex Roots \xm“‘3 Real Roots r-x / \, 4 ‘3 ~\ . 3’. l \ \ ”up—u”.— . \xmm’ \ \ F \ \, ‘\ FIGURE 2.4 POSSIBLE SOLUTIONS OF EQUATION (2.17) C "1" nip-g} -31-}? ‘ =~- ~12. / - \ " 1,594; 1 “13’ l J ' .. .. r 1 "2...... .... (it, .1“. FIGURE 2.5 PLOT OF P, SD FLOW, FOR ml = .85, k = .55 17 and two negative - while the dash line indicates one positive real root and two complex roots. Each curve will of course result in dif- ferent solutions for F. The following subsections will consider the solution of equa- tion (2.16) where C(F) has three real roots for SD, SC, and N flows, and then the solution of (2.16) where C(F) has one real and two complex roots for SDO flow. The velocity profile obtained when C(F) has one real and two complex roots will be referred to as SDCO flow. 2.4.1 Dimensionless Velocity Function for SD Profiles, For outward flow on the centerline,.which applies to. SDO, SD2, SD4, --- etc. profiles, the velocity and therefore the dimensionless velocity function FL must be positive for .9 equal zero. Therefore, from- equatiOn (2.19) C(F) is > 0 only if F < e1 since (eé-F) and- (e3-F) are both~ < 0. So 0\ 0 there are limits on the values of k and m2. From equation (2.58) m2(2k2 - 1) 2.3 (2.61) Therefore k > -% (2.62) and 3 m2 >, (2.63) 2k2-1 A plot of F for some typical values of k and m2 is shown in Figure 2.11. The effect of changing values of k and m2 on F is simi- lar to the case of three real roots discussed in Section 2.4.1. Items of interest are: 28 FIGURE 2.11 PLOT OF F FOR SDCO FLOW, FOR VARIOUS m2 AND R 29 1. For k = l, cn(o,k) = sech(¢) and the curve becomes asymp- totic to a constant value (i.e., for m = 25, F.» 10.33). This again defines the dividing line between the solu- tion with 3 real roots and the solution of 1 real root and 2 complex roots. 2. ForN/g-< k < 1 there are also solutions with 2, 4, or more interior zeroes. However, the lower portion of the curves, which is not shown, reaches values of 1000 to 2000 and the resulting flow profiles with 2 or more interior zeroes have large negative flow rates. Therefore, only the primary solu- tion (i.e., no interior zeroes) was considered. Again the function F is periodic and k and m2 have similar roles for stretching and adjusting the shape of the curve as with previous cases. 2.5 Application of Boundary Conditions In Section 2.4 the velocity functions were derived for each type of profile being considered. In each case F is a function of the flow parameters k and mi (i= 1,2,3, or 4) which determines the Shape of the velocity profile. The values of k and mi used in Section 2.4 were arbitrary values chosen for purposes of illustration. However, normally a problem would set Specific boundary conditions of o and e as given in Section 2.3. F(ja) = 0 (2.12) (1 f F(6)de = e (2.13) .a 30 Application of these boundary conditions to each dimension- less velocity function results in two expressions relating a and e to k and mi. These expressions can then be used to determine neces- sary values of k and mi to affect Specified values of a and E. For each case the application of (2.12) is a straightforward substitution and the application of (2.13) requires an integration of the dimensionless velocity profile between the limits discussed in Section 2.4. The resulting expressions for each flow profile are sum- marized below: SD Flow fill-1 sn2(—q—,k) = -§—(1 - ) (2.64) )MEI k2 le £0.[3 __ EFL...) - m1+ [(2 - 2] (2.65) Where E is the elliptic integral of the second kind, i.e., o E (¢,k) = f (1 - k231n2¢)d¢ (2.66) o SDC Flow 2m2% [112-81 Gnu-3") 9,k] = m2+el (2.67) e % 2 E 5- = a(e1-m2) + 2(6m2) EEa I? ,k] (2.68) 31 where 2 e1 = 3(2m2k2 - 3 - m2) (2.58) W 3m32(1 + k2) dn2(m3a,k) = (2.69) l-m32(k2-2) k23n(m3oz,k)cn(m3(x,k) e = 4a[m32(k2-2)-l] + 12m3E(m3oz,k) dam a k) ] (2.70) 3 9 N1 Flow 0. = xvi; (2.71) where K' = complete elliptic integral of the first kind (also the half period of sn as noted in Section 2.4.1), i.e., n/2 do K'(k) = f 1 (2.72) o (l-kzsinzo)§ E = i—HkZ-m-zw +3E'] (273) V15 ” ° where E' = complete elliptic integral of the second kind, i.e., 11/2 _1_ E' = f (1 - k2sin2¢)2d¢ (2.74) 0 Also, the values of 61 and 62 shown in Figure 2.10-(a) can be determined as: - Inn-l. _ 91 = mg K Sin 1[%(1- ) l 2 k2 ] ,k (2.75) 92 = 1,1111, 2K' - 91 (2.76) 32 where K = elliptic integral of the first kind, i.e., (19 do 1 o (l-kzsinzo)E K(¢,k) = f (2.77) N3 Flow oz = 210qu (2.78) m ll 7:: [(1.2 - m. - me + 312'] (2.79) V... So, a and e for N3 flow are simple twice a and e for N1 flow. 91 and 92 as shown in Figure (2.10-b) are given by equations (2.75) and (2.76) reSpectively. 2.6 Summary The resulting expressions for each flow condition presented in Section 2.5 represent a set of transcendental equations relating desired values of a and e to the flow parameters k and mi. Pre- vious authors, for example, [9] and [13] have developed similar expres- sions for the symmetrical flows in terms of Reynolds number. The non- symmetrical profiles have not been discussed as widely although the limits were discussed by Rosenhead [13]. However, in all cases pre- vious authors have stepped at this point and used the equations in a form similar to that Shown in Section 2.5 for calculations. Therefore, either a graphical or an iterative method of solution is required. As a result the so-called "known exact solutions" are very difficult to use for determining a velocity profile for set values of a and e. Also, unless one is quite familiar with the equations, their limits are not readily apparent and it is difficult to determine which flow 33 profiles are mathematically possible for particular values of a and e. The representation of these solutions in Chapter III will solve this problem. III. FLOW PARAMETER GRAPHS 3.1 Introduction The application of the boundary conditions to each of the velocity distribution functions in Chapter II results in two expres- sions for each profile relating a and e to the flow parameters k and mi, where i equals one, two, three, or four. However, Since these equations require a graphical or an iterative method of solution to determine values of k and mi for given values of a and E, their direct application is difficult. It is necessary to determine, for a particular set of a and e, which solutions are possible and what the values of k and mi are for these solutions so the velo- city profile may be determined. Also, it is not readily apparent where dividing lines exist between two types of flow; e.g., where an SDO flow changes to an SD2 flow. The above difficulties can be reduced by plotting curves of constant k and mi on an a-e grid to be called a flow parameter graph. Then for any value of a and e the graph is read to deter- mine corresponding values of k and mi. Since only curves that represent solutions are plotted, the boundaries of each type of solu- tion are readily apparent. In some cases these boundaries are defined by an analytic expression, and for others the plotted curves serve as the defined boundaries. For each of the symmetrical profiles to be considered the set of equations derived in Section 2.5 contains a transcendental equa- tion. This indeterminacy arises Since for each case one of the 34 35 equations involves an elliptic function sn, on, or dn which is a function of two parameters. By relating these elliptic functions to trigonometric functions, which are dependent on one parameter, expli- cit expressions can be deve10ped for a and e in terms of k and 1111 allowing for straightforward calculation for the constant k and constant m curves. Once the curves have been plotted for each pro- file on a set of coordinate axes, then the graph can be used to deter- mine the boundaries of each type of flow and the values of k and mi for any Set of a-e coordinates within these boundaries. The values of k and mi can then easily be used to plot the velocity profiles. The flow parameter graphs are deve10ped and the boundaries of the solutions for each type of flow profile are discussed in this chapter. In each case the transition from an elliptic to a trigono- metric function is described and the resulting expressions and limita- tions are discussed. Then the computer program used for calculating and plotting the curves is discussed and a symbolic flow chart of the program is Shown. These flow charts do not include printing or the so-called "book-keeping" Operations Such as that required to plot dotted rather than solid lines. (Dotted lines are used for all con- stant m curves, and solid lines are used for all conStant. k curves.) However, all the basic logic used in solving the expressions is included. Then several typical points are read from each graph and the resulting velocity profiles are shown. 3.2 SD Flow _ The case of symmetrical flow with diverging, or outward, flow on the centerline is complicated by the_fact that there are two types of solutions - one assuming three real roots in the cubic 36 expression for C(F) as covered in Section 2.4.1, and one assuming one real and two complex roots in the cubic expression for C(F) as covered in Section 2.4.4. These two cases will be considered separately. 