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This is to certify that the

thesis entitled

A THEORETICAL INVESTI'EATION OF ANNULAR NAGI‘JETO-
HYDPODYNANIC FLOW M Im A NOVING BOI'NWARY
presented by

DENT"? IS C. KUZNA

has been accepted towards fulfillment
of the requirements for

DOCTOR OF PHILOSOPHY degree in APPLIED mam cs

‘ WALWH"

Major vatofessor

Date "AL/(II
/ /

0-169

LIBRARY

Michigan State
University

 

Ԥ

 

ABSTRNCT

A THEORETICAL IIVESTIGATIOI OF
AIIULAR.IAGIETOHYDRODYIAHIC FLOR
‘UITH.A HOVIIG BOUNDARY

by Dennis C. Kuena

The problea.considered is the laainar, steady flow of a viscous,
incoapressible, conducting fluid in the annular space between two in-
finitely long circular cylinders under the action of a radially iapressed
aagnetic field and an axially iapressed electric field when the outer
cylinder is given a unifora angular velocity. The conditions of the
problea.reduce the aagnetohydrodynanic equations to three equations in
pressure, velocity, and aagnetic field. One equation gives the pressure
variation in the radial direction and the other two equations are coupled
equations for the velocity and the aagnetic field. These three equa-
tions are functions of one variable, and may be solved in closed fern.

In the liaiting case‘vhere the radii becoae infinite but their differé
ence reaains finite, and there is no velocity of the outer cylinder, the
solution becoaes Hart-ann's flow between infinite parallel plates with a

transverse nagnetic field and a unifora applied electric field.

'A mum. marrow or
m mmmc m
urn! A 1:0an mm

By

C5

Dennis CB‘AKuzna

A THESIS

Sub-itted to
iiichigan State University
in partial fulfillnent of the requireaents
for the degree of

Dacron OF mnemmv
Depart-eat of Applied Mechanics

1961

ms

The author wishes to thank Dr. J . E. Lu for his help and guidance '
in the preparation of this thesis.

Acimowledgeaent is also given to Dr. K. Brenkert, Jr. for his help
in the choice of a problen and to Dr. C. 0. Harris for his help and
encouragenent throughout the entire course of study.

This research was aade possible by two Cooperative Fellowships iron
the National Science Foundation.

VITA
Dennis C. Rum
candidate for .the degree of

Doctor of Phi losopty

Dissertation: A Theoretical Investigation of Annular Magneto-
to'drodynuic Plan with a loving Boundary

Outline of Studies:

Major Subject: Applied Hechanics
iiinor Subjects: Hatheaatics, Electrical Engineering

Biographical Itens:

Born: June 27, 193?, River Rouge, iiichigan

UndergShracSiauate Studies: Hichigan State University, -
l9 -

Graduate Studies: Hichigan State University, 1958-61

Experience: Instructor in Applied Mechanics, iiichigan State
University, 1958-59

leaber of Pi Tau Sign: and Tau Beta Pi

-iii-

LISTOFFIGJRES......

LISTOFSYHBXS......

WIGH.......

TAM OF CONTENTS

REVIEVOFHREVIOUSRESEARCH. . . . . . . .

EWHWSeeeeeeeeo

flWIOU.........

LEITIIGMSOFTHEFLOJ

Olga“ O I O O O O O 0

LIST OF REWBS

- iv.

11
23

31

LIST OF‘FIGURES

FIGURE ' EDGE
l. AnnularWChannel . . . . . . . . . . . . . . . . . . . . . . 2
2. Method of Obtaining the Magnetic Field . . . . . . . . . . h
3. velocity Profiles for a - 2 . . . . . . . . . . . . . . . . 1h
h.VelocityProfilesforn-h................ 15
5. Velocity Profiles for n.- 12 . . . . . . . . . . . . . . . l6
6. velocity Profiles for vb - 0.h . . . . . . . . . . . . . . l7
7. Velocity Profiles for v - 0.2 . . . . . . . . . . . . . . 18
8. velocity Profiles for v - 0.0 . . . . . . . . . ... . . . l9
9. velocity Profiles for v - - 0.2 . . . . . . . . . . . . . 20
10. velocity Profiles for vb - - 0.h . . . . . . . . . . . . . 21
ll. Rectangular Channel for Hortnann's Plow . . . . . . . . . . 25

12. velocity Profiles for an Inviscid Fluid . . . . . . . . . . 27

‘6’

13. Velocity Profiles for no Electric Field . . . . . . . . . .

L 0

MI?!

