Exact solutions of steady plane MHD aligned flows using (ξ,ψ) or (η,ψ) coordinates

(1)

Internat. J. Math. & Math. Sci.

VOL. 20 NO. (1997) 165-186 165

EXACT

SOLUTIONS

OF

STEADY

PLANE

MHD ALIGNED

FLOWS USING

(,ly)--OR (r/,Iv)--COORDINATES

F.LABROPULU(*)_and_O.P._CHANDNA(**)

(*)Department

_{of Applied Mathematics}

University of

Western

Ontario

London, Ontario Canada N6A 5B7

(**)Department

ofMathematics and Statistics UniversityofWindsor

Windsor,Ontario CanadaN9B 3P4

(Received August 8, 1994)

ABSTRACT. Anewapproachfor the determinationofexact solutionsof steady plane

infin-itely conductingMHD aligned flowsispresented. In thisapproach, the

(f, )-

or the

(r/,

)-coordinatesisused to obtain exact solutions of these flows where

(x,

y)is the streamfunction andw

f(x, y)

+

i7(x,

y)is ananalyticfunctionofz x

+

iv.

KEY WORDS

AND

PHRASES. Magnetohydrodynamics(MHD),

_aligned,_{streamfunction,}

exact_solutions,_{steady, plane, infinitely}_conducting.

AMS

SUBJECT

CLASSIFICATION

_CODE. ₇₆

1.

INTRODUCTION.

M. H. Martin

[4]

developed anew approachin the study ofplaneviscous flows of incom-pressible _{fluids by} _introducing _a _natural _{curvilinear coordinate}

system

(, )

in the physical plane

(x, y)

when constant arethestreamlinesand constant is an arbitrary familyof curves. _Following_Martin

[4]

_{and taking the arbitrary family of}_curves

(x,

y) constant to be x

constant,

_Chandna_and_Labropulu

[1]

_{studied exact} _solutions_of_steady

plane ordinary

viscous andmagnetohydrodynamic

(MHD)

flows.

In

thispaper,wepresentanapproach for the determinationof exactsolutions of steady plane

infinitely conducting

MHD

aligned flows andwe let

(x,

y) constant tobe either

(x,

y)

constant or

?(x,y)

constant wherew

N(z)=

(x,y)+

_iq(x,y)is_an_{analytic function of}_z andstudy flows_{when the streamline pattern}_is_of_{the form}

r/-

f(’)

constant or

f- k(r/)

constant

(2)

166 F. LABROPULU AND O. P. CHANDNA

Inthe caseswhen

f()

0 and

g()

I or

k(r/)

0 and

re(r/)

I, the problem iscalled an isometricflowproblemor Hamel’sproblemandwasfirst raisedbyJeffery

[3].

However, Hamel

[2]

wasthe first togive completesolutionsofthe permissible flow patterns forordinaryviscous

incompressible plane flows. Asexamplesto illustratethemethod,weusetwoanalyticfunctions

N(z)

v/

and

N(z)

lnz.

The planof this paperis asfollows: insection2,werecapitulatethe basic equationsgoverning the steady plane motion ofinfinitely conducting

MHD

aligned fluid flows. This section also containstherecastingofthe equations inanewformby employingsomeresults fromdifferential

geometry. Insection3,weoutline themethod of determining whetheragiven familyofcurves canbe the streamlines. Section 4 consistsof applicationsof thismethod.

Examples

I,

II,

VIIandXarefour streamlinepatterns for the Hamel’sproblemforourflows.

Two of theseflow patterns are different from the four well known flow patterns for Hamel’s probleminordinaryviscousfluiddynamics.

2. FLOW EQUATIONS.

Thegoverningequations ofaviscousincompressible and electricallyconductingfluidflow,in the presenceofamagneticfield,are

[5]

divv 0

1

curl

(curl

H)=

curl

(v H)

()

wherevisthe velocity vectorfield,Hthemagneticvectorfield,pthe pressure function, andthe

constantsp,/,

t*

andaarethe fluiddensity,coefficient of viscosity,magnetic permeabilityand

theelectricalconductivity respectively. The magneticfieldH satisfiesanadditional equation

divH 0

(2)

expressingthe absence ofmagnetic polesin theflow.