3.2.1 Solution where C(F) has Three Real Roots. The expressions developed in Section 2.5 by application of the boundary conditions to the dimensionless velocity function are ml-l sn2 0 .k = l—l - (2.64) In1 3 k2 = %[3 —-? EQ-E-T’k) - m1 + k2 - 2] (2.65) As pointed out in the Introduction the difficulty in working with these expressions is that the elliptic function an is dependent on two parameters, e.g., a and k. However, the function is really 'V-ml quite Similar to the trigonometric function sine. In fact, for k = 0, sn(¢,0) = sin(¢). Although the parameter It has the effect of Stretching out the period, sn(o,k) still only has values between plus and minus one. Finally, in the limit, sn(o,l) = tanh(o) and the period is infinite. Because of these relationships sn can be defined in terms of Sine. For this case consider sin(A) = sn(‘fi%f,k) (3.1) where _ . -l[ ( o A -- Sln sn W’k)] (3.2) 1 Or a = \Iml sn-1[sin(A),k] (3.3) Further sn-1[sin(A),k] = K(A,k) (3.4) where K is the incomplete elliptic integral of the firs t.kind. Therefore, 37 01 = le K(A,k) (3.5) With this substitution equation (2.64) can be solved for m1 to give m1 = l + k2[1-3 Sin2(A)] (3.6) The system of equations does not lend itself to explicit equations for k and m1 which could then be used in calculating the velocity profile. However, equations (2.65), (3.5) and (3.6) can now be used to calculate directly values of m1 and e for various values of k and A (with corresponding values of a). A series of curves for k-= constant and a series of curves for m1 = constant will be combined to produce the flow parameter graph. For the present consider only the constant k curves. The flow chart in Figure 3.1 represents the subroutine used to plot the constant k curves for 0° < A < 90° from the above equations. It will be noted later that when A = 90°, where sin(A) and, hence, sn(‘]%T3k) is a maximum, SDO flow changes to SD2 flow. The constant k curves as plotted by the subroutine represented in Figure 3.1 are shown in Figure 3.2. Items of particular interest are: 1. As k approaches zero the curves approach the e = 0 axis. 2. As k approaches 1.0 the curves approach an area on the left side of the graph which will be covered later with the solu- tion for SDCO flow. Therefore, k = 1.0 is the boundary.. 3. The area within the plotted lines k = 0.0 and k = 1.0 represents the only area for solutions of these equations for symmetrical diverging flow with no interior zeroes (except SDCO flow). CST?” ) < = .1, .2, .3,---1.6>—— -—— —-] m1 = 1 + k2(1-3 911.211) YES NO m1>0 l a a K(A,k)fi 1247—: ,k) e = 1 - 40‘(Z-kz-tm ) SE1 “I 1 FIGURE 3.1 SYMBOLIC FLOW CHART FOR CONSTANT k CURVES, SDO FLOW LON ER? 1 Ludéi) XX 39 II II o. FIGURE 3.2 CONSTANT k CURVES, SDO FLOW 40 ' Note, for large 6 all curves approach the a = 0 axis. Therefore, for smaller values of e a wider range of angles, a, can be used. For very large values of e the angle a must be very small to insure purely divergent flow. The curves are a maximum for a value of a that corresponds to A = 90°, e.g., for k = 0.0, the elliptic function sn is equal to the trigonometric sine function, and therefore, a = 900 is this maximum. The equation for this boundary can be obtained from equa- tions (2.65), (3.5), and (3.6) by setting A = 90°. sn ,k = sin 900 = 1.0 .42. (rm—1 Therefore k2 - .3 (1 - m ) 3 7 - 2 1 ( ' ) e = -—-l§—- [sn-1(l),k] - (1 - k2)sn'1(1) (3.8) Vl-Zkz or e -- ~13— [E(11/2,k) - <1 - flue/2,101 (3.9) 1-2k2 where K(n/2,k) E K' = Complete elliptic integral of the first kind. E(n/2,k) E E' = Complete elliptic integral of the second kind. This curve is also plotted on Figure 3.2 and labeled as A = 90°. One may then state that to the right of this curve any solution must encompass some backflow. To have a real solution for a from equation (3.3), the value of m1 must be positive. From equation (3.6), m1 is 41 always positive if k éfiJg: If k >\J§T then sin A must be less than‘\/%-+--l—-. For example, if k = .9, then sin (A) < q [1 81 3k2 '5555 : i.e., A < x 59.70 or A > 2 120.30. Then ‘a will approach zero for any value of k > '% as 6 increases since a = \fmT'K(A,k) and ml_.0 while I( remains bounded. Figure 3.3 shows the same constant k curves for 0°< 90° represents SD2 solutions with backflow shown in Figure 2.3 Items of interest for this graph are: 1. For Space conservation the curves were stOpped at {€12 8.5. For the curves with larger values of k (i.e., curves with k2> .60) the curve is discontinuous. (For k§\/%f this dis- continuity is required by conditions on m1 as pointed out in item 6 above.) 2. For any particular value of k the curve becomes tangent to the boundary on the right side of the graph then continues to the left of this boundary. The result is that a second solu- tion with two interior zeroes is superimposed over both the SDO region to the left of the A = 90° line and the SD2 region to the right of the A = 900 line. This secondary solution actually has solutions all the way to the a = 00 axis even though only a few lines have been plotted to reduce confusion due to the over-lapping families of curves. Therefore, to the left of the A = 90° line, for example, a = 50° and € = 2.0, there are two possible solutions - the primary with no interior zeroes and a secondary with two interior zeroes. To the right of the A = 90° line, for example, a = 85° and EPSI- 7.0 1'5th ,—.. 42 K:.9 K:.8 0 A=9O .5 . ‘ K:.'|\ ‘ l '3 '1 I 1 l ‘ Lu FIGURE 3.3 CONSTANT k CURVES, SDO G: SD2 FLOW 43 e = 2.0, there are two possible solutions each with two interior zeroes. The existence of the second solution with two interior zeroes simply implies that there are two combinations of k and m1 which will both cause F, the velocity profile func- tion, to be zero at i;a. Referring back to Figure 2.5, SD2 flow occurs if a =‘: 92. Also, referring to Figures 2.6 and 2.7 it is easy to see that two different curves could have the same values of 92. For example, the m1 = 1.05 and k = .55 curve on Figure 2.6 and the k = .9 and m1 = .55 curve on Figure 2.7 both have values of 62 very near 1600 even though the shape of the curves and, therefore, the shape of the velocity profile is quite different. As the curves continue for larger values of A they drop below the e = 0 axis and in this region represent solutions that are symmetrical, have outward flow on the centerline, have two interior zeroes, and have a net flow back toward the apex. Therefore, even though the equations were derived for so-called diverging flow, physically they can actually repre- sent negative or inward flow. This condition can be easily visualized by referring first to Figure 2.5. It was noted in Section 2.4.1 that if a is chosen to be 62 then there will be two interior zeroes. Therefore, choose a = 92 for each case and refer to Figure 2.6 or Figure 2.7, where the area between the curve and the axis represents the amount of fluid flow. Then if the positive area is greater than the negative area the net flow will be outward: e.g., k = .5 on 44 Figure 2.6. If there is more negative than positive area then the net flow will be inward: e.g., k = .9 on Figure 2.6. Also, a value of k can be found so that the negative and positive areas just cancel so there is no net flow. For example k = .7 on Figure 2.6 is close to this condition. 4. A portion of the curve for A = 1800 is also shown on Figure 3.3. This curve represents the second time that sine and hence sn is a maximum. The equation for E obtained from equation (2.65) by setting A = 180° is e = JEL-[E(n/2,k) - K(n/2,k)] = 35L (E'-K') (3.10) vml le Area to the right of this curve represents solutions with four interior zeroes, and the solutions could be carried to six interior zeroes, etc. However, any area to the right of a = 180° is not physically possible and will not be consi- * dered. 5. For positive 6 regions to the right of the curves (at e = 0 this is about 128°), there is no solution. For example, at a = 1300 it is not possible to obtain a solution of SDO or ** SD2 flow. For the constant m curves the same set of equations, (2.65), (3.5), and (3.6) are used, but (3.6) is solved for k giving: __ * The fact that secondary solutions, which are symmetrical, have four, six, or more interior zeroes, and have positive flow at the center- line, would exist over much of the region described in Figure 3.3 was not considered in this investigation. ** Other velocity profiles to be considered in later sections will have solutions in this region. 45 k = / ml-l l - 3 sinzA (3.11) 'The negative root is not meaningful. Also, the value of A is limited by the fact that if m1 > 1, then sinA {\/g- (3.12) or, for the range of 0 to 180°: A < 35.2° or A > 144.8° if m1< 1, for the range of o to 180°: 35.20 < A < 144.80 :if m1 = 1, then k = 0, therefore, e = O and there is no meaningful flow. With these considerations, the flow chart on Figure 3.4 'represents the computer program used to obtain the constant m curves. 