1‘

5' :=L¢;n

a
"I
b

9 E 9 E

:I:

t..
L033

'1

”< <L 0 ML 0L w

s Vb: Vi

' 2

LIST OF’SNIBOLS

radius of inner cylinder

radius of outer cylinder

ratio of radii of outer and inner cylinders
electric field vector ' ‘
coaponents of electric field

applied electric field

aagnetic field vector

diaensionless aagnetic field

coaponents of aagnetic field

applied aagnetic field

current density vector

coaponents of current density

distance froa center of channel to plate
‘Hartnann.nuaber

cylindrical Hart-ann nuaber

pressure

dimensionless pressure

Iagnetic Prandtl nuaber

radius

diaens ionless radius

position vector

area vector

tiae

velocity vector

coaponents of velocity

dinensionless velocity
strength of aagnetic source
distance froa aidchannel
axial coordinate

aagnetic per-eability
kinenatic viscosity
density

electrical conductivity
angular coordinate

angular velocity

-vii—

WHO]!

When an electrically conducting fluid aoves in the presence of a
magnetic field, electric currents are induced in the flow. Electric
fields applied to the flow also produce electric currents. These elec-
tric currents interact with the aagnetic field and produce aechanical
forces which aodify the flow. The friction forces due to the viscosity
of the fluid also Iodify the flow.

Although the equations which describe these phenoaena are very coa-
plicated nonlinear partial differential equations, soae special problels
m be solved in closed fora. The flow of an electrically conducting
fluid in an ammlar chamel with loving boundaries an be solved in closed
fora in acne special cases. Very few probleas with aoving boundaries
have been considered in cylindrical coordinates, other than that of the
‘iapulsively accelerated flat plate with a transverse ngnetic field as
the liaiting case of the cylindrical problemuh The solution presented
here pertains to the case of an annular channel with a aoving boundary.

In this stucu, a viscous, incoapressible, conducting fluid is con-
sidered in an infinitely long annular channel of inner radius a and
outer radius b (Pig. 1). i aagnetic field is applied to the CWI
in the radial direction such that no - w/r, where w is a constant.

A unifon electric field £0 is applied to the channel in the axial

direction and the outer cylinder is given a unifora angular velocity on.

 

*Superscript iii-hers in parentheses refer to the List of
References.

 

 

 

 

 

on
'

\\\\\\\\\\\\\\“

Fig. l Annular Channel

-3-

The applied ngnetic field is a line source aagnetic field located
on the canon axis of the inner and outer cylinders. Although a line
source aagnetic. field does not exist in reality, it is possible to obtain
a good approxiaation to such a field by the use of a core of nterial
with high perneability within the annulus and a cylindrical shell of
nterial with high peaaeability outside the annulus. The flip: lines
could close through these per-sable paths at long distances fro. the
region of interest (Pig. 2). The source of the flux could be the core
itself, if the core is aade of a per-nantly ngnetised uterial. Since
the purpose of this study is the theoretical investigation of the flow
and not the experiaental probleas involved, the details of providing the
necessaq ngnetic field will not be discussed further.

I; there were no aagnetic field, this problea would be the ordinary
flow in the annular space between a rotating outer cylinder and a
stationary inner cylinder.(2) This case can be derived as a liaiting case
of the annular ngnetohydrodynaaic flow.