Taking the flow tobe aligned

(or

parallel) so that the magnetic

field

iseverywhere parallel tothe velocity field,wehave

H=

(3)

where/

is someunknownscalar functionsuchthat

(4)

In

this paperwe study planemotionin the

(x,

y)-planeofaninfinitely conductingfluid

(i.e.

a

--

oo)

andhave the velocitycomponents u, v,themagnetic components H1, H2,the pressure

functionpand thefunction asfunctionsofx, y. We define the vorticity functionw, current

densityfunction and energyfunctionh given by

(3)

EXACT SOLUTIONS OF STEADY PLANE MHI) FLOWS 167

Using

(3)

to

(5)

insystem

(1),

wefindthataninfinitelyconductingsteadyplaneMHDaligned

flowisgoverned by thefollowingsystem ofsixpartialdifferential equations:

Ou

Ov

+

--

0 (continuity)

oy

O’-Oh

+

#-ffffyow

pvw

+

#[fluQ

0

Or/

tt-z

+

puo., #

uf

0

(linear momentum)

(solenoidal)

Ov Ou

w (vorticity)

Ox

Oy

(current

density)

(6)

for thesixfunctions

u(x,

y),

v(x, y),

h(x, y),

w(x,

y), f/(x, y)

and

#(z, y).

Onceasolutionofthis

systemisdeterminedthe magneticvectorfieldH and the pressure functionp(z, y)arefoundby usingequations

(3)

and

(5).

The equation of continuityinsystem

(6)

imphes the existence ofastreamfunction

(x,

y)

such that

0

-v, u

(7)

Oz

We take

(z,y)

constant to be some arbitrary family ofcurves which generates with the

streamlines

(x,

y)

constant acurvilinear net sothat inthe physical plane the independent

variablesx, y can bereplaced by

,

.

Let

(,),

u

u(,)

(8)

defineacurvilinear net in the

(x,

y)-planewith the squaredelement ofarc length alongany curvegien by

ds2

E(,)

de

2

+

2F(f,q,)dCd

+

G(,

)d

2

(9)

where

+

F

0"-

0"---t-

O-

0-’

+

(10)

Equations

(S)

can be solvedto obtain

(x,

y),

(x,

y)such that

0

Oz

0

Oy

joe

Oy

0

Ox

j

(11)

0

0’

0

s,

0

’

0

s

provided0

<

[J[

<

oo, whereJisthetransformation Jacobianand

S

Ox Oy

Ox

Oy

4.v/EG_

F2 +W

(say)

(12)

o

o o

FollowingMartin

[4]

and ChandnaandLabropulu

[1],

wetransformsystem

(6)

into -plane

(4)

168 F. LABROPULU AND O. P. CHANDNA

THEOREM 1. Ifthe streamlines

(x,

y) constant ofa viscous, incompressibleinfinitely

conductingMHD alignedflow arechosenas oneset ofcoordinatecurves inacurvilinear coor-dinatesystem

,

inthe physical plane, then system

(6)

in

(x,

y)-coordinatesmaybereplaced by thesystem:

(linear

momentum)

0

(Gauss)

(current

density)

1 0 E

(vorticity)

0

(solenoidal)

0

(3)

ofsix equationsforsevenunknownfunctions

E, F, G,

h,

,

w

and/9

of

,

.

O

h

O

h Ifwe usetheintegrability condition

69

4 69

0 04

in the linear momentum equations of

Theorem1,wefind that the unknown functions

E(b,

),

G(,

), F(, ), w(, ), ft(,

)

and

must satisfy thefollowingequations:

1

(4)

E d/

(15)

f=/9o;

j2

de

(16)

(18)

Equations

(14)

to

(18)

form an underdetermined system, the reason being the arbitrariness inherentin the choice of the coordinatelines constant. Thissystemcanbe made determinate

inanumber ofwaysandoneplausibleway is to assume

(x,

y)

(x,

y)

or

(x,

y)

(x,

y)

where

((x,

y)

and

r(x,

y)

arethereal and imaginary part ofananalyticfunctionasoutlined in

the next section.

3. METHOD.

Let w

+

ir

be an analytic function ofz x

+

iy where

(x,

y) and ?(x,y).

