ABSTRACT SOME GEOMETRIC PROPERTIES OF COMPRESSIBLE FLUID FLOWS AND CERTAIN CLASSES OF SUCH FLOWS OBTAINED BY INTRINSIC METHODS by Lester B. Fuller Steady nonviscous nonheat—conducting flow of a compres— sible fluid in the absence of external forces is discussed in this paper. In particular the geometry of such flows is studied, the dynamical equations which characterize these flows are reformulated, and some classes of flows are obtained. In connection with the geometry, a condition is ob— tained which is necessary and sufficient for the existence of stream surfaces that contain the vortex lines. These are called Lamb surfaces. Then, two necessary and sufficient conditions are found; one for streamlines to be geodesics on Lamb surfaces and the other for streamlines to be asymp- totics on stream surfaces. More generally;two necessary and sufficient conditions are obtained: one for the existence of stream surfaces on which streamlines are geodesics and the other for the existence of stream surfaces on which streamlines are asymptotics° Some of these conditions mentioned involve the magni- tude of the velocity vector, and so, relationships between it and the geometry are observed. Thus, the geometry and the dynamics are related. Concerning the dynamical equations and their solutions, the following has been done. A system of equations equiva— lent to the dynamical equations is obtained using two families Lester B. Fuller of stream surfaces and the family of constant pressure sur- faces as coordinate surfaces. From this reformulated system of equations, two classes of plane flows are found, and a single ordinary differential equation is obtained whose solu— tion leads to a class of three dimensional flows. SOME GEOMETRIC PROPERTIES OF COMPRESSIBLE FLUID FLOWS AND CERTAIN CLASSES OF SUCH FLOWS OBTAINED BY INTRINSIC METHODS BY 3,2 K( x.) Lester B. Fuller A THESIS Submitted to Michigan State University in partial fulfillment of the requirments for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1968 I .’ I. [I c:\.£7fiQ;J'9'/ ../"I (V - ‘ __ a’ . .I ‘r ,, ./ l ,. ACKNOWLEDGMENT I am deeply indebted to numerous peOple for their kind- nesses during the days in which I was preparing this thesis. There are the administrators who provided me with an instruc- torship that supplied for the financial needs of my family and me; there are the professors who took time to read a rough draft of this paper and serve as members of my graduate committee; there is Dr. Charles Martin who often offered words of encouragement and counsel; and there is my wife who typed a preliminary copy of this thesis. To each of these I express my sincere thanks. It is, however, to Dr. Robert Wasserman, my major pro- fessor, that I am most deeply indebted for his many helpful suggestions, his praise when my accomplishments were few, and his sharing many valuable hours with me. To him I ex- tend my deep heart-felt appreciation. Furthermore, believing that ideas come from the living God, I am very grateful to Him. ii CHAPTER I. II. III. IV. V. VI. REFERENCES INTRODUCTION THE GEOMETRY OF COMPRESSIBLE FLOWS CONTENTS RELATIONSHIPS BETWEEN A REFORMULATION OF THE DYNAMICAL EQUATIONS 5x5) AND'I‘HE MAGNITUDE OF THE VELOCITY VECTOR TWO CLASSES OF PLANE FLOWS . A CLASS OF THREE DIMENSIONAL FLOWS iii 29 33 51 64 76 CHAPTER I INTRODUCTION 1.1 Preliminaries As the title of this paper indicates, it is our pur- pose herein to make some observations concerning certain fluid flows, the equations which characterize them, and their geometry. Consequently, it seems appropriate that we begin with a discussion and explanation of some of these terms. A fluid flow is a set of functions which satisfies a certain system of nonlinear partial differential equa- tions, and so, we first consider these equations which are referred to as the dynamical equations throughout this paper. We write the system using the standard summation convention (i = 1, 2, 3) and, following it, explain the notation and discuss its derivation. (1.11) VEPV = 0 i _ l - _ (1.12) v Vivj — — P vgp (3 — 1,2,3) (1.13) Vivin = o In these equations we are using the customary notation of tensor calculus, so v1 and v3. are the contravariant and covariant components respectively of a vector field called the velocity vector field, and 'VE represents the 1 2 covariant derivative. The quantities p, p, and q are scalar point functions known as pressure, density, and entropy, respectively. Consequently, the terms 'Viq and V59 are gradients of scalar point functions. The dynamical equations are derived from the basic principles of conservation of mass, momentum, and energy. These derivations are given in most standard fluid mechanics texts, such as in the first chapter of H. Lamb [8]. The form of equation (1.13) we are using is nicely developed in R. Courant and K. O. Friedrichs [4, p. 14-16]. These equations are the mathematical model for steady nonviscous nonheat—conducting flow of a fluid in the ab- sence of external forces. By the term steady we mean that the quantities appearing in these equations depend on posi- tion only and are independent of time; by nonviscous we mean the force an element of fluid exerts on an adjacent element is normal to their common surface; by nonheat-conducting we mean there is no flow of heat from a hotter portion of the fluid to a cooler portion except that which takes place by convection, that is, by the motion of the fluid itself. A set of functions consisting of a vector function G, with components vi or Vj’ and three scalar point func— tions n, p, and p, which identically satisfy the dynamical equations in some region of three dimensional space, is called a flgw, Briefly, then, a flow is a solution of equations (1.11) to (1.13). In the special case where the density, p, is constant, the flow is called incompressible. 3 Otherwise, it is known as compressible. If the vector func- tion 5 is such that curl v — 5 throughout a region of three dimensional space, the flow is irrotational in that region. Otherwise, it is called rotational. Throughout this paper we use I to denote curl v and call it the vorticity or vortex vector. If the vorticity and velocity vectors of a flow are parallel at each point in a region, that is G x I = 5, then the flow is called Beltrami. Each curve of the family of integral curves of the velocity vector field is called a streamline, and, similarly, a vortex line is a member of the family of integral curves of the vortex vector field. From the physical vieWpoint, a streamline in a steady flow is the path of a fluid particle. Thus we see that certain families of curves may be associ— ated with a fluid flow. It is also possible to associate various families of surfaces with a flow, as the following remarks indicate. (We tacitly assume that any requirements such as continuity and differentiability of functions is satisfied.) The presence of the pressure gradient in (1.12) brings to mind the concept from vector calculus of level surfaces of a scalar point function. In this case, these are surfaces on which p is constant and which have the property that, at each point, a surface normal is collinear with the pressure gradient. The partial differential equation vivie = 0 has two linearly independent solutions which we shall denote by 61(X1, X2, X3) and 92(X1, X2, X3), where X1, X2, and 4 X3 denote independent variables. If we let C1 and C2 be arbitrary constants, then 61(X1, X2, X3) = C1 and 92(X1, X2, X3) 3 C2 represent families of surfaces known as Stream surfaces. From the two preceding paragraphs, we notice that we may associate with a flow geometric objects such