If there were no angular velocity, this problea would be a cylin-
drical analog of iiartaam's flow between infinite parallel plates with
a transverse ngnetic field and a unifora electric, field applied parallel
to the platen”) It will be shown that Hartaann's flow can be derived
as a liaiting case of the annular aagnetotvdrodynuic flow.

flux lines

cylindrical shell

COI‘C

 

Fig. 2 Hethod of Obtaining the Magnetic Field

x
REVISE 0P PREVIwS REM

Because of the great aaount of*work that has been done in aagne-
totvdrodynaaics, no atteapt will be ude to undertake a coaplete review
of previous research. This review will be confined to works which per-
tain in soae way to the problea presented here.

The results of the first theoretical investigation in the field
which is now known as aagnetolwdrodynaaics were published by
J. Hartaann‘ 3) in 1937. Hart-um considered the laainar flow of an
electrically conducting liquid in a hoaogeneous aagnetic field. This
investigation is considered the classic work.in.aagnetohydrodynaaics.

A cylindrical analog of Hart-Inn's flow has been considered by
S. Globe.(h) Globe considered the laainar, steady flow of an electric—
ally conducting, incoapressible fluid in the annular space between two
infinitely long circular cylinders under the action of a radially iapressed
aagnetic field and a constant longitudinal pressure drop.

I. G. CheknrevG) also considered the laainar, steacw flow of an
electrically conducting, incoapressible fluid in the annular space
between.two infinitely long circular cylinders under the action of a
radially iapressed magnetic field. <Chekaarev considered three such
probleas. The first problea concerned the flow under the action of a
constant longitudinal pressure drop and a radial injection of a liquid
‘with constant velocity at the surface of the inner cylinder. The second
problea.concerned the flow under the action.of a uniforn.axial electric
field and a radial injection of a liquid with constant velocity at the

surface of the inner cylinder. The third problem concerned the flow with

-s-

-o-

a uniform axial electric field and a constant longitudinal pressure
drop.

Few problems in cylindrical coordinates have been considered with
aoving boundaries. One of the few such problems that has been con-
(1)

sidered is thejwork of G. F. Carrier and H. P. Greenspan concerning
the flow past an impulsively accelerated flat plate with a transverse

magnetic filed as the liaiting case of the cylindrical problea.

muons

In order to determine which equations to use, some assuaptions aust
be rude m the nature of the fluid. It will be assumed that (l) the
fluid is incoapressible, (2) the free charge density and the displaceaent
current are negligible, (3) the permeability, conductivity, permittivity,
and viscosity are constant scalar quantities, and (it) the Lorentz force
is the only body force acting on the fluid.

Under these assimptions, the equations of nagnetohydrodynanics in
rationalized nks units are(6’ 7):

in- T (l)
vx?- ~|Aér§ . (2)
9.??- o. (3)
v.?- o (h)
T- “rum; _ l (s)
v.Tr‘- o (6)
.ggqvfivfi‘ - -15vpevv2v’eg3‘x'fi (7)

Since the physical aspects of the problem suggest the use of
cylindrical coordinates, these equations will be used in their coaponent
fora in cylindrical coordinates. These equations sinpl ify considerably
for the following reasons:

(1) Because of sy—etry around the axis, (a/ap) - 0.

-7.

-5-

(2) Because of steady flow, (B/Bt) - O.

(3) Because of the infinite dimension in the z direction and the
fact that none of the applied fields are functions of 2, 'fi‘ and 7?
cannot be functions of z.

(u) It is assumed that the applied field Ho - w/r fixes the
radial component of the magnetic field at r - a and r - b for all
values of 2.

(S) It is assumed that there is no flow in the 2' direction. This
may be accomplished by making (op/52) - O.

(6) An electric field can only arise from an applied voltage, a
time changing magnetic field, or free charges.

The above simplifications lead to the following results when
applied to the equations:

(3) Since 7;- .\;(r), equation (6) become:

d(rVr)
dr

 

- O (8)

When the conditions that Vf(a) - Vr(b) - O are applied, it is seen
that V ' 0.
r
.3 ...X
(b) Since H - H(r), it follows from equation (3) that

d(rHr)

- Q ‘
dr o L)

 

Applying the conditions that Hr(a) - u/a and Hr(b) - u/b, it is seen
that Hr - w/r. Thus, the radial component of the magnetic fitld is not
affected by the fluid flowing in the annulus.