Since w is an analytic function ofx, y, then its real and imaginary parts must satisfy the Cauchy-Riemann equations, that is

(5)

EXACT SOLUTIONS OF STEADY PLANE MHD FLOWS 169

The equations

f(x,

y) andr/=

r/(x,

y)canbesolvedtoobtain

z

x(f,

),

y y(f,

r/)

(20)

such that

0x

j.

0__

0x j.

0_.

0y j.

0__

0y j.

0_

(21

provided0

<

J*!

<

c, whereJ*isgiven by

j.

a(x,

y)

ax

oy

Ox

oy

(22)

o(

o

OV

o o

Using

(19)

and(21)in

(22),

weobtain

Using

(20),

(21)

and

(23)

in ds2 _dx

+

_dy

,

_we _get

(23)

Method for the

(f,)-coordinate

net.

Toanalyzewhetheragivenfamilyofcurvesr/-

f(f)

constantcanorcannot bestreamlines,

()

we assumethe affirmativesothat thereexists somefunction

7()

such that r/--

f()

(),

’() #

0

(25)

where

7’()

isthe derivative ofthe unknown function

7()

and we take the coordinate lines constant tobe constant.

Employingequation

(25)

in

(24)

and simplifying the resulting equation,weobtain

+

2J*

{J"

()

+

g’ ()

7

()}

g

()

7’() d

de

+

j’g2

()

7,2()

(26)

Comparing

(26)

with

(9)

after taking

,

weget

J"

’/

+

[/’()+

’()()l’,

’

E

F ]"

[:’()

+

’()()]

()’(),

G

J*()7’()

W

/Ea- r

J’()7’()

(27)

Since

then

j

o(, )

o(,

)

o(,

)

0(,)

o(,)

0(,)’

andtherefore

J W

J*()7’()

(28)

F

(6)

170 F. LABROPULU AND O. P. CHANDNA

Using

(27), (28)

and in

(14)

to

(18),

wehavethefollowing theorem:

THEOREM2. Ifasteady,plane,viscousincompressible fluid ofinfiniteelectrical conductivity

-

f()

flowsalong constant in thepresence ofan aligned magnetic field, then theknown

()

functions

f(),

g() and theunknownfunctions

()

and

’()

must satisfy

02w

1

+

f,2()

2f’()g’()

g()’7’()’__

2

[f’()

-F

g’()()]

00

-F g()

+

’()

,,

o

f’()’()+

+

()

t)

,(e)

oe

+

-f"()

+

()

-"()()

(

+

I’()

"(e)

f’()’()

()"()

’()

(e)"(e)

1

o

()

’(0)

()

’()

9()

’()

f

pOw

*

;

+

-z)

o

1 2

g(’)’7’(’)

{1

--t-[f’()-I--

g’()’7(’)]

}

(:-:-)2j.0

2

and

02

j.

2

[f’()

+

g’()-y()]

00

wherewand aregiven by

=hz

()

P(

J’()+

+

()

’()

9()

’()

2g’2()

7()

()

’()

()"()

}

a()

,()

(30)

(31)

d

7()

is somenctionof

su

that

7’()

#

0.

A

ven

fily ofcurves

f()

const:t is a

rmissible

fily of strel if d

()

oy

if thesolutionobtned for

7()

is

su

that

7’()

#

0.

Method for the (y,)-coordinate net.

To

Myze

whetheragiven fily ofcurves k(y) consttc orcnotbestrehnes,

()

we:sumethe mativesothat thereexists somefunction

7()

su

that

k(y)

7(),

7’() #

0

(33)

m()

where

’()

is the deritive of the unknofunction

()

d we te the crdinate fines constt tobe y constt.

Employingequation

(33)

in

(24)

andsimpnfying the resting equation,weobtn

.