as families of curves and families of surfaces. Furthermore, for a given flow, these curves and surfaces may have special properties. For example, the streamlines may be straight, or the streamlines and vortex lines may coincide as in a Beltrami flow. Hence, we describe a flow in terms of these geometric quantities and their properties and speak of the geometry of a flow. In case the streamlines are plane curves, and there exist stream surfaces which are planes with the property that all quantities of the flow do not change in the direction normal to them, we say the flow is a plane flow. 1.2 Background Concerning Geometry We would like to mention some of the previous research in the area of the geometry of fluid flows which, to some extent, motivates the geometric considerations undertaken in this paper, or has a direct bearing upon them. D. Gilbarg [5] poses the following question: In what way is the flow pattern (i.e. the streamlines) related to the velocity, and to what extent does one depend on the other? Or, we might state the question this way: to what 5 extent does the geometry of a flow determine its dynamics? Some partial answers to this question have been obtained. In the case of steady, incompressible, plane flow, Gilbarg determines all incompressible flows having the same flow pattern as an arbitrary given flow of an incompressible fluid. In fact, he shows that, if the given flow does not have a constant velocity magnitude along each individual streamline, the only flows with the same streamline pattern are those having velocity fields that are proportional to that of the given flow. R. C. Prim [12] extends these find- ings of Gilbarg to the three dimensional case. R. Wasser— man [21] shows that, if the velocity vector v = qE (IEI = 1), corresponding to each incompressible (compressible) flow with ‘fiaté = 0 there is a compressible (incompressible) flow having the same streamlines and constant pressure sur- faces. He also proves the converse, namely, that if a compressible and incompressible flow have the same (non- straight) streamlines and constant pressure surfaces, then they both have xyiti = O. R. c. Prim [13] points out that among all flows having the same streamlines and constant pressure surfaces, there is a flow containing a special family of surfaces, which is not necessarily present in all flows. We shall elaborate more fully upon this in Chapter II, but, at present, simply point out that much of Chapters 11 and III are motivated by the articles mentioned in this sec- tion. 6 In other areas of flow geometry,v and apply the vector identity (\7 ~v>G= (1/2)V(\7 - \7) —\7x (vxx’r) with G = qt. We get (2312) p(\7 x I) - pv(q2/2) = VP- We form the curl and obtain VpX(\7xI)+p\7x(\7xJ>)—Vva( ((1)2/2 =5 Forming the scalar product with G x I and transposing gives 5x5) - Vx (6x50) =1/p[(\-IX(I)) ~vpx v(q2/2)1. From the equation of state, we see this may be written - - - --1- - .3 %% 912 vxw-vx(vxw)--6vx (D‘ Vp+ v1] XV2 . Equation (2.312) shows that V(q2/2), vp, and v x I are coplanar, and hence, their scalar triple product is zero. Therefore, 16 (2.315) vxI - vx (\7xI) =%%%[(VX0—J)'VT]XV%2 :1. If, to the right side of this equation, we apply the ident- ity of Lagrange, which says that (5x16) - (6x3):(a-E)(E-a)-(a-a)(E-E), and drop the term involving v °§7n (by (1.13) this is zero), then (2.315) becomes 2 (chb) - vx (5x25) =-%§~%(&3~ VUUV avg-L It is to be inferred from Section 2.1 that p depends on n inauchawaythat 3133750. Thus (5x513)- vx (\7xI) ‘Vqu/Z) = 0, and, conversely, <1! =0 if I°Vn=0 or if (vxI) 'Vx (vxI) 0, then either I-Vq=0 or ° V(q2/2) - V(q2/2) 0. Equation (2.312), however, implies that <| 0 if and only if v . VP = 0, and the = Q‘Vfl for some scalar point function a. Hence, VB = (a + T)vn. and the gradients of B and n are parallel, so their level surfaces coincide. By introducing the term Bernoulli surface for a level surface of B, we may draw the following conclusions from Theorem 2.31 and its corollaries, concerning a flow in which pressure is not constant on streamlines. If Lamb surfaces exist, then Bernoulli surfaces and constant entropy surfaces 18 coincide and in fact are the same as the Lamb surfaces. Or, if the Bernoulli surfaces and constant entropy surfaces differ, then the flow cannot contain Lamb surfaces. Two articles, one by R. C. Prim [13] and another by P. Smith [15L motivated our investigation. For this reason, we use the term Prim-Smith flow, throughout the remainder of this paper, to designate a flow with the following two properties: (a) it possesses Lamb surfaces, (b) its constant pressure surfaces are not stream surfaces. 2.4 A Special Congruence of Curves Lying on the Lamb Surfaces Theorem 2.41: At each point of a Prim-Smith flow or a flow in which constant pressure surfaces are stream surfaces, the vectors VT)! VP: VP and vq are coplanar. Proof: Case I - Prim—Smith flow: The equation of motion in the form (2.42) qu/z — \7 x (13 = - -Vp implies that Vq, v x I and vp are coplanar. In a Prim-Smith flow, however, 5 x I and V7n are collinear. So vq lies in the plane of VT] and vp. From the equa- tion of state, it is clear that ‘Vp is also in the plane of Vp and V1], and hence, the four gradients, VT]: VP! Vp. and Vq are coplanar. 19 Case II - Constant pressure surfaces are stream surfaces: In this case t - vp 0. This, together with equation (2.42), implies that E ~vq = 0. Since E ~ VT) = o in the flows under consideration, the equation of state implies that t - VP = 0. So at each point of the flow VT], VP' Vp and Vq lie in a plane normal to the streamline through the point. We observe that in Case II, n, p, p, and q are constant on the streamlines and extend this property in the following corollary. Corollary 2.43: In a flow with constant pressure surfaces and Lamb surfaces that are distinct, n, p, p, and q are constant on the integral curves of the unit vector field given by (2.44) 2 2 [‘E-wgm‘: - [v’J-WSZIHE i = _ _ Z _ T _ f ._ 2 .. _ ~/[t-V<% >12+[w- mg )12- 2[t-v(-§l )1[w-v(%)1w - t where I = ww with l Q1 = 1. Proof: Case I - Prim-Smith flow: By the symmetry in (2.44), X - Vq = 0. Since E ' V7] = 0, according to the energy equation, and §'- {7r[= 0 in a Prim—Smith flow, i ° V7n = 0. According to Theorem 2.41, §7p and §7p lie in the plane spanned by VT) and Vq. Therefore, )2 ° Vp X - §7p = 0 and the corollary's conclusion holds for a Prim-Smith flow. 20 Case II - Constant pressure surfaces are stream surfaces: As previou'sly noted E - Vp = 0 if and only if E - V (q2/2) = 0. By hypothesis the constant pressure surfaces are dis- tinct from the Lamb surfaces, so that a - <7(q2/2) # 0. Therefore 2 = i t, and as observed immediately preceding this corollary, q, p, p, and q are constant on the streamlines, in this case. Thus the corollary is proved. 2.5 Geodesics and Asymptotics on Lamb Surfaces We have already observed in Case II of the proof of Theorem 2.41 that, when streamlines lie on constant pres— sure surfaces, n, p, p, and q are constant on the stream- lines. Since the equation of motion, (2.22), implies that 5 ° v7p is invariably zero, the additional property of p being constant on the streamlines (E - V7p»= 0) implies that the streamlines are geodesics [24, p. 99] on the con- stant pressure surfaces. Conversely, if a streamline is (a geodesic) on a constant pressure surface, E - §7p = 0, and as we have shown in Case II of Theorem 2.41, q, p, p, and q are constant on the streamlines. We summarize these remarks in a theorem. Admittedly, the term geodesic could be omitted from the statement, but we insert it because we wish to draw attention to this geodesic property. Theorem 2.51: The functions q, p, p, and q are constant on the streamlines of a flow if and only if the streamlines are (geodesics) on the surfaces of constant pressure. 