(c) Since neither a time changing magnetic field nor free charges
exist in the flow, the electric field must be given by the applied field.

Thus, Er - E¢ - O and E2 - E0.

A

.y.-
_L

(d) Equations (1) and (5) may be combined to eliminate J. Then
_L A .L A. I
VxH . <*(E+u‘JxH) (10)

The radial component of this equation, including previous assumptions

is

v H - o (11}
93

But V¢ is not zero, so Hz - O.

..L
Combining equations (1) and (7) to eliminate J and introducing
the above simplifications, three scalar equations are obtained from

equation (10) and the combination of (l) and (7).

2. .
d v! avg? v d(rhé)
. 2 r dr 2 2 dr
dr t On?
I’SSEEQZ £2” r \
'E dr ’ 0 E0 - r ‘35 (13’

2
oV H d(r )

£32 . ...?— .. :1 ...:h. (11;)
dr r r dr

The boundary conditions for the above equations are:

V¢(8) ' 0 (15)
V¢(h) '- wb . (16)
H¢<a> - 0 i "(17)

The boundary conditions for Vfi are the no slip conditions at the walls.
The boundary condition for H¢ is obtained by requiring that no current

flow in the inner cylinder. Equation (1) in integral farm becomes

§.d§-fff.d§ (18)

-10.
The path of integration is taken as the circle r - a. In the area

.5

bounded by r - a, J is equal to zero. Thus

2na H¢(a) .‘ O ' (19)

SLUTIOI

The equations and boundary conditions an be obtained in a note

convenient fora by the introduction of acne diaensionless quantities.

Let

R - r/a (20)

c - b/a - (21)
v - (Vglsoa)(m/o)l/2 (22)
n - ii¢(o 1:01;)"1 (23)
P - “150230)“ (2h)
. - “(o/m”? (25)
P - cw ' (26)

In the above diaensionless quantities, a is a cylindrical analog of
the Hartnann nuaber and P“ is the aagnetic Prandtl nuaber.

Introducing the above diaensionless quantities into equations (12),
(13), and (11;), the equations becoae:

«ii-”h - 1-51 (27)
2 .

.92 . .3... 3.1.9533

on R PaR an (28)

.23! +«lch..-!— I ..lL»<lFu‘ (29)

daz RdR Ra R2 an

-11-

~12-
The boundary conditions froa equations (15), (16), and (17) becoae:

V(1) - 0 (30)
we) - (wb/Eofl)(PV/O)1/2 - v0 (31)
Mi) - o (32)

When equations (27) and (29) are combined, an equation involving only

v is obtained:

2

D.
<

.%%-(m2+1)-:-§ - -5 (33)

M

(iii

This equation an be solved by the introduction of a new variable.

Let
R ' e" (314)
Then
fig - ems-5 (35)
2 2
d ~2n d -2n d
—— - -—- - (36)
(“,2 e dn2 e dn

When this new variable is substituted into equation (33), the
following equation is obtained:

2
g";- (menu - -ne" (37)
dn
The solution to this equation is well imovn. It is
v - C1 sinh pn + 62 cosh 5n + en/a (38)

where

B - (.2 + 1)”2 (39)

-13-
The constants are found by applying the two boundary conditions on
v. The constants are:

avo s oosh(p In C) - c

snails in E) “‘0’

 

 

..l
Cl a

02 - - 1/: (hi)

when these constants are substituted into the equation for the ve-
ocity and n is written in terns of R, the following equation is
obtained:

v - 1/: [A sinh(p in R) - cosh(fl ln R) + R] (1:2)

where
av 4- cosh(p 1n c) - c

o slnhii in c) (1‘3)

The velocity profiles are plotted for a range of values of the
velocity of the outside cylinder with a - O, 2, h, and 12 (Figs. 3-10).
The ratio of the inner and outer radii is arbitrarily chosen as 2.