[

+

{’(.)+

’()()}]

()

+

2]"

[’

()

+

(7)

F_.XACT SOLUTIONS OF STEADY PLANE FLOWS 171

Comparing

(34)

with

(9)

after taking

,

weget

E

J"

{

1

+

[k’(y)

+

m’(r)7()]2},

F J*

[k’(r)

+

m’(?)7()] m(r)7’(),

G

J’.()’()

w

v/G-

F=

s’.()’()

(35)

Since

then

o(,)

o(, )

o(, ) o(,)

a(,

)

0(,)

o(,

)

0(,)’

j

-]’.()’()

and therefore

J -W

-J’m(y)7’()

(36)

Using

(35), (36)

and

r

in

(14)

to

(18),

wehavethe following theorem:

THEOREM3. Ifasteady, plane, viscous, incompressiblefluid ofinfinite electrical

conduc-tivity flowsalong

k(r)

constantinthe presence ofalignedmagnetic field, then the known

-()

functions

k(r),

m(r)

and theunknownfunctions

(),

7()

must satisfy

(37)

and

2k’.(w)m’(r)]

1

[m"(W)

m2()

+

re(W)

(39)

f

()w

j.m2(r)

{1

+

[k’()

+

()()]

}

’(---)

and

7()

is somefunctionof such that

0"()

#

0.

(8)

172 F. LABROPULU AND O. P. CHANDNA

4. APPLICATIONS.

We use analytic functions w

+

i/

N(z)

in the first seven examples and w

+

i

N(z)

Inz in the otherfourexamples.

4.1. Examples forw

2

v/.

_Then,_we _have

Let

z=w

orw=

1

(

)

(41)

or

* V/+’/’

+v=

,

V/-z

Usingequation

(41)in

(23),

weobtain

(42)

j.

2

+

2

(43)

ExampleI.

(Flow

withr/=constantas

streamlines).

This example gives us a streamline pattern for Hamel’s problem for infinitely conducting

MHD

aligned flows. Thestreamlinepatternobtainedisnotoneofthefourwell knownpatterns

forordinaryviscousfluidflow. ThispatternisgiveninFigure 1. Welet

r/=

7();

7’() #-

0

(44)

where

7()

is anunknownfunctionof

.

Employing

(44)

in

(43),

weget

.

+

()

(45)

Comparing

(44)

with

(25),

wehave

f()

0, g() 1

(46)

Employing

(31), (32), (45)

and

(46)in

equations

(29)

and

(30),

we findthat equation

(30)is

identically satisfied and

(29)

reducesto

2

E

A,,()

f"

0

(47)

n’-O

where

A0()

4

4()

+

()

2

7"()

2p*

+

A()

[- *()]

v’()

()’()

v’()

1

Equation

(47)

is aquadraticin withcoefficientsasfunctionsof only. Since

,

are

indepen-dentvariables,it follows that equation

(47)

canholdtrueforall values of if all the coefficients

ofthisquadraticvanishsimultaneously andwehave

(9)

EXACT SOLUTIONS OF STEADY PLANE MHD FLOWS 173

Integrating

A()

0 fourtimeswithrespect to

,

weobtain

() + a()

+

()

+

0

(49)

whereal, a2, a3 anda4 arearbitrary constants that arenotzerosimultaneously. Usingequation

(49)

in

A0()

0,weget

a2 --0

(50)

Employingequation

(49)

witha2 -0in

A1()=

0 andintegratingtheresulting equationonce withrespect to

,

weobtain

z()

(3el

V/X

2

+

_y2

3alx

+

a3)

(51)

where

as

isanarbitrary constant of integration. Substitutingequation

(44)

in

(49)

witha2 0, wefindthat

air/3

4-a3r/4-a4

(52)

where_r/isgiven byequation

(42).

Forthisflow,the exact solutionsaregiven by

1

[-3al

3a

V/x

2

+

_y2 _y2

u

2

V/

x2

+

_y2

x+

4-a3

V/

x4-

V/

x 2

4-v

2

Z

+

y2

H,

(),

H

(),

=

+

4x2’+

a

()

-a

+

.

+

’(e)

2

+

wherep is

bir eons

d

()

is given byequation

(1).

If

a

g,

thengheflow

isiotagionN. _The, _we_have_the_foog_hrem:

NEOREM4. Steady pleflow

Mong

eonsg ispermissibleforinfinitely conducting

MHD

Nixed

flow d he

exae

solutionsfor he rotationM flow e given byequations

()

dfor theirrotaionNflow byequations

(g)

with 0.

Nxample II.

(low

with ( eonst

stretches).

his exple Msodens with astreamlinepattern for Hel’s problemd this pattern is notoneof he four

we

known patterns. Pigu2 shshisflow pattern.

Wele

(e);

’()

0

()

where

7()

is unknownfunegionof

.

