HODOGRAPHIC STUDY OF NON-NEWTONIAN MHD ALIGNED STEADY PLANE FLUID FLOWS

A study is made of non-Newtonian HHD aligned steady plane fluid flows to find exact solutions for various flow configurations. The equations of motion have been transformed to the hodograph plane. A Legendre-transform function is used to recast the equations in the hodograph plane in terms of this transform function. Solutions for various flow configurations are obtained. Applications are investigated for the fluids of finite and infinite electrical conductivity bringing out the similarities and contrasts in the solutions of these types of fluids.

INTRODUCTION.Transformation techniques are often emplo:y-d for solving non-linear par- tial differential equations and hodograph transformalion method is one of these techniques which has been widely used in continuum mechanics.W. F. Aes [1] has given an excellent survey of this method together with its applications in various other fields.This paper deals with tle ap- plication of this method for solving a system of non-linear partial differential equations governing steady plane incompressible flow of an electrical conducting second-grade fluid in the presonce of an aligned magnetic field.Recently, A. M. Siddiqui et al [2] used the hodograph and Legcndre transformations to study non-Newtonian steady plane fluid flows.O. P. Clandna et al [3] has also applied this technique to Navier-Stokes equations.Since electrical conductivity is finite for most liquid metals and it is also finite for other electrically conducting second grade fluids to which single fluid model can be applied, our accounting for the finite electrical conductivity makes the flow problem realistic and attractive from both a physical and a mathematical point of view.We have also included electrically conducting second grade fluids of infinite electrical coaductivity to make a thorough hodographic study of these fluid flows and to recognize the dawn an,t future of superconductivity in science.
We study our flows with the objective of obtaining exact solutions to various flow configura- tions.We start with reducing the order of governing equations by employing M. l:I.Mtrtin's [4] perceptive idea of introducing vorticity and energy functions.The plan of this paper is as follows: In section 2 the equations are cast into a convenient form for this work.Section 3 contains the transformation of equations to the hodograph plane so that the role of independent variables :. y and the dependent variables u, v (the two components of the velocity vector field) is interchanged.
We introduce a Legendre-transform function of the streamfunction and recast all our equations in the hodograph plane in terms of this transform function in Section 4. Theoretical development of section 4 is illustrated by solutions to the following examples in section 5: (a) flows with elliptic and circular streamlines (b) hyperbolic flows (c) spiral flows (d) radial flows.These applications are investigated for the fluids of finite and infinite electrical condlctivity bringing out the sinfilarity and contrasts in the soluti,ns of these two types of flfids.
2. EQUATIONS OF MOTION.The steady, plane flow of an inconpressible second-grad' flid of finite electrical conductivity is governed by the following system of equations: Ou +o Oy oo 0,,) where u, v are the components of velocity field ]7 , H, H2 the components of the magm.ticvector field ]E?, and p is the pressure function: all being functions of x,y.In this system and c2 are respectively the constant fluid density, the constant coefficient of viscosity, the con- stant magnetic permeability, the constant electrical conductivity and the normal stress moduli.Furthermore, K is an arbitrary constant of integration obtained from the diffusion equation We now introduce the two dimensional vorticity function w, the current density function j (2) of seven partial differential equations in seven unknown functions u,v, w, Ha, H2, j and e as functions of z,y.This system governs the motion of second-grade fluid of finite electrical con- ductivity.For the motion of second-grade fluid of infinite electrical conductivity, we only replace the diffusion equation in the above system by uH2 v Hi K.
ALIGNED FLOW.A flow is said to be an aligned or parallel flow if the velocity and the magnetic fields are everywhere parallel.Taking our flow to be an aligned flow, there exists some scalar function f(z,y), called the proportionality function, such that it f(,v)V Introducing this definition of the magnetic vector field in the above system, the aligned flow is governed by the following system of seven equations + _x--0 (4) Oe O pvw I--a--a,vV2w I*fvj (5 of " +'N =0 (s) Of Of f + i =w (10) Oz i i ,,k-o fu-tio-,,(:, ), ,,(:, ),,,,(:, ), f(, ),./(:, u), (:, ) ,d = bit,-y o.t,t K. Once a solution of this system is determined, the pressure and the magnetic functions are obtained by using the definition of e in (1) and the definition of in (3) respectively.
) is deterned, we are led to the solution of u u(x,y), v v(x,y) and therefore w (u(x,y),v(z,y)) w(x,y), e e(x,y), j j(x,y), f ](x,y) for the system (4) (10)   governing the fitely conducting flow.
The above anMysis Mso holds true for infinitely conducting second-grade fl,fid flows.How- ever, for these flows, the arbitrary constant K 0 and equation ( 7) and its transformed equation (18) are identicMly satisfied.
COROLLARY" II.lf L*(q,O) and f*(q,O) are the Legendre transform of a streamfimction and the proportionality function of the equations governing the motion of steady plane aligned flow of an incompressible second-grade fluid of in/nite electrical conductivity, then L'(q, O) and f'(q,O) must satisfy equations (47) and (49) where J*(q,O), w*(q,O), W(q,O), W(q,O), F(q,O), F(q,O) and x*(q,O) are given by (40) to (45) and (50).
Following the determination of velocity components u + iv qe i in physical plane we get f(z, V) and the other remaining flow variables.
5. APPLICATIONS.In this section we investigate various problems as apI)lications of Tl,'(,rem and II, and their corollaries. APPLICATION I. Let L(u,v) Au + Bv + Cu + Dv + E (52) be the Legendre transform function such that A, B, C, D, E are arbitrary constants and A, B are nonzero.Using (52)in equations (31) to (33), we get 1 A+B Wa =0, W2=0.