The aagnetio field is obtained by inserting equation ([12) into
equation (27) and integrating. flaking use of the fact that h(l) - O,
the following is obtained:

R
h(R) - l/R fl (x- av) dx (his)
Then
h(R) ----1/n2 (1 + p A) [cosh(B in R) - l/R]

+ 1/a2 [(3 + A) sinh(|3 ln 12)] (£6)

The pressure distribution is obtained by integrating equation (28).
Thus

R 2 a
P(R)-P(l) - f1 i-dx+P] 33mm: (1:6)

Diaensionless Velocity - v

 

 

 

 

 

 

0.1:
003 )- VO -' Ooh

V I 0.2\
0.2 P'- o P

vo - 0.0

0.1 '-

V0 - - 0.2
0.0

V0 - - 00
-001 P
-002 - L
-003 "’
‘Ooh I l l L
1.0 1.2. 1.1: 1.6 1.8 2.0

Diaensionless Radius - R

Fig. 3 Velocity Profiles for a - 2

-1S-

 

Ooh k /

inviscid fluid y
/ '0 . Ooh
z”

003*. / /

/
/ v -0.2

0.2-

Dimensionless Velocity - V

Gal _

 

0.0

 

‘Oah
1.0

l I i
1.2 1.1; 1.6
Diaensionless Radius - R

Fig. h Velocity Profiles for a - h

1.8

 

2.0

Dimensionless Velocity - v

-16-

 

O.hfl

0.?—

0.2L

inviscid

fluid
0.].— ‘__, ’\’ I

’

 

 

 

O.

-001—

-..).

-003—

 

-00L ' l J

1.0 1.2 l.h 1.6
Disensionless Radius - R

Fig. 5 Velocity Profiles for n - 12

1.8

 

2.0

Dimensionless Velocity - v

 

0.5

o.»

0.3

0.2

0.1

 

0.0

i I

 

 

 

Dinensionless Radius - R

Fig. 6 Velocity Profiles for v0

1.0 1.2 1.1: 1.6

- 0.1;

2.0

Disensionless velocity- v

-18-

 

0.14

0.3

0.2

0.1

 

O
O
O

 

 

 

 

1.0

1.2 l.h 1.6

Dimensionless Radius - R

Fig. 7 Velocity Profiles for 'o - 0.2

1.8

2.0

Dimensionless Velocity - v

 

0.3+

0
a
N

0.1

 

O
O
O

 

 

 

 

...-s
a
O

1.02 10" 106

Diaensionless Radius - R

Fig. 8 Velocity Profiles for v0 - 0

2.0

Dinensionless Radius - v

 

0.2 e

...},

 

11-12

 

 

1 1 J

 

 

 

-002
1.0

1.2 1.14 1.6

Diaensionless Radius - R

Fig. 9 Velocity Profiles for ”o

.-002

1.8

2.0

-21..

 

0.2

 

0.1 )-

0.0

 

>
I
>:
a -001
0
O
H
s
M
N
U
3 -002
2
..i
D

-003

 

 

 

 

1.0 1.2 1.h 1.6 1.8 2.0

Dimensionless Radius - R

Fig. 10 Velocity Profiles for v0 '- - 0.1:

- 22..

Since the magnetic Prandtl number is invariably small (10'7 for

mercury), the pressure due to the term containing the mgnetic Prandtl

number is so sail that it cannot be measured. Therefore,

a 2
P(R)-9(1) - f1 fi-dx (h?)

The pressure is found by substituting for the velocity and coapleting

the integration. Thus

 

2 2
P(R) - P(l) - l/n2 [-A—Efi-l- sinh(2p In R) + 1 E A ln R

“g sinh2(p in R) - 3% (5 + A) sinh ()3 In R)
n
2
+3; (3A + l) cosh(fl lnR)*%---12-
"gait“ +1)1 (“8)

LIMITING ms 0)“ THE FLO!

In its liaiting forms, the flow approaches solutions that have
already been deterained. Limiting foras of the flow any also be deter»
ained for special cases, such as no applied electric field or no vis-
cosity, which have not been previously deterained. These special cases.
will help in mderstanding the flow processes. .