21 In the case of constant pressure surfaces, then, the streamlines need merely to lie on the surfaces in order to be geodesics. In general, of course, this is not the case, and we now obtain a necessary and sufficient condition for a streamline to be a geodesic on a Lamb surface. Forming the cross product of (2.24) with G, we obtain (2.52) GxI=q(g—g-qx>fi+q§%6. If v x J) ¢ 0, then (2.52) implies that aq/ab = E . Vq = o if and only if E x I is collinear with H, the principal normal of the streamline. But, at each point of a Lamb surface, 6 x I is collinear with a surface normal. So, a streamline is a geodesic on a Lamb surface if and only if 5 ° §7q = 0. That is, q constant on each curve of the E congruence is a necessary and sufficient condition for streamlines to be geodesics on Lamb surfaces. Theorem 2.41 and the fact that E ' <7p = 0 imply that n, p. p. and q are constant on each curve of the E congruence whenever q is. Therefore, we can summarize our remarks as follows: Theorem 2.53: A necessary and sufficient condition for streamlines to be geodesics on Lamb surfaces is that q, p, p , and q are constant on the E congruence. Since q being constant on each curve of the E con- gruence (or b congruence) implies n, p, and p are also, and conversely, we may consider Theorems 2.51 and 2.53 as 22 relating dynamic properties of a flow to geometric properties. In fact, they say that E ° §7q = 0 is a necessary and suf- ficient condition for streamlines to be geodesics on con- stant pressure surfaces, and 5 ° $7q = 0 is a necessary and sufficient condition for streamlines to be geodesics on Lamb surfaces (in a flow containing such surfaces). As a consequence of these remarks, we have the following corollary. Corollary 2.54: E - Vq = b ' Vq = 0 if and only if the Lamb surfaces and constant pressure surfaces coincide. A further consideration of Theorems 2.51 and 2.53 re- veals that the i vector of Corollary 2.43 is collinear with the E vector when streamlines are geodesics on con— stant pressure surfaces, and collinear with the 5 vector when streamlines are geodesics on Lamb surfaces. This leads us to aSk if, under certain conditions, i is collinear with 5. If it were, pressure would be constant on each curve of the H congruence. Equation (2.22) shows that H ' <7p = 0 if and only if K = 0, that is, the streamlines are straight. So, we may immediately conclude, that if the streamlines are not straight, the i and H vectors cannot be collinear. Since a straight line that lies entirely in a surface is an asymptotic line of a surface [5, p. 237], the remarks of the last paragraph draw our attention to asymptotic curves on Lamb surfaces. Equation (2.52) implies that 5 x I and E are collinear if and only if 23 -15 .-. — q n n. V71n q. At each point of a flow containing Lamb surfaces, 5 x I is in the direction of a normal to the Lamb surface through the point. So, at each point of a flow, a unit normal to the Lamb surface and the 5 vector coincide if and only if K = H ' V7ln q. Thus we have proved the following: Theorem 2.55: In a flow containing Lamb surfaces, the streamlines are asymptotics on these surfaces if and only if K = 5 ° V7ln q. E. R. Suryanarayan [17] has shown that (excluding straight streamlines) K = 5 ° <7ln q is a necessary and sufficient condition for Lamb surfaces and constant pressure surfaces to intersect orthogonally, provided entropy is con- stant throughout the region of flow. His assumption of con- stant entropy is not necessary, however, as his conclusion follows if H - VT] = E ° vq = 0. So, a slight modifica- tion of his statement would be this: in any Prim—Smith flow with non-straight streamlines, K = 5 ° <7ln q is a neces- sary and sufficient condition for Lamb surfaces and constant pressure surfaces to intersect orthogonally, as well as for streamlines to be asymptotics on the Lamb surfaces. We close this section by making a few remarks concern- ing some similarities between Beltrami flows and flows con- taining Lamb surfaces. R. C. Prim [13, p. 434] has pointed out that V 7), VP! Vp, and vq are collinear in a Beltrami flow. From (2.52), however, 5 x I = 0 implies b - <7q = 0. Hence, for a Beltrami flow, q, p, p, and q are constant on the E congruence. From equation (2.52), 24 it is also clear that K = 5 ° <7ln q in a Beltrami flow. Consequently, the conditions mentioned in Theorem 2.53 con— cerning geodesics on Lamb surfaces, and the condition in Theorem 2.55 for asymptotics on Lamb surfaces, are both en- joyed by a Beltrami flow. Of course, we do not have Lamb surfaces in a Beltrami flow except, perhaps, in a degenerate sense. 2.6 Geodesics on Stream Surfaces In the last section, we considered two cases in which streamlines were geodesics on stream surfaces. It is our purpose here to obtain a necessary and sufficient condition for the existence of stream surfaces on which streamlines are geodesics. Equation (2.24) is not only valid for the vorticity vector of a fluid flow but, also, for curl E of an arbi- trary vector field E = qE where [E] = 1. We choose 5 = E, an arbitrary unit vector, and substitute. This gives a '-H°S)E+Kb. <1 X (H II ’0‘“ 0210/ :3 CH U‘Ir‘f’l Thus fi'vxE=0. Theorem 2.61: If the pressure gradient is not tangent to the streamline throughout a flow, then a necessary and suf- ficient condition for the existence of surfaces on which streamlines are geodesics is that E ‘ V7p be constant along the E congruence. 25 Proof: For suitable scalar functions a and B, the equa- tion of motion may be written as dE+BH = Vp. Forming the curl yields (2.62) Vaxt+avxE+Vf5xfi+f5vxfi=0. Dotting with H and using the fact that 5 ° V7x E - 0 yields Therefore, (an ' Vx n = — b ' VOL. By assumption B # 0, and so 5 ° V'x H = 0 if and only if a = E - twp is constant along the B congruence. But 5 - V7x H = 0 means the H congruence is a normal con- gruence, so there is a family of surfaces cutting it orthog- onally. On these surfaces the streamlines are geodesics. 2.7 Asymptotics on Stream Surfaces In Section 2.5 we obtained a necessary and sufficient condition for streamlines to be asymptotics on Lamb sur— faces. More generally, one might seek a necessary and suf- ficient condition for the existence of stream surfaces on which streamlines are asymptotics. We do this in this sec— tion. Before embarking on such an investigation, however, we note that the problem could be rephrased by saying we seek a necessary and sufficient condition for the 5 x firp congruence to be a normal congruence. Furthermore, since 26 V'p lies in the osculating plane of the streamlines, we are seeking a condition for which there exists a scalar point function (h such that E ° V (p = 0 and 5 ~ V (D = 0. Hence we must satisfy the integrability conditions [2, p. 186-187], namely j£_j13 : (tn nt)vjb£ 0. Using the Frenet formula tJ §7jb£ = - THE, where T is the torsion of the streamlines, and the fact that tgnj <7.b£ = - bfinj<7.t the integrability condition becomes 3 32’ = 133' (2-71) ’1' b n vjtg' To investigate (2.71) further, we make use of the vec- tor identity V(-u--\7) = (E1 °V)\7 + (\7 °V)G + G x (Vx \7) + E x ( V'XG). Replacing G by 5 and E by E, we have (2.72) 6:03-v)E+(E~v)fi+fix(vxE)+Ex(vxfi). Using the Frenet formula (E ° V7)H = - KE + 15 and forming the scalar product of the right side of (2.72) with 5, we obtain 0:5'(H'V)E+T-E'th+r-1°vxn. Substituting from equation (2.71) yields (2.73) 21=E°VxE-H°vxfi. From equation (2.62) and the fact that H ' V7x E = 0, we 27 observe that E -‘v'x E = and SI <1 X s n' Substitution into (2.73) shows that —-.l .1. _Q-. - 2T—b (aVB-tfivd) at Vx n. By a well known vector identity, V'E=V-(txn)=r-1'VxE—E°Vxn=-E°vxn, and we have 2 2 V9+Vg — 2 - 2 2 g - b'VIVpl p - = - V7- = e + - , 21 b 016 +0. b 2013 aVb Hence, we have proved the following theorem. Theorem 2.74: If firp is not collinear with either E or n, a necessary and sufficient condition for the existence of stream surfaces on which streamlines are asymptotics is .that T, the torsion of the streamlines, be given by the ex- pression 28 We conclude this chapter with a few remarks about Theorems 2.61 and 2.74. In both theorems, it is assumed that H ' S7p is not zero. This simply means that the special case of straight streamlines is not considered. It happens that a straight line is both a geodesic and an asymptotic on a surface containing it. Theorem 2.74 also fails to take into account the situation in which a = E ' :7p = 0. In this case, equation (2.62) implies that 5 ° V'x H = 0, and from (2.73), we see that a necessary and sufficient condition for streamlines to be asymptotics on stream surfaces is that 21 = E ° V x t. CHAPTER III RELATIONS BETWEEN x7 x I AND THE MAGNITUDE OF THE VELOCITY VECTOR 3.1 Introductory Remarks It has been proven by M. H. Martin [9, p. 470] that for plane flow, a necessary and sufficient condition for an irrotational flow is that q, the magnitude of the velocity vector, depends on pressure only. In this chap— ter, we would like to extend this remark by showing rela- tionships between q and E x I. We assume throughout this chapter that surfaces of constant pressure are not stream surfaces, and we denote by w = constant and m = constant, where w and w are scalar point functions, two distinct families of stream surfaces. Thus, we consider q to be a function of p, m, and ¢. 3.2 Beltrami Flows The equation of motion may be written in the form \7 X <3= V(q2/2) + 1/p VP~ Expanding the term involving q yields (3.21) \7 x I 3 (qqp + 1/p)Vp + qqcpva) + qquv¢° By assumption E °vp5£0, t °V=O, and E -vzp-0, so (3.21) implies that 29 30 (3.22) qq + 1/p = 0, and consequently, (3-23) ‘77 x I = q(q¢v¢ + qu 2L"). It follows immediately from this equation that 5 x I = 0 if q depends on pressure only, since then q¢ = qw = 0. On the other hand, suppose E x I = 5 (q # 0) throughout a region of space. For the coordinate system under consider— ation, V <1) and v (0 are not zero, and so \7 x I = 0 implies one of the following two possibilities: 32 (a) V¢=- Vgl/ or (b) q¢=qw=0. Case (a) implies that the surfaces on which ¢ is constant, and the surfaces on which w is constant coincide, which contradicts the known independence of o and w. There— fore, q¢ = q -= 0, and q depends on pressure only. In 2 summary we state Theorem 3.24: If constant pressure surfaces are not stream surfaces of a flow, the flow is Beltrami if and only if the magnitude of the velocity vector depends only on the pres- sure . 3.3 Prim-Smith Flows Proceeding to flows in which streamlines and vortex lines do not coincide but form Lamb surfaces, we observe a special property of the velocity magnitudes, which we now state. 31 Theorem 3.31: If constant pressure surfaces are not stream surfaces, a compressible fluid flow contains Lamb surfaces if and only if q, the velocity magnitude, is a function of pressure and entropy only. Proof: We recall that Lamb surfaces are stream surfaces on which q, the entropy, is constant provided E - vrp # 0. Therefore, we may replace ¢ by n in equation (3.23) and obtain (3.32) ExI=q(qnvq+qd/Vz,l/). By Theorem 2.31, if pressure is not constant on streamlines, I - vrn = 0 whenever Lamb surfaces occur in a flow, and therefore, forming the scalar product of (3.32) with I yields (3.33) o = q q¢I ~ :72). Since q # 0, equation (3.33) implies either = 0 or qw I ° V'w = 0. The latter equation can not hold, however, since I°Vzp=0, E°vn=0, I-vq=0,'and v ° Vgl/ = 0 imply that (E x I) x Vrn = (E x I) x V'w =‘O. Hence, v q and Vz/J are collinear, which contradicts the independence of n and w. Therefore, constant represents any family of stream surfaces distinct from the family of Lamb surfaces, and thus in a flow con- taining Lamb surfaces, q = q(p. n)- 32 Conversely, let us consider a flow in which q = q(p, n), and the constant pressure surfaces are not stream surfaces. Then, by a method analogous to the one for de- riving equation (3.23) we obtain ExI=qqnvq. Hence, I ° V72 = 0, and according to Theorem 2.31, the E x I congruence is normal. In other words, the flow is of Prim-Smith type. 3.4 A Classification of Steady Compressible Flows In the table below, we have attempted to picture a classification of all flows of the type mentioned in our introduction for which E ° <7p # 0. The left portion of the interior of the table represents flows for which E x I = 5, the upper part for irrotational flows and the lower part for flows in which streamlines and vortex lines coincide. The right side of the table represents flows for which E x I # 0, the upper portion for flows in which streamlines and vortex lines form surfaces and the lower part for all others. We have also indicated the restric- tions on the magnitude of the velocity in these cases. E x I = 5 E x I # O I = U streamlines and vortex _ lines form surfaces q — q(p) _ q = q(p.S) streamlines and vortex . lines coincide others q =q(p) q =q(p.w.S) (E'VP#0) CHAPTER IV A REFORMULATION OF THE DYNAMICAL EQUATIONS 4.1 Preliminaries A transformation of variables is frequently helpful in putting a differential equation or system of equations into a more desirable form. In the area of fluid mechanics, the hodograph transformation is an example of this. We illustrate it in the case of compressible, irrotational, plane flow. First, we introduce two functions, ¢(x,y) and ¢(x,y), where w(x,y) is a constant on each streamline, and ¢(x,y) is constant on each orthogonal trajectory of the streamlines. This can be done in the case of plane flow, and under the assumption of an irrotational flow, q(x,y) may be considered as a potential function, so V7¢ = E. A function, such as ¢(x,y), which is constant on each streamline, is called a stream function. Next, we let G be the angle measured counterclockwise from a positive x-axis to the velocity vector, and as usual, q = [51. The other quantities p and c, which appear below, are de- pendent on e and q. The function p, again, represents density, and c is the sound speed defined by the equation c2 = EE (it is a fundamental property of all actual media that, entropy remaining constant, the pressure increases 33 34 with increasing density). It can be shown [23, Chap. 4] that a compreséible, irrotational, plane flow can then be characterized by the following pair of equations, where the intrinsic variables w and ¢ are used as independent variables rather than x and y. 22-23- 29.: anw( 1)acp 0 (4.11) Interchanging the roles of the variables q, 9 and w. w by using 33-- 1.5.2. (p D59 56 = 1_5- 574? D q Biz-1.312 a) Daq (D $9 where D = q , we get from (4.11) ¢q $9 . 