2AB
We now consider finitely conducting and infinitely conducting cases separately by applying theorem and theorem II respectively.
CASE (III).In this case, L(u,v) A(u v) + Cu + Dv + E and -](u,v) C2.Flow variables for this case are: where C5 is an arbitrary constant.
3ax + 2a2 INFINITELY CONDUCTING FLUID.Using the expressions for L, J, , W, W2, F, F2 as given by ( 52), (53), (37) in equations ( 29) and (36), we find that st(u, v) must satisfy Solving equations (64) for f(u, v), we find that f(u, v) (u +v2) if A B and f(u, v) C if A : B, where is an arbitrary function of its argument and C6 is an arbitrary constant.
Therefore, we have the following two cases: (i) L(u, v) A(u 2 + v) + Cu + Dv + B, "(u, v) q(u + v), where is an arbitrary function of its argument.
We now consider these two cases separately.
CASE (I).Without loss of generality, we take j(u,v) u + v. Using L(u,v) A(u + v 2) + Cu + Dv + E in equations (24), we obtain and therefore.
Suing up, we have the following theorems: TnEoaE III.H L(u,v) Au + Bv + Cu + Dv + E is the Legene trsform of a streunction for a steady, ple, gned o of incompressible second-grade d of nite ectricM conductity, then the ow in the physicM ple (a) a vortex o ven by equations (58) to (51) when A B in L(u, v).
(b) a o with hyperboc strenes with o viables given by (63) when B -A in L(u, v). a vortex flow with flow variables given by ( 65)-(68) when A B in L(u,v).a flow with flow variables given by equations (69) with the streangincs when B A in L(u,v).
APPLICATION II" We let L(u,v) (Au + B)v + Cu + Du + E (70) to be the Legendre transform function, where A,B,C,D,E are arbitrary constants and A is nollzero.
(ii) L(u,v)=(Au+B)v+Du+E,f(u,v)=Dx-Using L(u,v) and ](u,v) for the two cases and proceeding as in application I, the flow variables in the physicM plane are obtained to be: given by equations (77) having Cx + Axy ABy + (AD 2BC)z constant as its streamlines when C 0 in L(u, v). a flow wih recangd hyperbol (z B)(y + D) constan its streamlines d is given by equations (78) when C 0 in L(u, v).THEOREM VI.HL(u,v) A(u + B)v+Cu + Du + E is the Legendre transform finction of a streamfunction of a steady, plane, ah'gned, inconpressible, intinitely conducting second-grade fluid flow, then the flow in the physical plane is given by equations (81) with Cz + Azy-ABy + (AD 2BC)z constant as its streamlines.
Proceeding as before, we have the following results: ./=0 where N4 is an arbitrary constant.
Summing up, we have the following theorem.
TnwORWM IX.If L*(q,O) AO + B is the Legendre transform function of a streamfunction /'or a steady, plane, aligned, incompressible, Iniiely conducting second-grade auid Sow, then the aowin the physical plane is given by equations (117) with tan -1 () constant as its streamlines.
TrlEORIM X.If L(q, O) At + B is the Legendre transform function of a s/reamfunction for a steady, plane, a/igned, incompressible, in6nitely conducting second--grade Buid ttow, then the flow in the physical plane is given by equations (119) having tan -() constant as its stream/ines.