'The curves for 0° < A < 90° are shown in Figure 3.5. Items of interest are: 1. There are two separate families of curves, one for m1 < 1.0 and one for m1 > 1.0. 2. The curves are drawn as dotted lines so they would appear different from the constant k curves. 3. As ml.» 1.0, the curves approach the E = 0 axis. 4. As ml.» 0.0, the allowable values of a.» 0.0. Figure 3.6 shows the same constant m curves for 1 00 < A < 180°. (Except that the curves for m > 1 and A > 144.80 1 are not included.) Items of interest are: 1. As with the constant k curves there is a corresponding second solution which represents flow with two interior zeroes overlapping the graph.. 2. The procedure used in the computer program to produce the 46 C 3 1 l—(ml - .2, .25, .3, .35, .4, .5 --- .9, 1.005, 1.01, 1.05 1.2, 1.8, 2.0 > 1-3 sinZA q - «mm/u? 121*,— an.) “1 4a e - - _ ymi m1 A-A+l° I I I I I I ‘ I I__.________ P-J—‘ YES FIGURE 3.4 SYMBOLIC FLOW CHART FOR CONSTANT m CURVES, SDO FLOW EPSILON 4.0 3.0 5.0 "3 C 4.1.0 -j.O 47 "I'I‘II II II rain 40 in (It: $0700.10 11011010013501 :LmALPHA M':|.OOS FIGURE 3.5 CONSTANT m CURVES, SDO FLOW E’F’S ILON I-8—-I_ ' FIGUM 3.6 CONSTANT m CURVES, SDO AND SD2 FLOWS 49 dotted line effect is related to the increment of A. There- fore, the length of the dashes is not constant. Figure 3.7 is the combination of the constant k and the constant m curves to give the complete flow parameter graph for SDO and SD2 flow. This graph includes the lines for the second family of solutions with two interior zeroes. Figure 3.8 is the same as 3.7 except that the second family of solutions is not included and the resulting graph is clearer and easier to use for determining the pri- mary solution which is usually the one desired. The computer routine used to obtain Figure 3.7 is called subroutine DIVERGE and is shown with some sample printout of calculated points in Appendix A. The routine is in FORTRAN and is written from the flow charts given on Figures 3.1 and 3.4. In addition there is a graph subroutine for the axis and two subroutines to determine the elliptic integrals and func- tions. The graph and function subroutines are used with each flow parameter subroutine and therefore appears in the back of Appendix A. Figure 3.8 can now be used to determine the flow parameters (m1 and k) for any value of a and 6* within the range of solutions and thereby determine the velocity profile. If the point is to the left of the A = 900 line the flow will be SDO. If the point is to the right the flow will be 302. If the point is on the A = 90° line, then the correSponding value of 6 represents the maximum flow rate that can be attained with SDO flow for the correSponding value of 0. Fur- ther increasing 6 for this value of a is only attained by allowing * If an a-e point is between the constant k lines or the constant m lines then extrapolation must be used for intermediate use. In actual application more intermediate lines could be shown in the area of interest. Fewer lines were used here simply to reduce com- puter time. EF’SILON 50 ........ 5.0 C) O ‘7‘. t... "J FIGURE 3.7 FLOW PARAMETER GRAPH, SDO AND SDZ FLOWS bf) EPSILON 4.0 I o 3.0 ---~~, IIIIIII 0‘0 0 {0 a c 1601101i01§01®>l$140ll°l€¥°$ ALPHA . _. I M.=I.01 ~5.0 ~b.0 -7.0 O m I FIGURE 3.8 FLOW PARAMETER GRAPH SDO AND SD2 FLOWS (PRIMARY SOLUTIONS) 52 backflow as in SD2 flow. Three points with the above characteristics are selected and shown on Figure 3.8. The values of k and m1 read from the graph for these points are approximate values as scaled from the graph. 1. a = 31° k = .45 e = .2 m1 = 1.05 2. a = 60° k = .525 6 = 4.1 m1 = .445 3. a = 92° k = .45 e = 1.95 m1 = .65 These values are then used in equation (2.34) to determine the corres- ponding velocity profiles. The profiles are plotted with the computer program shown in Appendix B and the resulting profiles are shown in Figure 3.9. A fourth profile is also shown in Figure 3.9. This pro- file is obtained by plotting point 3 on Figure 3.7 and determining the following values of k and m1 from the family of secondary solutions. 4. a = 920 k = .75 E = 1.95 m1 = .35 It should be noted that profiles 3 and 4 both correspond to the same value of a and e and represent two possible solutions at that point. Items of interest on Figure 3.9 are: 1. Each velocity profile is scaled so that its maximum velocity at a radius of 1.0 inches is CHM! inch. The scale for each plot is as shown. 2. Each velocity profile is calculated and shown at two radii - 53 [SCALE 1 INCHn 0.00111 FT/SEC SCALE 1 INCH: 0.01025 FT/SEC m I A) ALPHA = 31.000 K EPSILON = 0.200 M 0.450 ALPHA = b0.000 K 0.525 1.050 EPSILON = h.100 M 0.h45 SCALE 1 INCH: 0.00500 FT/SEC SCALE 1 INCH: 0.02bhh FT/SEC AN ALPHA = H2 EPSILON = 1 5| H H H (h) .000 K .HSO M H H 0.450 ALPHA = 02.000 0.b50 EPSILON = 1 950 II II CDCD as FIGURE 3.9 TYPICAL VELOCITY PROFILES, SD FLOW 54 1.2 inches and 3.0 inches. Then to fit in a reasonable space the profiles were plotted half size. 3. The plotted values represent a velocity profile where v was assumed to be .000159 ftZ/sec. However, the plot can also be considered to represent the shape of the dimensionless velo- city profile F. 3.2.2 Solution for SDCO Flow. The expressions deve10ped in Section 2.5 by application of the boundary conditions to the dimensionless velocity function are mz‘el 21112);— ] m2+e1 - en --3 .k (2.67) NI (0 I Q (D “i 8 N v + N A O\ B N v NIH :11 I Q | B [ 6(1+cosA) 1 M2 '3 2 7 4k (l+coaA) + cosA- 5 . 2 2 £13 5(2m2k - 3 ~ 1112) 3 a 4%.; K(A,k) NO YES LA-A+.05] 83'31'; P A T b “W22 " £12+ 2821 e - 2{1(21-m2) + 2V6m2 M[-6-2-_: “I I I I I I I I I I______ FIGURE 3.10 SYMBOLIC FLOW CHART FOR CDNSTANT k CURVES, SDCO FLOW (2 a 5,10,15,20,25,50,100,200,300>__ l 6 [ QUAD = m [5): (1+COSA) + 5 " COS A] l O‘\/%T: 3. For k = l, the curve exactly coincides with k = 1 curve for three real roots. For k = 1 equations (2.58), (2.59), and (2.60) can be used to show b, which is the imaginary part of e2 and e3, is zero. Therefore, along the k = 1 curve the complex roots become real roots and on this curve are equal, i.e., m2 e2 = e3 = a = - (2-+-3-) (3.21) The computer routine to calculate and plot the constant k and constant m curves is called "Subroutine Complex" and is shown in Appendix A with some sample print-out of the calculated points. The programming represents the flow charts on Figures 3.10 and 3.11. Three points are chosen on Figure 3.12, the values of k and m2 are read from the graph, and the values are then used in equation (2.57) to determine the velocity profiles shown in Figure 3.13. The coordinates and values of k and m2 are shown below each velocity profile where the scaling is the same as described in Section 3.2.1. The subroutine used for Figure 3.12 is shown in Appendix B. The flow parameter graph for the SD solutions shown in Figure 3.8 and the graph for the SDC solution can be combined and the resulting graph is shown in Figure 3.14. This graph represents SDO and SDCO profiles on the left of the A = 900 line and SD2 profiles on the right of the A = 90° line. Other families of curves for solu- tions with 2,4,6,--- interior zeroes have not been included. 60 SCALE 1 INCH2 0 02504 FT/3EL SCALE 1 INCH. 0.03049 FT/SFL I [I (23) 4: ALPHA 1 10.000 K 1 0.020 ALPHA = 20.000 K r 0.025 EPSILON r 2.000 M = 34.300 EPSILON = 0.000 M = 22.700 fl SCALE 1 lNCHr 0.02510 FT/SEC (3) ALPHA = 05.000 K a 1.000 EPSILON a 5.000 M = 12.000 FIGURE 3.13 TYPICAL VELOCITY PROFILES-SDCO FLOW 61 M'zoz 01 1301}0.I.fo11+0110..1$1$3 1 II'IIOIIU I.O| ALPHA \ .. u p H w x. 7... m .. . ma a A. 5 m. .. I = = . .a I .. . M. .._u mKK 3. I - . . O M 3 s I I I = I I I I I I- 01' I 1'. i. I! 5 I ”I! O 5 . . II II pp 11 i I 11' I I! I: 1.. - . dd 2 I I I 111111111111“ ...... . / 0 M .II II I .I I I: I. I I I II1 111111.111... .......; nU _ . . . . llh H n”... IHIIM! 1’ I. All!!! ”1th kn Huh: ..I. I O _ . — — I— — — . _y P I— Qm 0H“ 0.3“ o m 0.7. 0 m QN 04. 0.0 04.- 0.....- .m1 0.1.1 Oh: 0.3 OK. .m. 5 5 I n 2 m w 7 .Hmnm 7. a I 6. FIGURE 3.14 FLOW PARAMETER GRAPH FOR DIVERGING FLOW 62 3.3 SC Flow The expressions deve10ped in Section 2.5 by application of the boundary conditions to the dimensionless velocity function are 3m32(l + k2) dn2(m3a,k) = (2.69) 1-m32(k2-2) k28n(m 3C1,k) cn(m3a,k) dn(m§1,k) e = 4a[m32(k2-2)-l] + 12m3[E(m33,k) - ] (2.70) To obtain equations for plotting the flow parameter curves again relate the elliptic functions to trigonometric functions but first note dn2(0,k) = 1 - kzsn2(¢,k) (3.22) Then let sin A = sn0n31,k) (3.23) Substituting into (2.69) and solving for m3 one obtains _ 2 - 2 m3 =V l + k Sin A (3.24) (k2-2)k2sin2A - 1 + 2k2 . where l -l . l O! = -- sn (SinA,k) = -- K(A,k) (3.25) m3 1113 Then equations (2.70), (3.24) and (3.25) can be used to obtain values of a, m3, and e for various constant values of k and A. For curves with constant values of m3 equation (2.69) can, be solved for k, (again using (3.22) and (3.23)) to obtain 63 .1 '- -B i [132 - 4m37-sin2A(l-m32)]2 k = (3.26) 2m3281nA where B = 2m32 - sin2A(1 +-2m32) (3.27) Then (2.70), (3.25), (3.26), and (3.27) can be used to obtain values of a, k and e for various constant values of m3 and A. The limitations on these solutions are extensive and they will be discussed after presentation of the flow parameter graphs. The symbolic flow charts of the computer programs using these equations to plot the constant k and constant m curves are shown in Figures 3.15 and 3.16, reSpectively. The flow parameter graph with the curves for 0° < A < 900 is shown in Figure 3.17. These curves represent SCO solutions (i.e., all flow is toward the apex). Items of interest with respect to these curves and the gov- erning equations are: 1. As k approaches 0.0 the curves approach the e = 0 axis, and as k approaches 1.0, the curves approach negative infinity. 