22-93__ 232: anq (C2 1)59 O (4.12) 84> a - f3 55' - q q - 0 35 These are known as the hodograph equations. We note that equations (4.12) are linear while equations (4.11) are not. Furthermore, we can, if we wish, eliminate either ¢ or w and get a second order linear equation for a single variable. For applications of these equations see R. Courant and K. O. Friedrichs [4, p. 248-259]. A second type of transformation of variable occurring in the field of fluid mechanics is mentioned by R. von Mises [20, p. 433] and employed by M. H. Martin [9, p. 465-484] in the case of plane flow. This change of variable amounts to using the pressure, p, and a stream function (intrinsic quantities) as independent variables, and considering the other variables and the coordinates x, y as unknown func— tions of them. In other words, under the assumption that pressure is not constant on streamlines, the streamlines and isobars (curves on which pressure is constant) are taken as curvilinear coordinates rather than the streamlines and their orthogonal trajectories as in equations (4.11). Using this approach, M. H. Martin started with the system of dif- ferential equations introduced a stream function w, and considered p as a 36 known functions of p and w. He then reduced the problem of finding u, v, x and y as functions of p and ¢ to the integration of one quasi—linear partial differential equation for a single unknown function. It is our purpose in this chapter to extend this tech- nique, which was fruitful for M. H. Martin in the case of plane flow, to a similar technique for three dimensional flows. As in Martin's approach, the streamlines will be coordinate curves. The other coordinate curves will con— sist of two distinct families of curves on which the pres- sure is constant. Such coordinate curves are realized by introducing a family of constant pressure surfaces which are not stream surfaces and two distinct families of stream surfaces as coordinate surfaces. Using this coordinate sys— tem, we shall reformulate the dynamical equations, (1.11) to (1.13), and obtain an equivalent system. 4.2 Reformulation — Step One We shall assume, now and throughout the remainder of this paper, that a separable equation of state holds (see Section 2.1) and rewrite the dynamical equations replacing n with S in the last equation. 37 The quantities vi (1 = 1,2,3), p, and S are considered as the five unknowns of this system, and it is assumed that P(p) is known. Hence, p is determined from the equation of state. Since vi = qti and IE] = 1, we may also con- sider the five dependent variables as q, p, S, and two components of E. We commence our reformulation of system (A) with its first equation (the continuity equation), by substituting qtl for v1 and obtain i _ Vipqt ‘ 0' Expanding this yields i i _ quit + t Vipq _' 00 Since V. in .. L 1 pg. — Vipql pq dividing by pq results in i i _ Vit +t Vi ln pq—O. Substituting from the separable equation of state for p gives Vitl + t1 Vi ln PSq = 0. Using this equation and the equation of motion in the form of (2.22), with p replaced by PS, system (A) becomes the equivalent system 38 (4.21) Viti + tivi ln qu = o I (4.22) PquKnj + PS[thi(q2/2)]tj = -Vjp, j = 1.2.3} (B) (4.23) viViS =0. 4.3 Reformulation - Step Two We avail ourselves of the coordinate system mentioned in the last paragraph of Section 4.1, letting X1 = p, the pressure, X2 = w, and X3 = 4, where w is constant on each member of one family of stream surfaces and ¢ is con- stant on each member of another (distinct) family of stream surfaces. In this coordinate system, the streamlines are coordinate curves along which w and ¢ are constants and p varies. Furthermore, we let Ni denote a unit normal to the family of stream surfaces on which ¢ is constant and let 'n1 denote a vector cross product of t1 and N1 such that t1, 'nl, and N1 form a right hand orthogonal system (see Figure 1). ¢ = constant Figure 1. 39 From equation (4.22) we obtain the components of the pressure gradient in the directions of these three orthogonal vectors. Since tjn. = 0 and tjtj = 1, the dot product 3 of (4.22) with t3 yields (4.31) t3 Vjp = - PStl Vi(q2/2) . Making use of the Frenet formula Knj = tl<7it. and the fact that 'n3 and N3 are orthogonal to t], the scalar product of (4.22) with 'nJ and NJ produces the follow- ing two equations. j In vp - 2 I 3 i j PSq n t Vit. 3 j - 2 j i N V' - - PS N t ltn. 310 q v1 3 According to the energy equation, th7iS = 0. Therefore, thi(Sq2/2) = (q2/2>tlvis + Stl vi, and (4.31) may be written as j : _ i 2 t Vjp P t Vi(Sq /2). Consequently, system (B) may be written in the following form. 4O (4.32) viti + tivi ln PSq = 0 E (4.33) tjvjp = - PtiVi(Sq2/2) (4.34) 'njvjp = - PSq2 'nj tiVitj F (c) (4.35) Nj vjp = - PSqZthivitj (4.3s) tiViS = o - 1 J commonly called the geodesic and normal curvature, respec- It is true that the terms 'njtlvitj and thl<7.t. are tively, and given abbreviated notations. For future compu- tational purposes, however, it is more convenient to leave them in their present form, so we do it. 4.4 Reformulation - Step Three Upon examining system (C), we observe the term qu appearing in three of the five equations. The first equa- tion of this system may also be written in terms of qu, and then (4.32) to (4.35) may be considered as a system of four equations with four dependent variables, p, Sq2, and ti. We now rewrite (4.32). Multiplying (4.32) by two and using a property of log- arithms, we get 2 Vitl + t1 Vi ln P282q2 = 0. Using another property of the logarithm function and the fact that tl‘Vi ln S = 0, this equation may be written in 41 the form 2vitl + thi ln 132qu = 0. Consequently, by replacing Sq2 with u, system (C) may be written as follows. 2 Vjtj + tj Vj ln P2u = O - tj Vjp = - PtiVi(u/2) 'nj vjp = - Pu ’nj tivitj D (D) Nj vjp = - PuthiVitj tj‘7jS = 0 A 4.5 Reformulation - Step Four We now observe a few facts which allow us to simplify system (D). In the coordinate system we are using, X1 = p, and p is independent of X2 and X3. Therefore, 1 j = 1 ‘7jp = 0 j = 2,3 Consequently, tJVjP = t1 . 'nJVjP = .nl’ vajp = N1. The unit vector t1 is tangent to the streamlines, along which only X1 = p varies, and hence t2 = t3 = 0. As a result, such expressions as t1 Vitj and t1 ViS, in 42 system (D), become tlvltj and t1*v1s. In the light of these remarks, the coordinate system we are considering permits us to write system (D) (and hence the original dynamical equations)in the form (4.51) 2V1t1 + t1V1 ln qu = o 'l (4.52) t1 = — Pt1V1(u/2) (4.53) 'n1 = — Pu 'nj tlvltj ? (E) (4.54) N1 = - Puthl Vltj (4.55) t1 was = 0 _. 4.6 Reformulation - Step Five - Introduction of the Metric Coefficients and Final Form of (4.51) and (4.52) Proceeding with our reformulation we let 51, E2, and E3 denote a set of base vectors for the X1, X2, X3 coor- dinate system chosen such that 5i is tangent to the curve on which Xi varies. Then the vector E may be written in the form E = tlél + t252 + t3é3. Since t2 = t3 = 0, E = tlélo But, E is a unit vector, and therefore, - 2 t ° t = 1 = (t1) e1 ° e1. LEtting 51 ° el = 911, 43 t1 = 1/V911 - We introduce the reciprocal base vectors 51 (i = 1,2,3) defined by El ° éj = Since N is a unit normal vector to the family of surfaces 6;, where 6; is the Kronecker delta. x3 = ¢ = constant, N1 = N2 = 0 and, N . N = 1 = (N3)2 53 - 53. Denoting e3 ° 53 by g33, we have N3 = l/Vgaa a From differential geometry one knows that g11, as we have defined it, is an element of the metric tensor and that g33 is the reduced cofactor of another member of the metric tensor, namely g33. Hence, we shall denote by gij (i,j 1,2,3) the elements of the space metric tensor and by g13 the reduced cofactor of gij' Since tj = gjitl and t2 = t3 = 0, we see that Similarly N3 = g31 Ni implies that j3 N3 - c—i—o .[533 We have defined 'n1 as the vector product of N1 crossed with t1, and consequently [2, p. 145], where g is the determinant of the gij' Substituting the expressions for Nj and tk in terms of the elements of the metric tensor we obtain "912 911 'n1 = -—-——-—- , 'n2 - , and 'n3 = 0. #9911933 ‘x/ 99119” Prior to substituting into system (D), we collect the formulas just derived in one place for ready reference and state a formula for the divergence of a vector v1 [2, p. 171]. _ g t3 -( 1 . 