Compi equation

(4)

wih

(),

weget

(10)

17h F. LABROPULU AND O. P. CHANDNA

Usingequation

(54)

in

(43),

weget

"

,

+ ()

(56)

Employing

(39),

(40), (55)

and

(56)

in

(37)

and

(38),

wefind that equation

(38)is

identically

satisfiedand

(37)

takes theform

B.()

=o

n=0

where

Equation

(57)

is a quadratic in r/ with coefficients as functions of only. Since r/, are

,independent variables, it follows that equation

(57)

can holdtrue for all values of

r/if

all the

coefficientsofthis quadratic vanish simultaneouslyandwehave

B0()

B1()= B2()=

0

Integrating

B2()

0fourtimeswithrespect to

,

weobtain

b1")’3() +/72()

+

b37()

+

b4

0

(58)

wherebl, b,

b3

and

b4

arearbitrary constants thatarenot zero simultaneously.

Usingequation

(58)

in

B0()

0,weget

b

0.

Proceedingasinthe previous example,wehave

+

,

+

v

+

,2

+

/

+

,

()

+

1

[3blx_.3bl%/,T,2_.12_.b] V/_x+

v/.r2.t.y

2

,,

+

v

+

ab

+

b

+

1

{

]}

1

a

()-2

+

=

’()

where

₀

dp0 e

bitr

constts. If

b

O,then the flowisirrotation.

(11)

EXACT SOLUTIONS OF STEADY PLANE MHD FLOWS 175

Example III.

(Flow

with r/-

f

constant as

streamlines).

Weassumethat

"r();

"r’()

#

0

where

"r()

isanunknownfunctionof

.

Comparing

(60)

with

(25),

we get

(60)

f(f)

f, g(f) 1

(61)

Using

(61),

equation

(43)

yields

J*

2

+

2’r()

+ ,/2()

(62)

Employing

(31),

(32),

(61)

and

(62)in

equations

(29)

and

(30),

we find that equation (30)is

identicallysatisfied and

(29)

reducesto

4

C,,,(’)

’’

0

(63)

."-0

where

.()’()

}

,=()

Equation

(63)

isafourth degree polynomialin with coefficientsasfunctionsof only. Since

,

areindependentvariables, itfollows that equation

(63)

can hold true for all values of if allthe coefficientsofthispolynomial vanish simultaneouslyandwehave

c,()

c()

c,()

c0()

o

(64)

Integrating

C4()

0 fourtimeswithrespectto

,

weobtain

v()

+ =v=() +

()

+

0

(65)

(12)

176 F. LABROPULU AND O. P. CHANDNA

Using

C4()

0 in

Ca()

O,weget

{

"/’()

()’()

)=o

[p-

()]

,()

+

"

,()

whi uponintegration implies that

[, ’()]

()

,

wherecsis bitryconsttd

7(@)

is

ven

implicitly byequation

(65).

Employing

(6)

a

(),

c,()

0

e, 0

(7)

Finny, using

(65)

to

(67)

in

C(@)

0d

C0(@)

0,wefindthat bothof theseequationse

identicay satisfied.

Hence,

the fily ofcurves

-

constt e permissiblestremlines

for the flow underconsideration dthe unknownfunction

7()

is given implicitly by equation

(65)

with

c

0.

Employing

(60)

in equation

(65)

thc2 0,weget

,

(,

)

+

(

)

+

,

(s)

where and r/asfunctions ofz _{and V} aregiven byequation

(42).

Thus,the solutionsforthe velocitycomponents, themagnetic fieldcomponents, thepressure, the vorticityandthe current

densityaregiven by

( ),

( )

-,.

+

4

+

-

+

(9)

2x=

+y

V:

+

,

+

u_

+

()

4:

+

whp0 i bi, ontt

()

i by qu,io

().

The streine patte forthisflowisshoin

Fi

3.

ExampleIV.

(Flow

with

r/3

constantas

streamlines)

Welet

-

();

’() #

o

where

7()

is anunknown functionof and

,

r/aregiven by equations

(42).

Proceedingasinprevious examples,wehave

1

V/-.

+

v’.

*

+

*

(

3)

2al

V/x

2

’+

_y2

(13)

EXACT SOLUTIONS OF STEADY PLANE MHI) FLOWS 177

whereal

#

0andp0 arearbitrary constantsof integration.