2. The line labeled A = 900 indicates the maximum value of a for any particular 6 where purely convergent flow may exist. To the right of this boundary the solutions have two or more interior zeroes with back flow. (Actually this so-called "backflow" is outward, away from the apex.) The equation for this boundary can be obtained from the above equations for A = 90°. 64 C sun 3 I I. __.__ —— ——Q - .1,---.7,.75,.8,.9) -1 +.k2sin2A (1c?- -2)k2sin2A - 1 + 2k2 QUAD " YES O W m3 " (QUAD)% a B K(A,k) “‘3 ’ e - 4a[m32(k2-2)-1] + 12m3[E(m3a,k) k28n(m3a,k) cn(m3oz,k) (“101130.10 ] A - A + .1° or 1.00 ——-sI C m D FIGURE 3.15 SYMBOLIC FLOW CHART FOR CONSTANT k CURVES, SC FLOW 65 C START D |.< m3 . .775, .8, .825, --- 1.00, 1.25, --- 2.0, 3.0 --- 10.0 > A - 54.° AINC - .25 II 3 2 B - 21:132 - SIN2A(1 +2m32) VJJ-JJ-I-l z--z AINC = -AINc 2 QUAD a a -4m32 311341-0132 k -VQUA02 a g K(A,k) m3 e - 4a m32(k2-2)-l] + 12m3[a(m3a,k) k sn (mad, k) cn (11130, k) ' dn(m3a,k) . ] PLOT e ,(1 ~I l A-A-I-INC YES _ CED) FIGURE 3.16 SYMBOLIC FLOW CHART FOR CONSTANT In CURVES, SC FLOW EPS‘ I LON 1'4 5‘ 7. 66 K=.| K=.2 \w~r'-=“'"‘~i§i‘ii .“161101101400101401l 15110111014: ' - ‘ ; ’11 143:.975 ‘ ' 113:.95 M3=.925_ M3=.9 M3: .375 3 01 ll g, N U! FIGURE 3.17 FLOW PARAMETER GRAPH, SCO FLOW 67 Thus sn(am3) = sin 90° = 1.0 and from (3.24) 2 5 m3 = (k '1) (3.28) k“-1 Then from (2.70) l l 2_ *2- It. '2' 2.. e = 4K'[(-—-k 1) (k2-2) -(-——--k 1) 1+ 12(k 1)E'] (3.29) k“+1 kZ-l k“-l and K! <2 = ---+r (3.30) (k2-1)§ k“-1 where K' and E' are the complete elliptic integrals of the first and second kinds, reSpectively. The values of e and a can then be plotted for various values of k. For the constant k curves using (3.24) the expression under the radical must be positive. If A = 0° then from (3.24) m3 ,VCI (3,31) -1+2k2 and k must be less than'\/%7 , i.e., k < g- (3.32) On the other hand if one specifies k then A is restricted. For example if k i\/g-, then m3 is not defined for A = 0°. For example, if k = .9 then A > z 53.5° (3.33) The angle a as given by (3.25) still approaches zero since 68 a = E;%LEL and m3 4 0.0 as A decreases to 53.50 and K 3 is bounded. Therefore, the constant k curves for k < as. begin at a = 0.0 and e = 0.0, but the curves for k > are asymptotic to the a = 0.0 axis. For the constant m curves equations (3.26) and (3.27) are used to determine allowable values of A and k. For m3 = 1.0, since k must be positive, one has A > sin-1f\/%f)z 54.50, and A < 90°. The plus sign for the square root term in (3.26) is used to give real values. For m3 > 1.0, A 3.0 and A # 90°, since at 90° k = 1, and a = infinity. For practical purposes of representing values on the computer the value of A is allowed to approach 90° until k.> 0.999. The plus sign in equation (3.26) is used to give real values. For m3 < 1.0 both the plus and minus signs give acceptable values for certain ranges of A. An analysis of equation (3.26) for tn3<21.0 shows there are two critical values of A, to be denoted by Ac and A;' These are the values of A when the inner quadratic of equation (3.26) is zero, i.e., when B2 = 4m32sin2A(l-m32) ' (3.34) For Ac < A < 900 both the positive and the negative roots in equation (3.26) give solutions. The value of A; becomes significant on Figure 3.19 where the curves include regions where A > 90°. Figure 3.18 is a sketch of the m33= .950 curve from Figure 3.17 with an indication of the values of A for each range of the curve. This duplication of solutions 69 FIGURE 3.18 PLOT 0F m3 = .95, 300 FLOW 70 causes the computer program to be considerable more complex, as reflected by the flow chart in Figure 3.16. Rather than solving (3.34) for AC the computer program is written to increase A from 90° toward AC using the positive sign in (3.26) until the inner quadratic in (3.26) is negative, indi- cating that A is now greater than Ac. Then the increment is reversed and A is decreased from AC toward 900 this time using the negative Sign in (3.26). Thus, Ac and fur- ther on A; are never explicitly calculated nor do they need to be in order to plot the desired constant m curves. With reSpect to computer time this procedure is just as efficient if not more so than explicitly solving equation (3.27) for AC * and then increasing A to this known value. The Flow parameter graph on Figure 3.19 includes solutions for SC2 and SC4 profiles. For this graph A has been allowed to increase until a is greater than 1800 or le‘ is greater than 8.5. The complete subroutine called CONVERGE and some sample printout of calculated values is shown in Appendix A. Items of interest with respect to these curves and the governing equations are: 1. The A = 900 line derived earlier divides SCO flow from SC2 flow. This general procedure is used several times in the accompanying com- puter programs. Even though explicit solutions for the limits are derived, very often it takes less computer time to simply increment until a resulting expression is out of bounds. Also at times the limits, when they are determined explicitly, are exact values which a digital computer can only represent approximately and the differ- ence between the exact value and the approximation may be infinite. EPSILON 0. CI. 7.0 ED 3.. 0 T7! C) (‘J 5.0 a!!! ‘\:“‘111“"' 01.“. \ . [ \ \ \‘ K:.' sq 1‘ ,_ , . , 5“\‘-g__-4 L \ \ g \ l.\5 M3: 0 \\\\\ FIGURE 3.19 FLOW PARAMETER GRAPH, SC FLOW 72 The A = 180° line divides sc2 flow from 304 flow. To the right of this line solutions have four interior zeroes. The equation for this boundary is obtained by setting A = 180° in equations (3.24), (2.70), (3.25), and (3.23). From (3.23) sin (180°) = sn(am3) = 0 (3.35) From (3.24) m3 = 2: (3.36) 3k -1 From (3.25) a = %E; (3.37) From (2.70) SK' 2 2 I = :Eg-[m3 (k -2)-1] + 24m3E (3.38) Then equations (3.37) and (3.38) with m3 defined by (3.36) can be used to plot values of a and € for values of k between 0.0 andfiv/;-. As k-41v/;—; m3.» 0 and a-» 0, so the curve becomes asymptotic to the a = 0 axis. For large 6 all curves become asymptotic to the a = 0 axis. Again, as for SD flow there is a second family of curves indi- cating possible SC2 profiles in all areas to the left of the A = 180° line.. Only a few lines have been plotted here 0n3= 3.0, 4.0, 5.0,---10.0), but corresponding values of k curves may also be plotted. (In this region k would be greater than the k = .8 curve plotted and less than 1.0.) 73 Similarly, a secondary family of curves of SC4 may be plotted that covers all parts of the graph to the left of the enve10pe of curves on the right side of the graph. (Both SC2 and 8C4 solutions would cover all values of a from zero to 180° for e < 0.) Other families of curves for 506, 3C8, etc. types of flow may also be deve10ped from these basic equations. 5. For the constant m curves, equations (3.26) and (3.27) again give an indication of the allowable values of A and R. For m3 = 1.0 for k to be positive A < sin-l(§) Z 125.5°. The positive sign in (3.26) is used to give real values for k. So for the entire rn3= 1.0 curve 54.5o z < A < 90° and 90° < A < 125.5°. For rn3> 1.0 the plus sign in (3.26) gives real values for k if A # 90°. For computational pur- poses A has been allowed to approach 900 until k > .999. For rn3<21.0 similar conditions exist as in the SCO flow. That is, the solutions of (3.34) determine two critical values of A, called Ac and AL. For 90° < A < A; both the plus and the negative roots in equation (3.26) give a solution. Figure 3.20 is a sketch of the complete m3 = .95 curve from Figure 3.19 with an indication of the values of A for each portion of the curve. Figure 3.21 is another plot of SC flow without the secondary solutions. This allows for easier use of the graph to determine values for primary solutions. Four typical data points are selected from Figure 3.21 and the values of k and tn3 are read and used in equation (2.45) to plot the velocity profiles shown in Figure 3.22. The coordi- nates and values of k and m3 for each profile are shown on Figure 74 FIGURE 3.20 PLOT OF m3 = .95, SC FLOW 7S _ K: .8 =‘7 cu _' K=.75 ' a. r5 a, —— l ‘I‘ o :— A=|8O D . .n— . 0._.. if“. \ _ \ -J 0___ , “x. _ ((33 .1 _ ‘ ‘\. a.) ‘ ' \ q__ \ h“. \ _ \ \ O \ cJ—— \ 97—- " , K=.I c: l ‘3 . l lEL‘Z‘l \ ., ‘ 3:140 we \‘ \ .. ‘. \ urns ‘ \ \ \\‘\\ 390 \ ..\\ ..