0, 0) t : __l_1_ V911 J V911 . 33 (461)4N-=(0,0. 1) N3=—9——- ~ 3 @255- J33"! I J - 1 ( n - ———————- -912, 911: 0) t— '“99119 (4.62) V.vj 1— ibrg VJ) 3 G 8x3 Using (4.62), the components of t3 given in (4.61), and the fact that ln P2u is a scalar, equation (4.51) may be written as a + 1 a 1n P2u _ 0 5x1 911 J— 5X1 - ° 911 We 45 Multiplying by V911 and recalling that X1 = p, this equa- tion becomes 5 ln P2ug = O. 5E; 911 Substituting for the component t1 in (4.52) and realizing that we obtain H _ P _ 5 (E). V911 V911 ‘85'2 Multiplying by 2 #911 yields 2 + p 3%. = o. 4.7 Reformulation - Step Six - Another Form of (4.53) and (4:54ZTH" In equations (4.53) and (4.54), we have the expression t1V71tj appearing. To expand this we use the formula at. (4.71) V t. =——-1 -1“? t, k j BXk jk Z where Pfk is a Christoffel symbol of the second kind ex- pressed by Zi a .. a . a . (4.72) Mk: 12.. .333 . 91:1 - 23.2: . 3 5x 5x3 5x1 Substituting into (4.53) for 'n3 and t1 as given in (4.61), we obtain “912 _ Pu ’912 911 ' '—"_"""" v1’51 + Vltz 49911953 ~19911333 4911 911 912 (4-73) 912 = Pu(\lg11 Vltz ‘ — Vltl) 911 As we shall now show, however, V71t1 = 0, and by means of (4.71) we have (4°74) 912 : Pu V911 (Eh _ Pg t2) 5X1 21 1,... To prove that Vltl = 0, we observe that tlvitjtj = 0. This implies that tjtlvitj = 0. Substituting from (4.61) for the components of E, we see that (1/g11)V71t1 = 0. Hence, V1t1 = 0. Expanding (4.54) in a manner analogous to that used in expanding (4.53) yields -Pu (4-75) 913 = "“— (923V1t2 + 933V1t3) V911 Since <71t1 = 0, (4.73) implies that 912 PW911 <71t2 = Using this expression for :71t2 in (4.75), we obtain 23 ‘9 912 Pu 33 911 ,fl§;; 47 Multiplying by gll/g33 and employing (4.71) gives 13 23 9119 + 9 912 at3 (4.76) = - Pub/911 —— - I‘fl tE Since g13 is the reduced cofactor of gij’ 913 = 921923 " 922931 . 923 = 921931 — 911932 9 9 2 911922 ' (912) gas = 9 and . 2 911913 + 923912 931[(921) ' 911922] 2 = — gal. 2 933 911922 ' (912) Therefore, (4.76) may be written — w— ata 2 (4'77) 931 — P 911 __ _ F 1 t OX1 3 K 4.8 Reformulation — Step Seven - Final Form of (4.53) and (4.54) In the last section, (4.53) and (4.54) of system (E) became equations (4.74) and (4.77), respectively. We write these two equations in the form at. : I l __J. _ ’e ' : glj Pu 911 8X1 le tg I j 213' and put them entirely in terms of u, P, and the elements of the metric tensor. To accomplish this, we first substitute 48 the expressions for tj (and t2) given in (4.61). This yields 9- g glj - Pu V911 ap(-J—1'_ - PEI wil- , j " 2,3 V911 4911 Solving for the term involving the Christoffel symbols, we obtain E _ 5 glj glj P. 9 ‘ 911 "‘ ‘ 1 1 5 P 3 E ‘ P (a: u ' .8 1 6911 Since P.1 921 =-§ . [5, p. 98], we wee that (4.53) and (4.54) of system (E) may finally be written 5911 q - g . j = 2[“911 %E(—ll-_ -§%l]l j: 2:3- BX 4911 4.9 Reformulation - Final Step Combining the results we have obtained following system of equations (E) and recognizing that (4.55) simply implies that 8 depends on X2 and X3 only, we see that from the dynamical equations we have derived the following system of equations. 49 (4.91) g—(Bfllfl) = o E P 911 (4.92) 2 + p 2; = 0 5911 912 912 (4.93) 2 = 2['~/911 33(J—) " Fl:— p (F) 5X 911 5911 ' a 913 913 (4-94) 8X3 = 2[”911 hp (F " 131—1— 911 (4.95) s = s(x2, x3) 3. On the other hand, each step in deriving (F) from (A) is reversible. So, suppose we introduce into Euclidean three space two independent families of surfaces and let the curves in which they intersect be curves along which p varies. One of the families of surfaces can be taken as a family such that on each member X2 is constant and the other as one such that on each member X3 is constant. Then, if system (F) has a solution consisting of u = f(p, X2, X3), S = h(X2, X3) and x, y, and 2 as func- tions of p, X2, and X3, in a region where the Jacobian is not zero, we may obtain a solution of system (A) consisting of q, p, S and ti as functions of x, y, and 2. It should, perhaps, be remarked that equations (4.93) and (4.94) of system (F) can be put in a more compact form provided g12 and 913 are not zero. Factor 912 and g13 from the brackets, replace 1/P by -(1/2)up (using equation 4.92), and introduce logarithms. Then these 50 equations become 89 ug2 (4.96) 11 = (312 %— ln( 12) 5X2 P 911 8911 a u9§3 (4.97) 5X3 - 913 SP- ln( 911 In the next two chapters, however, we analyze situations in which 912 = 0, and for this reason, leave (F) in its present form. CHAPTER'V TWO CLASSES OF PLANE FLOWS 5.1 A Coordinate System for Plane Flows In this chapter we obtain two classes of plane flows using syStem (F). So, we consider the streamlines as lying in planes parallel to an xy—coordinate plane. X232 X3 = S = constant A / x / Figure 2 (- 3: 3 p = constant Under these conditions, we may choose the family of stream surfaces composed of these planes as a family of coordinate surfaces. We do this and let X2 = 2 be a con— stant on each of these coordinate planes. As a second family of coordinate surfaces, we choose the stream surfaces that are cylinders, each of which has a streamline for directrix and a line parallel to the z axis as generator (see Figure 2). These are surfaces on which .X3 is constant. But these cylindrical stream surfaces are Lamb surfaces 51 52 because the vortex vector is perpendicular to the xy-plane in flows such as we are considering, and consequently, the vortex lines are generators of these surfaces on which X3 is constant. By Corollary 2.32, the function S that ap- pears in the separable equation of state is constant on each of these Lamb surfaces. Hence, S may be taken as the X3 variable. To complete our coordinate system, we take, as the third coordinate family, the family of cylin- ders each of which has an isobar in the xy-plane as directrix and a line parallel to the z axis as generator. This will complete our coordinate system as long as pressure is not constant on streamlines. 5.2 The Metric Tensor The rectangular Cartesian coordinate system with x, y, z coordinates and the coordinate system of Section 5.1 are related by the transformation x = x (p,S) y = y (p,S) z = X2, where we assume that the Jacobian J = Xp yS — XS yp # 0. Since 9.: 5X.-5-’-‘-.—+§1.§1.—+-5—?-.—§-z—. 13 5x1 5x3 5x1 5x3 5x1 5x3 I the metric tensor is as follows: 53 x2 + 2 0 x x + 9 Y9 p S ypys (gij) = O 1 O 2 2 xpxS + ypys 0 XS + ys Furthermore, the determinant of this matrix is _ _ 2 and consequently, -(X x + Y Y ) X2 + y2 913 = p S p 82 and 933 = pp p 2 . (XpyS - ypxs) (Xpys — prS) 5.3 Equations (F) in the Case of Plane Flow From the metric tensor of the previous section, we observe that 921 = 0 and 911 is independent of X2, so that equation (4.93) of equations (F) is identically satisfied. We assume g13 # 0, replace (4.94) by (4.97) and solve for the expression Sg-ln.