If a " O, the flow should approach the ordinary flow between ro-
tating cylinders with no magnetic field. Equation (1)2) an be written in
a slightly different fora to mks the liaiting process easier. The
equation then becomes

0 1/2 in R

I - 2 4
v sinh (a2 + 1)1/2 111 c [cosh (. 1)

mm 2+1
v s n ) 1/"’1nR--R] 1/n

cosh (g2 + l)”2 in c - c
mm (.2 + 0172 in c

+..1.[
I

sinh (n2 + 1)”2 In R) (h9)

The limit of the above equation as n '* 0 is

11. v _ "o "mu“ R) 11:- 1 2 1/2
a -' 0 siann c) * ir'O ;{R . cosh K. 1 1) in R]

2 1/2
,cosha+1 lnc-¢!m[(2+1)1/21R(SO
sinhKllz + 1)1 2 1:: c] s a n' I} )

In order for the liait in equation (50) to be finite, the quantity

 

in brackets aust be zero when a - 0 so thatthe liait will boot the
fora O/O. If a - O, the quantity in brackets becoaes the following:

6031;111:111: 6' ° sinh(ln R) - cosh(1n R) + R -

c+lc-2c£;§__l_ZB-L;.’_lfi+R-O (51)

c-lc

-23.

- 2),-

Since the liait is of the fora O/O, L'Hospital's rule nay be
applied. After applying L'Hospital's rule the liait becoaes:

111 v vo sinh(ln R)
a-'O . sinhfin c) (52)

 

The diaensional quantities say now be inserted. After rearranging,
the equation becomes:

2 2 2
“IV - cob 1‘ ’8 (S3)

._.0 b2_.2 r

 

But this is just the ordinary flow between rotating cylinders and is iden-
tical to Eq. 5.15 in Schlichting( 2) if the velocity of the inner cylinder
is set equal to zero and the necessary changes of syabols are made.

If a and 13 approach infinity so tint their difference is equal
to a. and w/r approaches Ho with ”0 - 0, the velocity distribution
should approach that of Hartmann's flow between infinite parallel plates
with a transverse magnetic field and a unifora applied electric field
parallel to the plates (Fig. 11). Applying the liaiting process, the

following is obtained:

.13.“, n/a - fill. (51.)

where H is the regular Hartaann number uHOL(o/pv)1/2.

Let y designate the distance from nidchannel so that

R . 2.1.3.1.! (55)

Then
311‘. (n2 + 19” in R" - nu + y/L) (56)

and
1‘“ (n2 + 1)”? in c - 2a (57)

8".

L
I

 

 

 

MI

\\\\\\\\\\\\\\\\\\\\\\

 

 

 

 

 

 

——_b a.

Iso

\\\\\\\\\\\\\\\\\\\X\\\

H T T I T 1 T T T

Fig. ll Rectangular Channel for Hartaann's Flow

 

 

 

 

-26-

ll-
VHR - 1 (58)

Equation (142) is written in terns of the diaensional quantities and
the above Units are applied. The equation then becomes the following:

LE
3-?- V¢ - 19 (ml/2 [W ...... nu + w

- cosh li(l * y/L) + 1] (59)

After substituting uHOL(c/pv)1/2 for ii and siaplifying, the
velocity distribution beconesi

E
ii: hi!
,-.v,-fi-u-£°—§;,—,—,%4=ll ‘60)

0

But this is Just the velocity distribution for Hart-arm's flow and is

identical to Eq. 1-29 of Cowling(6) if P, the pressure gradient, is

set equal to zero in tint equation. . I
For the case of an inviscid fluid, the velocity an be deterained

froa equations (12) and (13) by setting v equal to zero. These result
in the equation

o£0--%V¢ - o (61)

Although v is zero when v - 0, equation (61) nay be written
in dimensionless quantities if it is observed that

av - :4, ' (62)

O

The velocity distribution for. an inviscid fluid is found by substituting
equation (62) into equation (61). The velocity distribution then becoaes
(Fig. 12):

Diaensionless Velocity - v

1.2

0.0

- 27..