\ m“ \ \ N \ \ \\ \\ \\\‘ o ‘ ----- . 7 ‘ h A390 “\ \\\ K \\\ \\\\ I \‘ \\\\\\\ 1 \ \\\\‘\ I \ \\\ \x h.“ \ : 3‘1. \\\§‘ “ ‘\ \3 ‘ \ \\ \ 7 V. \ \ \\ \ . : \\ \ _/\\\\\\ \ \ I \ \ M3: I .0 \\\\\ \ ‘ . FIGURE 3.21 FLOW PARAMETER GRAPH, SC FLOW (PRIMARY SOLUTIONS) 76 SCALE 1 INCH. 0.01b85 FT/SEC (1) SCALE 1 INCH: 0.00531 FT/SFL l2] ALPHA 1 00.000 K = EPSILON = ~h.000 M a 0.850 1.b50 ALPHA = 112.000 K = 0.550 EPSILON = '2.850 M = 0.877 SCALE 1 INCH: 0.01700 FT/SEC SCALE 1 INCHx 0.01073 FT/SEC (3) (i4) \ / ALPHA = 111.000 K = 0.b80 ALPHA = 100.000 K a 0.550 EPSILON = 7.000 M = 1.500 EPSILON = 7.750 M : 1.500 FIGURE 3.22 TYPICAL VELOCITY PROFILES, SC FLOW 77 3.22. The computer routine used for this plot called subroutine PRO SC is shown in Appendix B. 3.4 Non-Symmetrical Flow Non-symmetrical flow (N flow) with one and three interior zeroes as sketched in Figure 2.10 parts (a) and (b) are now considered. As noted in Section 2.4.3 there are an infinite number of other possible non-symmetrical profiles. The N1 and N3 profiles are similar enough so that they are considered simultaneously. From sec- tion 2.5 the equations for the two types of flow are for N1 Flow a = K'Vm—q (2.71) z: 4 2 .. _ l I e V775: [(k m.* 2)K + 3E ] (2.72) for N3 Flow a = 21("~‘/mL+ (2.78) e = 43—- [(k2 - m1+ -2) K' + 312'] (2.79) Vmu where for both cases the values of 91 and 62 for F = 0 can be determined as . -1 1 "‘4'1 T}? 91 = W’muKSIH [§-( - )] ,k (2.75) k2 92 =\/mu 2K' - 91 (2.76) Also, for N3 flow 93 = 92 + 291 = 91 + 2K“"mu (3.39) 78 For non-symmetrical flows the equations are actually simpler than those for symmetrical flows. The same dimensionless velocity pro- file as given by equation (2.34) derived for SD flows is used for non-symmetrical flows. The application of the boundary condition 92 f F(O) d9 = E for the N1 flow -91 and 93 f F(9) d9 = e for the N3 flow -92 results in expressions for E as given in equation (2.73) for N1 flow and (2.79) for N3 flow. Similarly, the application of the boundary condition F(:;a) = 0 results in expressions involving ellip- tic functions which are equation (2.75) for 91 and (2.76) for 62. But, the angle which describes the inclination of the walls is a==(61+92)/2 for N1 flows and a==(92+63)/2 for N3 flows. From (2.76) it is shown that for N1 Flow a = -——7f——_ = 2 =‘le+ K' (3.40) for N3 Flow 92-F93 82-+(291-+92) (3.41) As 91 and 62 cancel, equations (3.40) and (3.41) involve only the complete elliptic integral which does not depend on a. (For previous cases of symmetrical flow the corresponding expressions involved either 79 sn, cn, dn, E, or F, which are functions of both a and k.) As a result there is no necessity to relate the elliptic functions with trigonometric functions as done in Sections 3.1 and 3.2. The general procedure for determining constant k curves for the N1 flow parameter graph and the N3 flow parameter graph is to assume a value of k. Then vary a within the allowable limits given by equation (2.75) and calculate ml+ from (2.71) or (2.78) and e from (2.73) or (2.79). Then (2.75) and (2.76) can be used to deter- mine 61 and 62 if desired. (They are not required for plotting the solution although the existence of 91 as determined by (2.75) is used to determine the allowable limits of a.) This general procedure is illustrated more clearly in the constant k symbolic flow chart for N1 flow shown on Figure 3.23.* The constant m symbolic flow chart for N1 flow is shown on Figure 3.24. Here the procedure is not quite as simple because an algorithim is not available for the inverse of the complete elliptic integral K', even though it is a function of the single parameter k. Rather than write the necessary inverse algorithim a simple interation procedure is used and is shown on Figure 3.24. In this way equations (2.71) for N1 flow and (2.78) for N3 flow can be used to determine k for assumed values of a and mg. The complete flow parameter graphs for N1 and N3 flows are shown on Figures 3.25 and 3.26, reSpectively. Items of interest are: * The equations and procedures for N3 flow are similar enough to those for N1 flow so that the flow charts for N3 flow are not shown; 80 C START D i k . .1, .3---.9, .92, .94, .96, .965, .970---.980, .981,---.990:> 7 —‘LJT - dun [‘K'V1 i __ __.<1 - 10,1800, 2 on 10> ' - 1 , NO a = 1/10. K ‘ Ki K' '- 1 cz/Vmu a=K u YES , A 3—1 -— ADJ- .1 3( k2 k - .0999999 YES Pdetermine Kb» based on k YES A < 0 NO A >11 k - k-ADJ N0 ADJ - ADJ/10. e = —-[K"k2-mu-2) + 38'] <38 e ”S «WWW [k-k+ADJ| 92‘2thK"91 ..;__J L______________ |_.._. PLOT 05, 6 FIGURE 3.24 SYMBOLIC FLOW CHART FOR CONSTANT m CURVES, N1 FLOW 82 Both graphs show that either "net inflow" or "net outflow" may be represented equally as well. For large +€ all curves become asymptotic to the a = 0 axis. For E = 0 equations (2.73) and (2.79) show that, assuming m4 # O U m1. = 9%— + k2 - 2 (3.42) Since mq > 0 for meaningful flow then ' 3%- + k2 > 2 (3.43) This limit is determined graphically to be between k = .980 and .981. Therefore, in both Figures 3.25 and 3.26 the k = .980 curve crosses the e = 0 axis and is asymptotic to the positive 6 axis at a = 00 while the k = .981 curve does not have a solution for e = 0 and as shown is asymp- totic to the negative 6 axis at a = 0°. For k = 0 equations (2.73) and (2.79) show that e = O. For all other constant k curves, and accordingly constant m curves, the upper and lower limits of the curves are restricted by the fact that if the flow is to be non-symmet- rical as described, then 61 must be > 0°. The equation for 91 is l . -1 1 “‘1 ED: 91 =VmHK Sin 3- - ],k (2.75) k2 For 91 > 0 the expression within the inverse sine function 83 K: .970 K: .975 K: .955 0 . K: -96 :0 I t“ K: .94 ..._ 1 ~ r\ 0. .0 O. . n “4.0 -5.0 ~b.0 47.0 . -E1.0 FIGURE 3.25 FLOW PARAMETER GRAPH, N1 FIN 84 FIGURE 3.26 FLOW PARAMETER GRAPH, N3 FLOW 85 must have a value between 0.0 and 1.0. Define this quantity as T 1 mq'l T ’ - l - (3.44) 3 k2 For the upper limit where 61 will be maximum and E > 0, T = 1.0. Therefore 1 mh-l 5(1 _ )= 1.0 (3.45) k2 or m. = 1 - 2k2 (3.46) Substituting the above into (2.73) and (2.79) one obtains for N1 Flow e = -—13- [K'(k2-l) + E'] (3.47) .1-2k2 for N3 Flow 6 = __Z&__ [K'(k2-1) + E'] (3.48) 1-2k2 Corresponding values of a are determined from equations (2.71) and (2.78) for N1 Flow 1 a = K'(1-2k2)2 (3.49) where 1 k (v; (3.50) 86 for N3 Flow ' 1 oz = 21(‘(1-21<:2)2 (3.51) where (3.50) also applies. Equations (3.47) and (3.49) for N1 flows and (3.48) and (3.51) for N3 flows can be used to determine the right most boundary of possible flow for e > 0. These lines are not plotted as they would interfere with the constant k lines which are very close together in this region. However, they are used to allow plotting of the constant k and m curves right to the limiting values. For the lower limit where 61 will be a minimum and E < O, T = 0.0, therefore 1 mu‘]. 3‘( -—T) = 0.0 (3.52) k or In1+ = k2 + 1 (3.53) Substituting equation (3.53) into (2.73) and (2.79) and using (2.71) and (2.78) one obtains for N1 Flow 12(E' - K') Vk2+1 07 = K'Vk2+1 (3.55) (3.54) 87 for N3 Flow 24(5' - KL) WIkZ + 1 a = 2K'Wlk2 + 1 (3.57) Equations (3.54) and (3.55) for N1 flows and (3.56) and (3.56) (3.57) for N3 flows can be used to determine the right-most boundary of possible flow for e < 0. Again, the curves are not plotted but are used in the computer programs for limits on the constant k and constant m curves. Figure 3.27 is a sketch of the dimensionless velocity function for typical values of k and mg, with the limiting cases described above. The curves are not meant to depict actual curves for fixed k and mu 'but merely to show what occurs as the limiting values are approached. In addition to the previous discussion of e and a it can also be shown from equations (2.75), (2.76), and (3.39) that (a) for upper limit (T = 1.0, € > O) _;_ 91 = 82 = K'(l - 2k2)2 (3.58) 1 93 = 91 + 92 = 2K'(l - 21(2)2 (3.59) (b) for lower limit (T = 0.0, e < 0) 81 = 0° (3.60) 1 83 = 62 = 2K'(k2 + 1)2 (3.61) One further point of interest is that on the upper limit the 6 equation for N1 flows given by (3.47) is identical to the A = 900 curve dividing SDO flow from SD2 flow 88 Typical value (0 < T < 1) r—- Upper limit ,' (T = 1.0) 27/ 91 4’14 Lower limit (T = 0.0) Ahy curve within shaded area will result in non- symmetrical flow FIGURE 3.27 SKETCH OF F FOR N1 FLOWS SHOWING LIMITING CASES 89 derived in Section 3.2.1 and given as equation (3.9). This is to be expected as this line denotes the limiting case when 91 = 62 and the non-symmetrical flow becomes symmetrical. Also, the equation for the upper limit for e and N3 flows given by (3.48) can be rearranged to become identical to equa- tion (3.38) derived in Section 3.3 as the dividing line between SC2 and 804 flow. Again this is to be expected as the N3 flow in the limit becomes symmetrical with two or four interior zeroes depending on the boundary conditions. Further, equation (3.30L‘which was derived in Section 3.3 as the dividing line between SOD and SC2 flow, can be simpli- fied to be identical to equation (3.54), which is the lower limit for N1 flow. Again this is expected as the lower limit of N1 flow is also the boundary between SCO and SC2 flow. 5. With respect to the allowable range of a, equations (3.49), (3.50), (3.51), (3.55), and (3.57) can be summarized as follows for N1 Flow Re[K' V1 - 2k21's a g m1h[180°,K'\/1 + k2] (3.62) for N3 Flow Re[2k' V1 - 2k2]g a g min[180°,2K' V1 + k2] (3.63) The computer routines for the N1 and N3 flow parameter graphs are called NONSYMl and NONSYM3, respectively, and are shown with some print-out of the calculations in Appendix A. These programs basically follow the flow charts in Figures 3.23 and 3.24. 