The flow pattern ofthisexampleisshowninFigure4.

(7)

Example V.

(Flow

with y

3

constantasstreamlines) Weassumethat

3

();

’() #

0

(72)

where

7()

is anunknownfunctionof and

,

r/aregivenby equations

(42).

Following the examplesabove,weget

2b

V/z

2

+

_y2

H

u,

H2=

P v, w=

+V/x2+y2,

f=

b

/x

2

+

_y2

/ +

"

-

+ /

+

a

+

+(

+)/

+1}

+

(73)

where

bl

-

0 and_{P0 are}arbitraryconstants ofintegration. Figure5 shows the streamhnepatternof thisflow.

Example VI.

(Flow

withr/-

3

constant asstreamlines). Weassumethat

-

();

’() #

0

(74)

where

7()

is anunknownfunction and

,

r/are given by equations

(42).

Followingthesame

procedure as in previous examples, we conclude that this family of curves is a permissible

streamlinepattern for infinitely conductingMHD aligned flowandthesolutionsaregiven by

(14)

178 F. LABROPULU AND O. P. CHANDNA

v -z

+

V/x

2

+

=

+

₃

+

2d,

V/x

2

+

=

H1

u,

H2

v, w=

y2

+

=

a

2

+

where

dl #

0,

d2

d_{P0 are}arbitryconsists. The flow pattern forthisexpleisshownin

Figure6.

4.2. Examples forw lnz.

Let z e or w lnz. Then, wehave

or

z e cos,7

(76)

y e _sinr/ n,

1/2n

(==

+

U=)

(77)

Usingequation

(76)

in

(23),

weobtain

j*

e2

(78)

Example VII.

(Flow

with r/=constantas

streamlines).

Thisexampleisapossiblestreamlinepattern fortheHarnel’sproblemforourfluidflow. This

patterningiveninFigure 7.

Weassumethat

r/=

7(b);

7’(t/’) #

0

(79)

where

7()

is anarbitraryfunctionof

b

andr/isgiven by equation

(77).

Comparing

(79)

with equation

(25),

weget

f(’)

O,

9(’)

1

(80)

Employing;

(31),

(39.),

(78)

and

(80)

inequations

(9)

and

(30),

wefind that equation

(30)

is

identicallysatisfiedand equation

(9)

reducesto

Integrating

(81)

withrespect to%b, weobtain

where/

isan bitrm’y constant and

(,)

is anrbitraxy function of

.

Thus, thisfinily of

streamlines is llowed by infinitely conducting MHD Migned flow and the exact solutionsfor

this roationNflowre

iven

by

,

"’(,)

+

,

,,

.,(,)

,,

+

,

o

+

.,(,),

H1

9(b)u,

H2

=/(%b)v,

a

=/(e)w

(15)

EXACT SOLUTIONS OF STEADY PLANE MHD FLOWS 179

where p0 is an arbitrary constant,

/()is

given by equation

(82)

and

’()

is an arbitrary

function of

b.

Example VIII.

(Flow

withr/-

f

f2

e2

constantas

streamlines).

Weassumethat

q

f

f2

e2

,();

7’()

:

0

(84)

where

0’()

isan unknown function and

,

q are givenby

(79).

The streamlines areshownin

Figure 8.

Proceedingasabove,wehave

1

()

-

[-

’0

=]

+

0,

()

0

[

( +

)

(

+

)]

[- ,.z0

]

=

+

2/ 1

[-

,’Zo’]

’

+

’

[-

(=

+

)

(=

+

=)]

H

o,

H

.

-[0

’g]

+

n

o

4

[

_(

+)]

s.

(s)

2p 1

[

,-Z]

=

’

+

’

{=

+

[ (=

+

’)]’

+

( +

=

+

=)

(’

+

=)

+4

(x

+

y)

}

+

Po

where0,

o

#

d Po e

bitr

constts.

Example

IX.

(Flow

alongy-

f2

constant

strelines).

Wete

v();

v’()

#

0

(86)

where

v()

is

=

unknomnction=df, =egiven by equations

(77).