(§-) as follows: 11 5g 1 11 a 3 5 2 = _ l 913 BS 59 ln(911) + 59 n 913 u a 1 5911 8913 ln (———) = -——— _ 2 55 911 913 55 5P Next, we substitute the metric tensor components and simplify. This yields 54 5 u _ 1 5 x2+y2) 6 (x x +y y ) . SPln In equation (5.33) we substitute for u from equation (5.41) and get a X2 +y2 Pf2 (S) 573(L5—E : -2. yp Therefore, 57 Pa x2 +y2 : _2 55 y; f2(S) But 2+2 X2 5 32.312 _L _2 _._5_ 35( 2 )‘ap(2+1)‘ap( yp yp @“NHE. ) , so that the previous equation becomes P35 0/ A k:}Ux N N V II HI I A mm In terms of the functions q(p), 5(p) and y(S), this is g, Lawpnz _._ -2 p .IB'(p)]2v2(S) f2(3) P Separating variables we obtain Pgi_ [a'figflz = 412(5) =R d I 2 2 P [a (p)] f (S) where R is a separation constant that must be negative. From the last equation, we obtain two equations which must be satisfied by q(p), fi(p) and y(S), namely, 2 L m = _L (5'45) dp (B' (p)) P(p) (R < 0) (5.46) 1%) = - %f2(S) Thus we have four equations, (5.43) to (5.46), im- posing restrictions on q(p), B(p), y(S), and the functions of integration, f(S) and h(S). We first determine q(p) 58 and 5(p) from (5.43) and (5.45) and then, determine y(S) from (5.44) and (5.46). From (5.43) we see that Q'(E) = k 5' (P) P(P) (33(9) Substituting this into (5.45) yields a differential equa- tion for 5(p) alone, that is, k2 g... 1 2 = R dp P(p) (Hp) P(p) Integration yields 2 (5.47) (3%» = 2 k . P (p)[R 1(9) + C1] where I(p) is an integral of 1/P(p) and C1 is a con— stant of integration chosen such that RI(p) + C1 > 0 in the region of flow being considered. We now proceed to find q(p). From (5.43), we see that . _._ ks'm) 0‘ (P) P(p) (up) Logarithmic differentiation of (5.47) followed by substitu- tion in this last equation for fi'(p)/fi(p) gives u : __:E__ u R a (p) 2P2(P) [2]? (p) + RI(p) + C1] and integration yields k k_R <19 2 (5-48) u(p) 3 - P2(P)[RI(P) + C1] + c2 , where C2 is a constant of integration. 59 Now, if we can choose the arbitrary functions f(S) and h(S) and the constants k, R such that equations (5.44) and (5.46) can both be satisfied, we will have a solu— tion to the system of equations (5.31) to (5.33). Taking the logarithmic derivathnaof (5.46) yields S_f'S y S f S Substituting from (5.44) and multiplying by k f(S), we have (5.49) h(S) =k f'(S). Consequently, choosing f(S) and k determines h(S). In summary, we may obtain a plane flow satisfying equations (5.31) to (5.33) by proceeding according to the following steps. (1) Pick an arbitrary function f(S) and two constants k and R (R < 0). Then y(S) is determined by (5.46). In order for the Jacobian (see Section 5.2) to be different from zero, f(S) must not be a constant function. (2) Choose constants C1 and C2, with C1 such that RI(p) + C1 > 0. Then, determine u(p) from (5.48) and B(p) from (5.47). Care must be ex- ercised in choosing C1. and R (of first step) so that B'(p) is not zero. (3) Finally, u(p,S) is determined from (5.41). 6O vIt is of interest to note that in these flows a con- stant value of p implies a constant value of x. Hence, the surfaces of constant pressure are parallel planes, each of which is parallel to the yz-plane. 5.5 A Second Class of Plane Flows Obtained byfa Separation of Variable Technique Returning to equations (5.31)tx>(5.33) we attempt to find another solution, namely one of the form x = q(p), y = 5(p) + y(S). The function u is again obtained from (5.31) and is given by 2 2 2 f (S) (xp + yp) 2 Y9 u: where f is an arbitrary function of S. As in the pre— vious section, we can start with equation (5.32) and obtain (5.42) which reads P (p) f(S) xpyS yp = h(S), where h(S) is an arbitrary function. Under the present assumption that y(p,S) = fi(p) + y(S), this equation leads to (5.51) P(p) f(S) CI. (p) Y. (S) = ES‘(p) “S” Separating variables yields 61 P(p) a'(p) : h(S) = A (5'(P) f(S) 7'(S) ' where A is a separation constant. From this, we have the following two equations. (5.52) 15'(p) = P0?) XWPL . = h(S) (5.53) V (S) Af(S) In precisely the same way as in the previous section, we begin with (5.33) and find that a Xz 2 P (p) -E = — BE y; f2 (S) But under the present assumptions, when we substitute for x and y , we obtain P P ~_2 .d_ ELLE). : _ 2 : P(p) dp [5. (9)] 132(5) D, where D is a negative separation constant. This requires that (5.54) f2(S) = - 2_ D and (5.55) P(p) dp [:g'(p)2 = D. Consequently, we see that to obtain a flow where X(P.S) and y(p,S) are of the assumed form, the various constants and functions involved must satisfy equations (5.52) to 62 (5.55) with u being given by the same equation, namely (5.41), as in the last section. Although P(p) was as- sumed to be a given function, we notice that (5.52) and (5.55) place a restriction on P(p) regardless of the form of u(p) and 5(p). We substitute from (5.52) into (5.55) and simplify as follows: 2 d A _ P(p) dp [P(p)] - D 2A2 _ = D + E I P (p) p where E is a constant of integration. Therefore, P(p) must be of the form 2A2 P(p) Dp + E ’ where A, D and E are constants with D < O. This in- cludes the equation of state used by Chaplygin, Karman, and Tsien [20, p. 278]. With this restriction on the form of P(p), however, we can proceed to obtain a solution of the dynamical equations characterizing plane flow by the fol— lowing procedure. (1) Pick constants A, D and E arbitrarily with D<0. (2) 63 Choose the functions u(p) and h(S) arbitrarily (not constant). Determine f(S), to within a plus or minus sign from (5.54). Determine B(p) and y(S) by integration of (5.52) and (5.53), respectively. CHAPTER VI A CLASS OF THREE DIMENSIONAL FLOWS 6.1 Introductory_Remarks In Chapter V one of the families of coordinate surfaces that was introduced was a family of cylinders each member of which has a streamline as a directrix. This family of surfaces was a family of Lamb surfaces, and furthermore, each streamline was a geodesic on one of its members. We would now like to generalize what was done in the preceding chapter by seeking a three dimensional (rather than two dimensional) flow in which streamlines are geodesics on Lamb surfaces. Since S is constant on the Lamb surfaces of a Prim- Smith flow (see Corollary 2.32), we may take it to be an independent variable, say X3. If we again let X1 vary along streamlines and take p = X1 and w = X2, the as— sumption that streamlines are geodesics on Lamb surfaces implies that g21 = 0. Thisfollows from the fact that the members of the b congruence of curves lie on the constant pressure surfaces (equation (2.22) implies b ‘ <7p = 0) and also on the Lamb surfaces, if the streamlines are geodesics on these surfaces. The requirement that g21 = 0 allows us to omit (4.93) from equations (F). 64 65 6.2 The Metric Tensor In the consideration of plane flows in Chapter V, there was a general relationship among the variables X, y and p, S, which we used (Section 5.2). In this chapter we assume a particular relationship among the variables x, y, z (rectangular Cartesian coordinates) and p, ¢, S, namely one of the form (6-21) X = X(p. w). y = Y(p. 8). z = z(p. ¢). in which the Jacobian, J, is not zero. The metric tensor is then of the form 2 2 2 X + + z 0 p yp p ypyS .. = O x2 + 22 0 (913) w w 2 ypys 0 yS where zeros appear in the first row second column and second row first column due to the requirement that g21 = g12 = 0. Computing 912 and equating it to zero yields (6.22) x X + zpz = 0. We note that the determinant g is given by _ 2 2 2 (6.23) g — yS (x; + z¢)(xp + z ). Before introducing the metric tensor into system of equations (F), we observe some of the geometric implications of (6.21). If w is fixed, from (6.21) we have x = x(p) 66 and z = z(p), which may be considered as the parametric equations of a curve in the xz-plane. So, surfaces on which w is constant are cylinders with generators parallel to the y axis. Similarly, surfaces on which p is con- stant are cylinders with generators parallel to the y axis. 6.3 Introduction of the Metric Tensor into System of Equations (F) As in Chapter V we seek a solution to system of equa- tions (F) for which g13 # O and replace equation (4.94) with (4.97). We substitute the expressions for the terms of the metric tensor into (4.97) and get 2 2 5 (x2 + y2 + ZZ) a u yp ys T9 p p=YYg-ln S p S p x; + y; + 2; Taking the derivative with respect to S and dividing by ypyS yields 2 2 y _ 55'1“ 2 2 2 ° S X +y +z P P P The left side of this equation is 5 2 531“ ys’ and therefore integrating with respect to p and solving for u we obtain 67 F2(2,1/,S)(x2 + y; + 2;) (6.31) u = E . p Y where F2(¢,S) is an arbitrary positive valued function. We now proceed to substitute for the metric coefficients and u in (4.91) getting [P2(p) F2(¢,S) y§(x: + 2;)(x; + 22):] _P, _ 2 — 0. P '55 Y Simplification yields P2