90 As illustrations two arbitrary data points are selected and marked on each flow parameter graph in Figures 3.25 and 3.26. The resulting values of k and mu are read from the graph and used in the dimensionless velocity function given by equation (2.45) to plot the correSponding velocity profiles. These profiles and their corre- sponding values of a, €, k, and mu are shown on Figure 3.28. Scaling is similar to that used for previous velocity profiles as discussed in Section 3.2.1. The computer routine used for this plot is shown in Appendix B. 3.5 Summary_ The flow parameter graphs presented in this chapter summarize the derivations and calculations. These graphs show the limits of the principal solutions for each type of velocity profile. The graphs can now be used to determine the flow parameters mi and k for parti- cular values of a and e. These parameters as used in the dimension- less velocity functions derived in Chapter II can then be used to quickly and easily determine the exact velocity distribution. The various flow parameter graphs can be superimposed on one graph, but the large number of lines would be difficult to interpret. However, one combination that is useful is the combination of Figures 3.14 and 3.21 which represent the primary solutions for SD and SC flow. The resulting graph is shown on Figure 3.29. This figure shows an area for positive e between the SD2 solution and SC2 solution where there is apparently no possible flow. However, referring back to Section 3.3 and Figure 3.19 one may note that this region (as well as others) is covered by the ”secondary" family of curves for SC2,SC4,--- etc. flow profiles. 91 SCALE 1 INCH: 0.17003 FT/SEC (l) V ALPHA < 50.000 K 2 0.053 EPSILON : 0.500 M t 0.080 SCALE 1 ALPHA = EPSILON INCHu 0.02Q5q FT/SEC 75.000 K = = 1.250 M e (2} SCALE 1 INCH 0. 3Hb7u .T/SEC (3) ‘v ) < J ALPHA = HQ. 700 K = 0.081 EPSILON z *5. 500 M - 0.040 SCALE 1 INCH? 0.0b353 FT/SEC ALPHA = EPSILONz FIGURE 3.28 TYPICAL VELOCITY PROFILES, N1 AND N3 FLOW M :300 .775'\ 202 K:1.O K=o8 .325—\ l K=.75 c; j l :1. l I I . ‘ o l I ' M'— 3 .1-1 ‘ .'\ \ l l l i. 0.4 I I l \ '53.. ‘1 41’ 5) 2'5 ‘\ loo—F 1 , | 1 I I I . w 0.. 1 1 l l L"! 1 I , ' N .1 m ”-4-” I 0 I x K __| O ' I ’ ’ 1—1 3' E I’ I . E .1 Fl} ’1 I |5 ‘ J I / ‘4. . ,_.. LL] (:3 1 1 ,1 , 7 ""5 . _ | 1’ I I \\ m ‘1 I 1' 1 K: . l I 1", 1 6 - 16.872; . “1’: 1,6 ~ \ (‘J I1' I 1' 7’ ‘ :1: ‘1. ' “l;- — -~ In . ‘1.’ . \ .\ c .' -:. .' 3 -- 5 .. ~ - --e --------------- . j} 7 . ‘ - —-.9—- ------------ . q ' (J . \ _ a “W - U :5“- 3“ “ ‘ ' 7‘ \ II I u ‘ ' . I 1 \ ‘ ‘ ”-— a—{ “1 |' lI ‘ “ ' ‘ : ’I‘ ' : ||1 ‘ |‘ “ ’7 em -.. ' ° \ 1 I i 1 1 1 \ (‘IJ lll'l" “. ‘1 \‘ ‘ ‘ .' I 1 \ ‘ ' ' I . I \ \ I C? I. I ll \\ 1‘ \ 1‘ ‘ \ | I. .2 4.. ”'1 I ‘ 1 ‘ \ ‘ " ' ' II ' \ 1 \ “\ 1 In 111 ‘ \ \ \ \\ c’ \ Q ' l \ 1 \ \ 1 : ‘ l \ \ \' |\ I .T 1 "ll 1 ‘ \ \ \ \\\\ \ : H l ‘1 \ \ \ \\ \\ I O 1 1 l 1 \ \ 1‘ \\ 1 I l /—|K=.I K=.7 /' t M = \| \ \ \ k \.\ x" \ \ \ \ \ . 1= 1 1‘ 1 1:31 \ \ r \ ‘1 \ ‘ \1 \ \ \‘\\ \ \ \ K \\ ~ \ \ \‘ A=90 \‘ ‘ \\ “\ 0‘“ k \\ \ \\\‘ \ \\\“ FIGURE 3.29 FLOW PARAMETER GRAPH, SD ANE SO FLOW (PRIMARY SOLUTIONS) 93 A further summary of the possible flow profiles available in any particular region is shown in Figure 3.30. This chart can not be used to determine the flow parameters or the velocity profiles, but does indicate further the range of each type of flow. Some of the values of e are read from the graphs, and some are interpolated between calculated points so these values (particularly for 8C4 where a < 80°) are approximate. However, the general run and the order of magnitude of the limits are as shown. 94 .oowH Mo 6 cm Om a: w HHme wow meoH03H0m manwmmom m3o:m xyoB m.vmmncomom .Boaw Nam HON mafia wcfluHEHH ozu smmsumn mucoummwwp .xoummmmu mwzu a“ mum: meowumuseEoo umusano mumpsuom muoE ecu on map muficammw m ma muozu we ou Emma .HMHH .mo>wzom «mqumomm VHHUOAM> mdemmom mo .Ummxcmmom kn vmcflahmuop mosam> msu cam mosam> mmmnu cwmsumn mmHocmmmHome mnu mo umoz h moz KREAD or & €\ 1 GUESS 002 00 = Q"0.S "i C = 1 1 sin(ZOzO) - 2.7.0 cos(20£0) 2 7012 C 2 N1 - 53'-Z§_ Sin(4d0) - C1200 C032(2a0) -'—§- 012 _ 1 N1+-2-—+012cos (20.0) +C1 cos(4a0)+010locos(2a0) “1 “ ' 8C, sin 2n0 48 sin(2a0) 4 .' _ , ()0 — O. " €01 N0 lad -9o l\ YES 7 10'-10 aa +90 m —-1 oz _ 0 2 <9 = -01 to +Ot>- - .4 1 N0 = 4C1 cos(2(xo) N0 f0 = Cl cos(20) - 7,- 1 012 2 2 C12 f1 2; N1+T+Cl cos (2010) + 7-,;- cos(40) 012 PLOT u,9 -+-3-cos(200) 9 sin(ZO) f = f0 ‘1‘ Efl l ' F = Cf . lZFv . u = ...... C .... D R J V FIGURE 4.1 SYMBOLIC FLOW CHART FOR MODIFIED PERTURBATION METHOD 108 graph it is obvious there is no primary solution for e > O at ac. In fact, referring back to Figure 3,7 it is clear that even including primary and secondary solutions there is no solution for SDO or SD2 flow at a = ac and e > O. In particular, the primary solution is for EH32 flow. From Figure 3°19 it is also obvious there is a secon- dary solution for SC2 flow in the same area, however, as will be verified later, the primary solution as can be determined from Figure 3.29 is the significant solution, With the aid of Figure 3.29, and the accompanying print-out of results where more accuracy is desired, four data points near ac are selected for comparing the dimensionless velocity distribution function, F, for the modified perturbation method with the exact solution for SC2 flow. The velocity distributions obtained from the modified perturbation method using the flow chart on Figure 4.1 are shown on Figure 4.2 with corresponding values printed for a, 6, a0, and al. The velocity distributions obtained from the exact solution, using Figure 2.29 to determine k and m3 and equa- tion (2.45) to determine the velocity distribution function, are shown on Figure 4.3 with corresponding values of a, E, k, and m3. Subroutine PRO CON used for the exact solution and subroutine PERT which follows the flow chart in Figure 4.1 are both shown with some of the calculated print-out in Appendix C. Figures 4.2 and 4.3 clearly verify the general procedure of the modified perturbation technique as the flow profiles on the two graphs are nearly identical. Exact comparison to determine the effect of larger values of e is difficult to obtain from Figures 4,2 and 4.3 due to the relatively small scale. Therefore, selected data values 109 SCALE 1 INCH-- 0.00057 FT/SEC (l) ALPHA 128.727 ego-127.92 a - -54.58 EPSILON z ~0.015 1 SCALE 1 INCH: 0.000b7 F T/SEC (2) ALPHA 2 122.0314 EPSILON = -O.l'l43 a0 - 120.23 a1 - ~12.62 SCALE 1 INCH-:2 0.00253 FT/SEC SCALE 1 INCH-- 0.00337 FT/SEC (L4) (3) ALPHA -= 125.778. (10 - 122.91 EPSILUN .-. ~O.h05 a1 - 44.48 ALPHA r. EPSILON = 128 .b83 "-0 .818 (10 - 118.79 FIGURE 4.2 VELOCITY DISTRIBUTIONS OBTAINED BY THE MODIFIED PERTURBATION METHOD 110 SCALE 1 INCHn 0.00057 FT/SEC ;§:::::::;\\\ ~. ,6 ll) \ % ALPHA + 128.727 K r 0.200 EPSILON a -0.015 M a 1.003 SCALE 1 INCH2 0 000b? FT/SEC _.~\“\ [9) C- V 9 ALPHA = 122.03h K = 0.200 EPSILON = ~O.lh3 M t 0.005 SCALE 1 INCHu 0.00250 FT/SEC (3) ALPHA < EPSILON e 128.778 K n ~0.¢05 M 1 0.h00 0.QHQ SCALE 1 INCH2 0.00330 FT/SEC (Q) LPHA = L28.b83 K a 0.u50 EPSILON r -0.815 M t 0 C377 FIGURE 4.3 VELOCITY DISTRIBUTIONS, SC2 FLOW 111 are obtained from the computer print-out and summarized in Figure 4.4. The approximate percentage of error on.Funme 4.4 was calculated as the difference between the two solutions divided by the maximum velocity for that profile. Analysis of this data indicates as expected that as 6 is increased toward 1.0 the accuracy of the overall profile is reduced. For data point two a was chosen a significant distance from ac, but the error is comparable to that for data point one where a is very near ac. Overall, the exact solutions clearly verify the correctness of the theory of the modified perturbation technique as applied to this problem. The use of one or two additional terms in the infinite series for a, f, and N would surely improve the accuracy, but the general procedure seems to be well founded. The possibility that the modified perturbation method might indeed predict solutions at a = ac for E > O has also been investi- gated by calculating values of a for values of do as do.» ac and for e > 0. However, in each case as 00 a QC, the resulting values of a are away from QC. This effect is illustrated by the results plotted in Figure 4.5, which is a plot of a vs. e and can be used to determine values of a for fixed values of a0 and e. Each line represents a value of a0 and it is obvious that as do.» ac from either direction the slope approaches zero and the value of a for a constant 6 becomes further and further from ac. Therefore, as none of the lines cross the ac axis for e > 0, no solutions are predicted for positive 6 when a = ac which agrees with the exact solution. However, all the lines eventually cross the ac axis for e < 0, so there are definitely solutions in this region, again veri- fying the exact solution prediction. 