 

 

 

 

 

Diaensionless Radius - R

Fig. 12 Velocity. Profiles for an Inviscid Fluid

2.0

~28-

v - R/a (63)

For the case of no applied electric field, the velocity distribution
m be deterained froa equation (112). In order to do this v and v0

aust be written in terns of V9, and to.

"v
12 b 121M! 1 R
EE;(m/O)/ - %;;(pv/a)/ ——)£——-(-:mh31:c

cosh(B In C) -

sinh(9 in c) c s‘“h(5 1“ R) - cosh(9 1n,R)+ R] (6h)

«in:

When. equation (6h) is aultiplied by an( pv/o)1/2 and 50 is set equal
to zero, the velocity distribution with no applied electric field becoaes

(Fig. 13): inh 1 R
V¢ " ”b —‘(£—")':inhp 12c (55)

Ratio of velocity to

Velocity of Outer*Cylinder -

 

1.2

1.0

0.6

9
gr

9
no

 

0.0

 

J l i
1.0 1.2 l.h 1.6 1.8

 

Diaensionless Radius - R

Fig. 13 velocity Profiles for No Electric Field

 

 

2.0

D1311 5510?!

Pros the plots of diaensionless velocity versus the dimensionless
radius (Figs. 3-10) it is seen that the velocity profiles approach nearer
and nearer to the velocity profiles for inviscid flow as the cylindrical
iiartaann nuaber increases. It is also seen that the viscous effects are
confined sore and more closely to the walls as the cylindrical iiartaann
nuaber increases. This substantiates the idea that the viscous effects
an be neglected when the Hart-am number is nuch greater than unity.

The cylindrical Hartman number is of the sane order of aagnitude as the
Hartnam nuaber if b/a - 2.

From the plot of the velocity profiles for an inviscid fluid (Fig. 12),
it is seen that the ngnitude of the velocity increases with decreasing
values of the cylindrical Hartaann number. In a viscous fluid the aaximn
velocity increases and then decreases with increasing cylindrical Hartaann
number. This also substantiates the idea that viscous effects are acre
iaportant at low Hartaann nmbers.

Froa the velocity profiles it is seen that the effects of the
aagnetic field soaetines oppose the viscous effects of the aoving wall.
when the wall drags the fluid faster than the current due to the electric
field would drive the fluid, then the magnetic field opposes the notion
of the fluid. For the case of no applied electric field, (Fig. 13), the
aagnetic field alqu opposes the viscous effect of the loving wall.

These results indicate the complexity of the aagnetoivdrodynaaic flow
with moving boundaries. They show the interaction of electroaagnetic and
viscous forces and indicate that aoving boundaries further coaplicate a

complex problea.
-30-

LIST OF REFEREKIES

Carrier, G. F., and Greenspan, H. P., "The Tine Dependent
Hagnetomdrodynanic Flow Past a Flat Plate,” J. Fluid Hech.,
Vol. 7, 1960, pp. 22-32.

Schlichting, 11., Boundar L er Theo , New York,

Pergamon Press, 193.

Hartmann, J., ”Hg-Dynmics 1," i531. Danske Videnshab. Selskab,
Hat.-fys. Hedd., Vol. 15, 1937, lo. .

Globe, S. , 'Laainar Steady-State Hagnetohydrodynaaic Flow in
an Annular Channel," Plus. Fluids, Vol. 2, 1959, pp. Wit-1:07.

Chehaarev., I. G., "Sone Probleas of the Stationary Flow of
a Conducting Liquid in an Infinitely Long Annular Tube in

the Presence of a Radial Magnetic Field,” Soviet Ems” Tec .
m3" Vol. 5, 1960, pp. 565-569.

Cowling, T. G., ngtomdrodxng ics, New York, Interscience
Publishers, Inc. , l9 .

Stratton, J. A., Elect netic Theo , Iew York HcGru-iiill
Publishers, Inc., 19111.

”753W USE CE LY