yg (x; + z;>(x; + 2;) ll 0 (6.32) éTp 2 yp Shifting our attention to (4.92) and substituting for u from (6.31), we obtain ( ) ( ) 5 (M ‘ 2 6.33 p p = —— 35 y; F2(w.s) Thus, to find a three dimensional flow (of the form indicated by equations (6.21)) with the property that stream- lines are geodesics on Lamb surfaces is reduced to finding x, y, z and u in terms of p, w, and S such that (6.22) and (6.31) to (6.33) are satisfied. 6.4 An Application of a Separation of Variable Technique In accordance with a standard separation of variable technique, we assume there is a solution to the equations 68 just mentioned of the following form. X = a1(p) 51(w) (6-41) y = a2(p) 7(8) 2 = a1(p) 52(w) Substitution into (6.22) results in a11zai(p)[ai(p)12[<51<¢>>2+<5;(¢))21[5i(w)+5§(¢)1] '55 [aé(p)]272(S) which means that (6.43) d_ (P(p)dz(P)G1(P)Oi(P)) :0, dp 02(P) provided the various functions appearing are not constant. Finally, we must substitute into equation (6.33) from (6.41). This gives a [5;(p)12[5§(w) + 5§(¢)1 _ _ 2 P(p) -- - ---—-- a? [a5 O in the region of flow, and the function I(p) is again an integral of 1/P(p). Let us call this 71 function k(p). With this notation, from requirement (3) we see that (11(P)C12(p) = FTP-7337B)— I and hence, d1(P)02(P) is a known function of p. Let us denote it by h(p). So, we have ai(P) (6.51) , = k(p) O12(P) and (6.52) a1(p)a2(p) = h(p)- From these two equations we obtain a single differential equation with dependent variable o2. If this differential equation can be solved, then a1(p) may be obtained from (6.52), and we have a class of three dimensional flows. We proceed as follows. Solving (6.52) for a1(p) we get = 2121. a1(p) 02(p) Differentiation yields h' (9)62 (p) - h(P)O(2 (p) ai (p) = 03(P) Substituting this expression for ai(p) in (6.51) and multiplying by a§(p)aé(p) yields h' (pm (p) - h(pm; (p) = k(p)a§(p)aé(p).~ 72 Collecting terms we find that (h(p) + k(p)a§(p))aé(p) - h'(p)a2(p) = o. If we multiply by 2a2(p), this equation becomes 2 [h(p)-+ k(p)6§(p)1 §ElQ2(p’] - 2h'(p)6§(p) = 0. Letting Y(p) = a§(p) and solving for dY/dp results in gx_= 2h'(p)Y (6'53) dp h(p) + k(p)Y In order to put this in a standard form discussed by E. Kamke [7, p. 24], we make the substitution l/r = h(p) + k(p)Y. Differentiation of this yields - —— = h'(p) + k'(p)Y + k(p)Y'(p) Hence, 1 dr k' Y + ‘—— . (9) 7r dp Y'(P) = "f(S) [h We substitute into (6.53), change the dependent variable from Y to r, and get —® h'(p) +11?!) (%-h(p))+ +;2 a-p]= %L(--h(P)). Solving for r'/r2 we obtain 6 — 76,) [6.156 6.6)]. 2.6.6). Multiplying by r2 we have 73 dr (6-54) 55': f1(p)r + f2(p)r2 + f3(p)r3, where f1(p), f2(p) and f3(p) are known functions of p with f1(p) = - E'p f2(p) = h£%g§'fipl- 3h'(p) f3(p) = 2h(p)h'(p)- Methods for solving (6.54) for various forms of the coefficients fi (i = 1,2,3) are given on page 25 of E. Kamke. In conclusion, the problem of finding a solution to system (F) where x, y, and z are of the form (6.41), and for which streamlines are geodesics on Lamb surfaces has been reduced to the problem of solving a single first order differential equation. 6.6 Some Remarks Concerning the Geometry of the Flows of This Chapter If the requirement (6.42) is satisfied by choosing 61(w) = cos w, 62(w) = sin ¢ and A = 1, as suggested in Step IV in the procedure for obtaining a flow, equations (6.41) become x = a1(p)cos¢ (6.61) - azlp)v(8) "< I z = a1(p)sin¢. 74 Let us consider the following three types of surfaces: (a) Lamb surfaces on which S is constant, (b) constant pres— sure surfaces, and (c) surfaces on which w = constant. (a) If S is constant, (6.61) is of the form X = a1COS¢ z = alsinw y = f(Ql) where f(al) is obtained as follows. Equations (6.43) and (6.44) yield a functional relation between a1 and a2 from which we can determine a2 in terms of a1. Sub- stituting this result in the equation y = a2(p)y(S) and holding S fixed yields y = f(a1)- Consequently, the surfaces on which S is constant (these are Lamb surfaces) are surfaces of revolution [5, p. 49]. (b) When p is held fixed, a1(p) is constant, say C1. Consequently, from (6.41) we obtain x = C1 cosw Z = C1 Sim; Y = C2 Y(S) where C2 is the value of a2 for some fixed p. Then x2 + 22 = Ci and the surfaces of constant pressure are coaxial right circular cylinders [5, p. 47]. (c) When w is constant, (6.61) implies that z KX [while y = a2(p) y(S)]. As a consequence, w - constant surfaces are planes. (See Figure 3.) S = constant ¢ = constant Figure 3. From (a) and (c) we observe that the streamlines (w = constant and S - constant) are meridian curves of surfaces of revolution, in fact, the Lamb surfaces. From (a) and (b) we conclude that the parallels of these Lamb surfaces are isobars. Furthermore, we note that the 2 vector of Corollary 2.43 is tangent to these parallels and is collinear at each point with the binormal vector of the streamline through this point. 10. 11. 12. 13. 14. 15. E. REFERENCES Beltrami, Considerazioni idrodinamiche, Rendiconti Instituto Lanbardo, 22(1889), 121-130. Coburn, Vector and Tensor Analysis, Macmillan, New York, 1955. , Intrinsic relations satisfied by the vor- P. ticity and velocity vectors in fluid flow theory, Mich. Math. J., 1, no. 2 (1952), 113-130. Courant, and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience, New York, 1948. P. Eisenhart, An Introduction to Differential Geometry, Princeton University Press, Princeton, 1947. Gilbarg, On the flow patterns common to certain classes of plane fluid flows, J. of Math. and Physics, 26(1947), 137-142. . Kamke, Differentialgleichungen Losungsmethoden und Lbsungen, Chelsea Publishing Co., New York, 1948. Lamb, Hydrodynamics, Dover, New York, 1945. . H. Martin, Steady rotational plane flow of a gas, Am. J. of Math., 72(1950), 465-484. . M. Milne-Thompson, Theoretical Hydrodynamics, Mac- millan, New York, 1957. F. Nemenyi, and R. C. Prim, On the steady Beltrami flow of a perfect gas, Proc. VII, Intl. Cong. of App. Mech., 2, part 1(1948), 300-314. . C. Prim, On the uniqueness of flows with given streamlines, J. of Math. and Physics, 28(1949), 50-53. C. Prim, Steady rotational flow of ideal gases, J. of Rational Mech. and Anal., 1 (1952), 425-497. Serrin, Mathematical principles of classical fluid mechanics, Handbuch der Physik , Bd. VIII/1(1959). 125-263, Springer-Verlag, Berlin. Smith, Some intrinsic properties of spatial gas flows, J. of Math. and Mech., 12, no. 1 (1963), 27-32. 76 16. 17. 18. 19. 20. 21. 22. 23. 24. E. 77 R. Suryanarayan, Intrinsic equations of rotational gas flows, Israel J. of Math., 5(1967), 118-126. , On the geometry of the steady, complex- lamellar gas flows, J. of Math. and Mech., 13(1964), 163-170. Truesdell, The Kinematics of Vorticity, Indiana U. Press, Bloomington, 1954. , Intrinsic equations of spatial gas flow, Zeitschrift ffir Angewandte Math. Mech., 40(1960) 9-14. von Mises, Mathematical Theory of Compressible Fluid Flow, Academic Press, New York, 1958. . H. Wasserman, A class of three dimensional compres- sible fluid flows, J. of Math. Anal. and App., 5 (1962), 119-135. , Helical fluid flows, Quart. of App. Math., 17(1960),443—445. , Fluid Mechanics, Unpublished notes. E. Weatherburn, Differential Geometpy of Three Dimensions, Cambridge U. Press, Cambridge, 1955.