112 ' Velocities at a : Plot a e 6 r = 1.2 inches times 105 A error SC2 Mod. Pert. Exact Soln. Soln. 1 128.727 -.015 120° 5.3592 5.3539 0.0223 900 14.5837 14.5660 0.0744 60° 4.5890 4.5851 0.0164 30° -14.4651 -14.4473 0.0748 0° -23.7867 -23.7552 0.1324 2 122.034 -.143 120° 1.1400 1.1640 0.0855 90° 9.7264 9.7629 0.1300 60° -.3621 -.3475 0.0520 30° -19.0067 -19.0270 0.0723 0° -28.0703 -28.0999 0.1054 3 128.778 -.405 120° 19.2354 20.3168 1.0372 90° 45.6877 47.1445 1.3972 60° -.5874 -.0453 0.5199 30° -72.0406 -72.8874 0.8122 0° -104.2626 -105.3270 1.0209 4 128.683 -.818 120° 21.0358 24.8678 2.7863 90° 43.7890 48.3188 3.2937 60° ~17.2405 -15.9384 0.9468 30° ~101.1347 «103.4543 1.6866 , 0° ~137.5295 -140.3150 2.0254 1 FIGURE 4.4 THE DIMENSIONLESS DISTRIBUTION FUNCTION FOR COMPARISON OF SC2 FLOW AND MODIFIED PERTURBATION METHOD 113 FIGURE 4.5 RELATION BETWEEN 0: AND 040 NEAR Cz'c FROM MODIFIED PERTURBATION METHOD 114 4.5 Summary An example of the application of the flow parameter graphs deve10ped in Chapter III is presented in this chapter. However, the verification of the usefulness of the modified perturbation methods in deter- mining velocity distributions for values of a near ac is but one possible example of the general usefulness of the graphs. For most applications it is advantageous to have the flow parameter graphs plotted to a larger scale for more accurate determination of the flow parameters, and in fact this has been done to determine the values used in this chapter. However, for purposes of presentation in this report each graph had to be reduced in size to fit on a standard size page. V. SUMMARY A method has been presented in this dissertation of using the exact solutions for the problem of radial two-dimensional viscous flow between non-parallel plane walls to determine the exact shape of the velocity profile for a Specified flow rate (e) and angle of inclination (a). The problem and the governing equations were described in Chapter II with reference to the forms of solutions that have been obtained by previous authors. Also, several velocity pro- files were described, an expression for each velocity profile was obtained, and the boundary conditions of flow rate and angle of incli- nation were applied. In Chapter III it was pointed out that due to the transcen- dental form of equations obtained in Chapter II, explicit solutions for the velocity profile in terms of the flow conditions are difficult. However, by relating the elliptic functions in the transcendental equa- tions to trigonometric functions the equations were solved in terms of two flow parameters (k and m), and the results were then plotted on a - € coordinates referred to as flow parameter graphs. The flow parameter graphs contain curves of constant k and constant m and were deve10ped for each velocity profile that was considered. These curves describe the limits of each velocity profile and can be used to determine the flow parameters for calculating the velocity distribu- tion for any a - € within the family of curves. Further, in Chapter III the limits of each velocity profile and the relations between the different profiles were investigated. 115 116 In Chapter IV an example of the application of the flow para- meter graphs deve10ped in Chapter III was presented. The example involved a modified perturbation technique used for calculating velo- city profiles for small flow rates near a critical angle of inclina- tion where standard perturbation techniques do not apply. The velo- city profiles as obtained by use of the flow parameter graphs verified the modified perturbation theory as applied to this problem and pro- vided a ready means of evaluating the accuracy of the solutions obtained for different boundary conditions. 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.2 :2; .2 32m: 2.2. .munz . c _ . mnmo.n nu oomv u n . .n . ummxuu unconncoo.o. n a .nnucoum> oomn.n. nu nomn.m- nwu MMMfl.M “mu mmwmwmm Mm.wm “Wumnu unmxuu mnnfinncoc.o. n a .nnupoqm> ounm.u nu omom.n- A nau Noam.“ nou numunuo oc.m~ nmuu oumxnu cannoncao.o. n a .»n_co4w> nnn°.u nu onnn.n. nnu mano.o nOu wumnuuo ca.o~ nwuuyn oumxuu concencoc.°. n a .nnuuo4m> nmnn.o nu m~mn.n. nnu onmn.o nou wumnuuo oo.n~. nm4u7n ommxuu nmmnmneo°.o. n : .nn.noum> nonm.o nu onmn.n- nnu nwmm.o nou wumnmua °¢.n~ nmuuyn oumxnu ounomnco°.°. n a .nnucoum> nnmn.o nu anon... nnu onn~.o nou mwmuuua oo.n~ nwuu7u uuwxnu nunnnnooo.c. n a .»nucouw> ounn.o nu muma.n. nnu oflno.o nou mumuuua no.9» nmumnn unmxnu nnnnnnoao.on n ; .nnuuoum> onuo.m nu omfin.n. nuu «non.» nou mumumuo ca.nn .mum n umm\nu anomnneoc.en n a .nnuuouu> mnmonm nu omeu.m. nnu onnm.m n nou mumunuo °=.~n u nmumnn onmxnu,~uon~naon.o. n ; .nnuuoum> ennn.m nu when.o. nnu nmn~.m nou mwmumuo oo.,n nwuuyn ommxuu unnomnono.o. n ; .nnuuo4m> «own.m nu mnnn.o- nnu onmo.m nou muwunuo on.wn nmum7< ommxnu aonnwdooa.an n 4 .nnuuo4m> nooo.n nu mmmu.o. nau moou.n , nou mmmunua co. m n an unmxuu vmmncaooo.oa n a n>nucouw> muom.n nu omoo.u. nflu mwon.v nou mwmanua oo.wn nwuunu oumxnu womwodeco.ou n a .>n_uo4m> =«nn.n nu nomn.u. nHu nn-.n nou mumuuua an.un nmuu uuw\nu ononooocc.o. n a .»n_noum> unoo.n nu nono.u. nnu mono.n nou nunmuuo 91.0» nm mnn onm\nu mononaoao.on n 4 .>n_cogm> nanu.n nu -00.u. nnu ammo.» nou wmwumua oa.on nmwwna ummxnu umcnnooo°.o. n 4 .nnuuoum> nmnn.» nu onnn.o. nau anon.» nou aumuuuo.oo.on nun“. owmxnu momenocoo.o. n.4 .>n~coum> nonm.n nu ounn.n- nau onfid.n nou wumuuuo on."nn .munyn unmxuu unanncnao.°n - ; .nnuuonm> nano.~ nu mmnm.n. nnu nnno.~ _ nou mumuuua a:.mn nmunyn omnxnu wnnnoooao.an n a .nn.coum> onoo.~ nu nouu.a. nnu mamm.~ nou mmmuuua °=.,n nmJMJu oumxuu nmnomcoo°.°. n a .nnuuouu> unon.~ nu nano.n- nnu noo~.~ nou muwnuua ao.wn nun“; uumxnu oncomocao.o- n a .»n_ooum> ooma.~ nu nmon.o- nnu ammo." nou mumuuua oa.mn .nmnnnn wwm“»“ MMMMMWWWWHM. ” u H»H“mmuw> comm.“ nu vmnm.on nau nmcu.a nou mmmamun o=.ov nmumnu . a . > cmmm. an coco. I u n . . n .. uumxnu munanooo°.o. n a .nnuuoum> onn~.n nu ”nun.m- nwu “MM".M ”mu mwwmmwm uw.m" ”wwmyn me\hm comnwocoo.al I J Q>P~LOJN> whoona flu “Hov000 ”flu mommnb "0k mmwmnwn canvv uwdm7< ummxuu nooonococ.c. n ; .nnuuo4m> nnfiu.o nu nnnm.o- nau monm.o nou mumunuo no.9m ..wunyn awmxnu canonoooo.a. n 4 .»n_uoum> nonn.° nu numm.n. nnu anom.o nou mummnun co.nm nmunn« uwm\nu wonnoooac.°. n ; .nnucoum> anon.° nu nanm.o. nnu on~°.a nou mumunwo on.nm n 4“ Ummlu numnoooac.e n a .nn.uo4u> ancn.o. nu na~m.o. nfiu com~.a. nou mumuuua cm.um .wuuw« uumxnu anomaoooo.= n a .»n_uo4w> oomn.o. nu nmun.o- nnu nnmm.o. nou mwmuuun oc.wm .nm4u7n unmxuu nonmnooco.a n ; .nnuuoum> monn.°. nu noon.o. nnu nonn.o. nou mwmumuo oo.mm nmum n uumxnu ofioflmoaac.= n ; .nnucouw> Nmn0.on nu omen.o. nnu nnmo.«. nou mummaun on. m nmuun onm\nu nannmoooo.o n ; .nnuuoum> oun«.n. nu monn.o. , nfiu finnn.n. nou mumuuua an.mm nwnu.<. owmxuu nNonnoooo.= n ; .>n_noum> nonn.n. nu ncmc.o. nnu noum.n. nou mumnuua ca.mm n unyn uumxnu unannocoo.o n a .>nuuo4m> adoo.au nu neon.o. nnu ammo.du nou mumcumo oo.om uanuyu ommxnu "meanoooo.a. n a .>n_oo4m> nano.nn nu ooou.o- nHu mono.~- nou mumauuo oa.oo n any uumxnu mnoumoooc.o n a n>nuuoum> cued.mu nu mmom.o. nau nnan.~n nou mumauuo oo.«o nW4m7« uumxnu nnnnmacoo.o n ; .nnuco4m> numn.~. nu Nun~.m. nuu comm.~- nou .mumunuo an.~o n Any. uumxnu.noo~oaooo.a n a .>n_uo4m> nnoo.~. nu mono.o- nau omnu.~- nou muwuuua 93.;0 ntunu omm\uu annoooaoo.o n ; .nnuuonm> anon.~. nu nuou.n. nnu oooo.n. nou mumnmuo on.no n 4“ unmxnu "monsooao.° n a .nnuuonm> munn.n. nu «cam.u. nnu no-.n. nou mumnnmo ca.mo nwumyn wwm“»“ mummmwuunnu u u u»»“ww“w» omnn.n. nu mmH~.u. nnu ownn.n- ncu mumzmua 93.00 nwuunu . _ nnnm.n. nu nmon.¢- n . . . . unmxnu omonoaoae.a n a .nn.coqm> n~nu.nn nu omnn.o. aw“ ”www.mu ”mm mwwmwmm “w.mw “w“myn wnm“»u cannooooonc n ; u»n.uogm> oomonnn nu con~.o. nau ~nno.n. nou mumumun oo.oo unmumnu a nu a cnooaeo a n a nnunoum> can“ n. nu nnno.n. nau comm... nou mumuaua oc.on nmumzn psmxnu onn~enooa.o .>n_oo4m> nmun.n._ nu nman.n. nuu ~5on.n. nou muwaama oo.nu unmunyn, HHflfiHfifldv-‘H cocooooooooooooo. fidddfldflfiflflfififlflflf‘fl ~~~m~~~ - ‘ ~~~~~~~~~~~~~ry~m~~mmm~~~m~m~mmmmmmmtuwmmm~~~~~~m~~mmmmnr flflfifidflflflfldfidflflfiflflfidflflfi erfiio-“vv no. ~ I I-.' .n’. --.. ufiu‘v ‘g' ”0.4...." v..- ..— “‘i~ 1I.