Following the

ave

procedure,weget

1

()

-

[- ,-z]

+

,

z()

z

x2

+

y2In

(x

2

+

x

y2

+

(

+

(87)

4,u 1

[In

(x +

y2)

+

I]

[-

,.go

]

=’

+

v’

2p,u2 ₁

2

+

where

o,

o

d poebitraryconstts. Theflow patternfor thisexpleissho

(16)

180 F. LABROPULU AND O. P. CHANDNA

Example X.

(Flow

with constant asstreamlines).

Theflow patternin thisexampleis apossiblesolutionofthe Hamel’sproblemforourflow.

Figure 10isshown this streamlinepattern.

Welet

();

’()

#

0

(ss)

where

3’()

isanunknown function and

f

isgiven by

(77).

Using

(88)

in

(78),

weget

J* e27()

(89)

Comparing

(89)

with

(33),

weobtain

k(r/)

0,

re(r/)

1

(90)

Employing

(39), (40), (89)

and

(90)in

equations

(37)

and

(38),

we find that equation

(38)is

identically satisfied and equation

(37)

gives

7’()

\’()]

+-’(i

0

(91)

Thus, constant can serve asstreamlinepattern for infinitely conducting

MHD

aligned flow

and the solutions aregiven by

1 y

,’()

+

1 x

7’()

z

+

H1

(),

H

()

-’)

’()

0

+

-’

[0- ’()]

,()

p

,()

,’()

+

-’(

[o

,’(]

.

lee

.(

+

"()

.(e)’

+

,(),

a

(e

+

,()

(92)

wherep0 is anarbitrary

constant,/()

is an arbitraryfunctionof and

7()

is a solution of

equation

(91).

Requiring the pressure to be single-valued,wemust take

7"()

]’

0

7,s()

which,uponintegration, gives

ale2"’)

+

2a7()

+

as

0

(93)

where al, a2 and

as

are arbitrary constants that are not simultaneously zero. Using

(93),

equation

(91)

isidentically satisfied. Employing

(88)

in

(93),

weobtain

(17)

EXACT SOLUTIONS OF STEADY PLANE MHD FLOWS 181

Usingequation

(77)in

(94),

weobtain

a,(

+

)

+

==n

(’

+

=)

+

=,

Hence,the solutions for this rotational flowaregiven byequations

(92)

with

0’()

givenimplicitly

byequation

(93).

Ifal 0,then the flow isirrotational. 1

[p

0,,2()]

andusing

(93),

the solutions

(92)

take theform

Letting

2

()

-:

[

’=()]

where

:

()

#-;--2 1.5 -0.5 0 0.5 1.5 2

(18)

-2

-1.5 -1 -0.5 0 0.5 1.5 2

Figure 2. Streamlinepattern for ExamlpeII

0.5 1.5 2

(19)

EXACT SOLUTIONS OF STEADY PLANE MHD FLOWS 183

10 8

-10

-5 -4 -3 -2 -1 0 2 3 4 5

Figure4. StreamlinepatternforExampleIV

10

-5 -4 -3 -2 -1 0 2 3 4 5

(20)

8 6

4 2

0

-2 -4 -6

-8

-4 -2 0 2 4 6

Figure 6. StreamlinepatternforExampleVI

-1 -2

-3

-5

(21)

EXACT SOLUTIONS OF STEADY PLANE MHD FLOWS 185

1.5

0.5

-0.5

-1.5 -I -0.5 0 0.5 1.5

Figure 8. StreamlinepatternforExampleVIII

30

20

10

-10

-20

-30

(22)

-I

-2

-3

Figure10. StreamlinepatternforExampleX

REFERENCES

1. O.P. Chandna and F. Labropulu, Ezact solutions

of

steady plane

flows

usin9

yon Mises

coordinates,J. Math. Anal. Appl.,

185(1),

pp. 36-64, 1994.

2. G. Hamel, Spiralformige bewegungen zaher flusigkeiten,, Ider. Deutch. Math. Verein, 25,

pp. 34-60, 1916.

3.

G.B.

Jeffery, On the two dimensionalsteady motion

of

a

visco

fluid, Phil.

Mag.,

29,pp.

455-464, 1915.

4. M.H. Martin, The

flow of

aviscous

fluid:

L

Arch.

Rat.

Mech. Anal.,41,pp. 266-286, 1971. 5. S.I. Pai, Magnetogasdynamics and Plasma Physics, Englewood Cliffs,

N.J.,

Prentice-Hall,