-o“"nc *x‘] w ('a-“§ifl""1.v"* ..., - “-. w v» "or. - 4' «iv f' .':'-'L'.'Z‘. a” 196 uuxxpu ounxpu oamxpu oxmxpu oumxpu onmxpu ummxpu uumxpu oumxnu oumxpu oumxpu ummxhu uumxuu ommxbu oum\pu ommxpu oumxhu ommxpu ommxpu ummxnu oumxpu ommxpu uumxbu ummxpu ommxhu ummxhu oumxhu ommxbu ommxpu uumxuu oumxuu oumxpu ommxpu ommxpu ommxpu oumxpu ummxpu uxmxpu oumxhu ummxhu umm\»u ommxpu ummxhu uumxpu ummxpu uumxpu oumxpu ummxpu ommxpu ummxpu ommxhu uumxpu oumxpk uumxbu ummxpu ummxhu ommxpu umwxpu uumxpu ummxpu ommxwu «yenmzcsc.ci “Hammooac.o mnonmccoc.o Honnmoooo.a onmmmoaoc.= «amomoooc.o manomaooo.c nowumoooo.o onenm°°=°.e «nonmoooc.o manomoaoo.o nomomco=°.° onwcmocco.a oommmocac.c “mommaaoa.o nouumoo°=.a monumoooo.o oncomoooo.c monomcaoo.o onummooco.c nnommooaono ommnmoooo.o «unnmoooonc ounumocoo.a anmnmooac.o oncomoooo.= oceanooao.c camunocoo.o «anoncoo°.o ndcvvoooo.o uncnncooo.o oavanoaoc.o mononoooo.c snoun==°°.a nooonococ.o «mnnnoco¢.o ~un~nocco.o nnflanocco.a nncmmca=°.c cnmnmoaoc.e muonmoaaa.= cannmoacc.= canonoo=°.a «onofioooo.o nonnnocac.a anonfioooc.c munooocao.o ”monooooo.a nnnnoocoa.o mflcoooooo.a mmmunmooo.°n onnnnmooo.cu "onunmoco.cn ammonmooo.on mmnmnmooa.on wvnnn~9o°.on camonmaoc.on nmc~n~ooo.cn nononmcoc.ou monommcao.en oonowuoooneu .nnunc_u> .>»_coJa> .>n_uo4w> n>»_Logm> .»»_coJm> .>»_coum> .>»_uo4m> .nnuco4m> n>p~L04w> .>»_UOJm> .>»_coam> .»»_coum> .>n_uo4m> .»n_uogm> .»»_uoum> .»n_co4m> .>»_L04m> .»»_oodm> .nnuu04m> .>»_cogw> .>n_qum> .»»_cOJm> .>».c04m> .>»_coJm> .>n_oo4m> .>»_coum> .>n.uoum> n>hucoqm> .»n.uo4w> n»»~L04m> .>»_cogm> .»»_uoum> .»»_uoum> .>n_uo4w> .nn.coJm> .>n_co4m> .nuuco4m> .»n.uo4m> .>n_LOJu> .>n_uo4u> .>»_Logm> .>n.uo4m> .»»_c04m> .>n_coJu> .»n_uo4u> .>n_uo4m> .nuucoum> .»n.uo4m> .>n_couw> .>n_uo4m> .>»~cogm> n>»_c04w> .>n.uo4m> .»n_uoum> .>»_uo4m> .>»_uodm> .>nuuo4m> .»n_uoum> n>pncogm> n>n_uoau> .>»_UOJw> na:n.n. nauo.n. ”sauna. ooou.nn moum.m. cano.m. uono.¢. “aco.ou o~°«.on ~nna.o. nmm«.o. -o«.on ooo«.on moo«.on -n«.o. onfi«.on aono.o. ammo.on mono.m. mnoo.mn nouo.mn oanu.n. nnno.m. oonm.m. nnmn.m. oflan.nn ooc«.mn oo~9.m. nouo.nn n-~.vn somm.vn ~nnn.n. n~o~.n. mnfio.vo nu~o.no on«o.nn vacc.nn moca.nn nuoo.~. non~.- omom.~n aon~.~. cmwo.~n annu.dn onmm.fin uno~.«. omoo.«n nonu.on nann.on oaco.on osmo.ou Nnmo.on mono.ou omno.on nano.o aono.o comm.o mmwo.o nnms.o mnno.o amon.o cun~.m. flouu.a. cnmn.«. Nono.°. ommn.o. omdc.° muon.o oouo.o noon.“ “can.” nnc~.~ nuou.~ nunu.» «nnm.n ~nao.n nonn.n ~«cm.n menu.“ name.» «ono.o nnon.o onuo.o mom~.u Nmno.u macs.» -nn.n modu.o onoo.o anon.o nonu.o nmoo.on nonn.on «moo.o« nuoo.on ooom.«a ounm.au nmnm.«« nama.~a nnmn.~a coco.~« nono.mn n~a~.nn ocmn.na afioo.nn nmmo.nu ~nma.nu noun.nn oaon.nu scam.n~ omn«.mfl nuou.~ «noo.~ ndoo.~ unoo.~ ~n~m.~ ~°~n.~ ~mo~.~ .oon«.~ onco.» anon.” anon.“ nuu nau an; nau nuu nnu nau nuu nuu nnu nuu nnu nau nau nuu nau nfiu nflu nau nHu nau nau «nu nau nau nnu nau nau nau nau nnu nuu momn.ma nuno.m. oomn.m. an0.m- conm.m. nono.mu unoo.o- onno.o. omen.o- ~N=H.o. fl-«.o- ou~«.o. «mwuno. «nofl.o. owoo.o. oono.o. unoo.o. uono.m. oomo.m. nmno.m. oomn.m. nuno.m. nonm.m- ommn.m. coon.m. .nnufi.n. «ovo.m. nnoo.n. «onu.n. o~nm.n. ~non.n- comm.n. mane... domm.n. oono.nn omnn.n. no-.n. coco.n. o~o~.~. oomn.~. nnfin.~. nono.~n comm.fln sonm.«- «can.«. nnmo.«. «ooh.o. nfimm.o. com~.o. vn-.o “moo.Ofi mnoo.0fi nmuo.ou unmo.ou mono.ou onuo.o oomo.o Nummno oncn.o «Hou.o ooco.o ncu ncu ucu ncu nou ncu «cu ncu uou ucu «on nou «on ucu non nou nou nou nou uou uou nou ncu nou nou uou «on nou non non nou non nou nou uou nou nou nou nou nou «cu nou nou nou uou nou uou nou ucu nou nou log nou «cu nou uou nou nou nou nou nou muu:mu: wuwcmu: mwuamuo mwmcmuo wwmamua mmmcmua mmwmwun mwmcmu: mwmcmwo mwmmnwn muwmmuo wmmamma muwmmwo wwwmmwn mwmmmuo mmmammn mmwcmwn mmmamuo mWwamwo mmwmmmo mmwmmmo mmwamuo mmwcoua mmwanwa wwwmmwo mmwamua mmwmmmo mmmcmmo mwmmmwo muwmmuo mmwamuo .mwmmmwn mmmmmwo mmwzmwo mumzmwo wmmanuo mwmzmma mwmmmun mwwmmwo mmmmmua mummmmol mummnua mwwmmun mmwumuo mmmmmuo mwmcmwo mumcmmo mmwmnun mwmcwwn mmwmmmo mmmmcmn mmmammo mmwamwn mmmmwwa mwmaMmo wwwzwmn mmwawun mwmcmwo wwwmmua mmwamwa mummmwo cc.nn o:.nm oa.«m on.~o 99.no oo.vm oa.mo oo.oo ca.so oa.mm oo.oo oo.co oo.«o oa.wo oo.no ao.vo oo.mo oo.oo ca.uo oo.mo oo.oo oc.oon. oo.naa oo.wo« oo.no« oo.voH oo.mo« o=.ooa oc.uog oo.wo« ao.oca oc.°HH oo.««a oo.ma« on.naa oa.n«« ca.maa o;.oaa on.n«a oo.w«a o=.uaa oa.c~« oa.«~n ca.-H ca.n~n o=.n~a oa.m- o=.owa ca.u~« nnnmma oc.o co." oo.~ oo.n oo.v. oa.n oo.o oo.s oa.o cone oa.ba 74 . 7< nm4m7< nwum7< nw4m7< nw4m7< nw4m2< nw4m7< nm4m7< nmdm7< uw4m7< nw4m7< nw4n7< nu (W LU ('1 .nm4m7< nm4m7< nw4m7< nm4m7< nw4m7< nm4m7< nm4m7< nm4m7< nw4m7< nw4m7< nw4m7< nm4m7< nm4m7< um4m7< nm4m7< nm4m7».Lc .u: .»».LO4m> .>».u04m> .>»_uogm> .>n.uoqw> .nnuuo4m> .>»~Logw> .>».co4m> .>n_co4w> .»n.uo4w> n>n_uo4m> .>»_pogm> n>puuogm> .>n_uogm> n>n~uo4m> .>n.uo4m> .>»_uogw> .>»~uoqm> .>n_uo4m> .»n_co4m> .>».uogw> .>»_uogm> .>»_cogm> n>pucoqm> .nnuuo4m> .>u.coum> .>n~uo4m> .>n_qum> .>n~coJm> .>»_uogm> .>n.uoum> .>n.uOJm> .>n~uouw> .>n~uo4u> .>n~uo4m> .»n.uoqm> .>n_co4m> .>n.uo4m> n>nucogm> .>n~co4w> .>»_cogm> .>»_qum> n>p_uoqw> .>n~uo4m> .>»_uogw> n>p~uo4m> .>n_cogm> .>».L04w> .>»_L04m> .>»_c04m> .»»_co4m> .>»_uo4m> .>n~uo4m> .>n_uo4m> .>n.uo4m> .>»_u04w> .>»_co4m> .>»_ooqm> n>hnuoqw> .>>~004m> .>»_oogm> xxjf I omen.» comm.» coco.» nmmm.u mnmo.n oomn.n oun~.u nnno.n n~n¢.¢ nonm.o .nvmn.o anna.o ovum.“ mu~0.m oonn.m ~°~«.m nooo.n muom.n cann.n «Hoo.n noou.n chum.» «on~.n “floo.~ onco.~ naon.~. m°~«.~ connuu comm.«. oun~.« muoo.° nofiu.o monn.o onoa.o noo«.o. oown.on mono.on «mno.o. ouua.«. nonn.«u oaoo.an nano.an ouma.~. mumv.~n nneo.~n nnoo.~n mnfla.n. omnn.n. nnnm.nn nmmu.n. oomo.nn o«n«.vu nmmn.nn nnom.nn ooao.nn homo... samo.vn ~n~a.mn ooo~.mn aoon.mn can»; c... mmfio.«. mmnn.«. ~n~u.n. Huuo.~- oomn.~- nmmo.~- omo~.o. omnm.o-. mmmo.n. onun.n- nnoo.n. csmo.m- eman.m. omeu.n. Nuca.o- mnnn.o. mnmu.o. omoo.u. unma.u- nono.u. -oo.u. omna.o- ouun.on momm.ou nonu.o. nnmo.o. n~oa.o. n~n~.on conn.o. nu~n.o. maon.o. annm.o- H~mm.o. nonm.o. uamm.o. nmnn.o. moan.o- omen.o- Nooa.o. oom°.n. ocoo.o. ooou.m. mocm.n. ~um~.o. mono.o. numu.u. nonm.u. mmn~.u. «moo.o- ooum.o. onn~.o. nnco.m. nmam.m- mon«.m. mmns.n. onnn.nn anno.n. oonm.n. «doo.nu nomo.~. "an nau nau uau nau nau nau nau .udu nau nau nau uuu nuu nau nah nau «flu nfiu «flu nwu nau nau nau nau nau nau nau nau nau «an uau nau nau nHu «flu nau nau nau hum nau nau nau nau nan nau mzvm x «mun.m moo~.m vndo.m namo.u camo.n onan.u mocwnu Naoo.o cnmu.o «mmm.o ovo~.o cano.o ~non.m oavm.m nmm~.m onmo.m Noon.n mmon.n unm~.n ~mno.n ammo.» nnon.n onaa.n onnm.~ mamm.~ noo~.~ omen.” nmon.fl sane.” ounH.« memo.o monm.o omo~.° ammo.o oom~.o. ndmm.o. “on“.o. enmo.«o finfin.«u nonm.«. onmm.a. souo.~. nnfln.~- oomm.~n omnu.~. oooo.n. nom~.n. omen.n: oono.nn «omm.n. ~nno.n- oou~.n. wnon.n. onnm.n. «onn.n. nnoc.n. dovo.m. nnna.w. coon.m. ommn.nn "Cu ucu uou uou «cu uou «on «cu uou uou non uou uou «on uou uou nou nou «cu nou uou nou nou nou nou uou uou «cu nou nou nou ncu nou nou uou nou ncu uou uou nou non nou nou nou nou non nou uou nou nou nou nou nou nou uou uou nou nou nou non nou Uuutvuc muwcmuo mumnmuo muwamuo muwmmmo mumamwn wwwqmmo mmwamwn mwmmmmn mwmammn mmmumwo mmmcmmo mwwamwa mummmwa mwmzmwo mmmcwmo mmwcmmo mwmcmwo mecmuo wwwamuo muwammq mmmmmmn mwmamwn mwmzmmo mwmamwn mwmumwa mwmamun mwmcmmo mummuwn mmwmmmn mmmanwa mmwamua wwwamun mwmmmun mmmmnmo mwmzmma mmwmmuo mmmcmwn mwmmmwn mwmmnua wwwanun mwmcmwo mwmamun mummmun mmmammo mmmmmwn mummmwo wumznun wwwamun mmwmawn mmmmmwn wwwamwo mumzmwo mummmmo mmmzmmo mwmcmmn mmwmmma mwmamuo mummmwo mmwmmmn mmmzawn cc aw 95.0H an.cm oa.«~ ca.- a=.n~ =a.v~ c=.mm °=.o~ on.nm oo.m~ eo.o~ canon ca.fin oa.~n oo.nn oa.nm co.mn oo.on oa.un no.0» no.on as.on .oanac ac.~n ca.nn o=.n¢ oc.mn on.on connv. oo.mn oo.on oo.am oo.wm oo.~m oo.nm oo.vm oo.mm on.om =o.um oc.mm o=.om oo.oo on.Ho ca.~o 99.no oo.vo oc.mo oo.oo on.no oo.wo oa.oo as.cu oa.an o=.~n oo.nn oa.vs oo.mn oo.os oa.ns aa.ou nu .074 nuum7< nmqmy< nw4m7< nm4m7c. nw4m7< nm4m2< nw4m7< nm4m7 .>n.co4m> .>»_uo4m> .»n_cogm> .nn.co4m> .>»_coum> .>n_couw> .>n_cogm> .>n.cogm> .>n.coum> .>n_uo4w> .nn.oo4m> .>n.coum> .>n_nogm> .nnuuo4m> .nnuco4m> .>»_oogm> .»».uogm> onm°.o« ~nmo.cu anno.=« amno.on namo.o oono.o .oomono mmmo.o nnmu.o mnno.o nmmm.o when.n moon.o nno~.o gona.o «599.0 omom.o nous.¢ an an at an an an an In an an an an uh an In nu nuou.~ nooo.~ ndco.~ unoo.~ ~n~n.~ ~o~n.~ ~mo~.~ oona.~ cano.n anon." mama.” moan.“ ~n~a.a ammo.a omnm.o n~o~.o nn~o.o. onnn.o. nau nu. nau nuu nau «flu nau nan ndu nau ndu nau can nau nau nau can nau Hooc.oa Nnoo.on nouo.ou nnm°.on mofla.on onuo.o .oamo.o Numo.o onmu.o «dau.o oaoo.o nnom.o moon.o nuu~.o mond.o naaono nnoo.o ~««~.o «cu uou uou ucu nou ncu uou uou uou ncu uou nou uou nou uou nou nou nou mumnmuo mumcmuo mwmnmuo muwummo mumnmuo mwmnmuo mmwnmua mwmumuo mwmnnua muwnmua mmwcnuo mwmummo mumumuo mwmnmun mmmnmuo mwmnmua mmwanuo mmmmmwa nm4m7< nm4m7< nw4m7u nw4m7< nm4m>( nw4m7< nmumyn nm4m7< nwumvn nm4m7< nm4m7< nm4m7< nw4m7< nm4m7< nmgmy< nm4m7< nm4n7< nwémyn oz— .2_ .2— .z_ .zu .2. .z_ .21 .z~ .2” .2” .2. .21 .z~ .2" nnflnnnnnnnnnnnnnnn nu um um um um um um ax um am «a um um am at am