HODOGRAPH METHOD IN MHD ORTHOGONAL FLUID FLOWS

Equations for steady plane MHD orthogonal flows of a viscous incompressible fluid of finite electrical conductivity are recast in the hodograph plane by using the Legendre transform function of the streamfunction. Three examples are studied to illustrate the developed theory. Solutions and geometries for these examples are determined.

1. Introduction.This paper deMs with the application of the hodograph transforma- tion for solving a system of non-linear partiM differential equations governing steady plane magnetohydmdynanfic flow of a viscous incompressible fluid in the presence of a magnetic field.W. F. Ames [1] has given an excellent survey to this method together with its plications to various other fields.Recently.O. P. Chandna et M. [2.3] used the hodograph and Legendre transformations to study non-Newtonian steady plane Migned and trans- verse MHD flows.O. P. Chandna et M. [4] Mso applied this technique to Navier Stokes eq,ations.In this paper we consider the magnetic and wh,city field vectr are nmt,Mly orthogonM and the dectficM conductivity of the fltid is taken to be fitfite.Since electricM conductivity is finite for most viscous fluid, our acc,unting for finite electficM conductiv- ity mes the flow problem realistic and attractive from both a mathematic md physicM point of view.We study our flows with the objective of deternfining exact solutions to vious flow configurations.The plan of this paper is fdlows: In section 2 the equations e cst into a convenient form for this work.Section 3 contns the transformation of equations to the hodograph pirate so that the role of independent vables x. y and the dependent vibles u. v are interchanged.In section 4 we introduce a Lee,dre transform function of the streamfunctim and obtn a system of thr equations in the Legendre trsform fimction md the proportionty fimction.In section 5. we demonstrate the use of theoreticM results found in section 4 by deternfining solutions to the following flows: (a) vortex flows (b) radiM flows (c) spiral flows 2. Basic equations.The steady, plane flow of viscous, incompressible fluid of fiidte electficM conductivity is governed by the following system f equations: 0,.,, +,, + , + *ja (.) (2.12) of seven partial differential equations in seven unknowns u. ,,.Hi.Hz.w.j and h as functions of z.y.
We consider our flows to be orthogonal flows.A plane flow is,said to be orthogonal when the velocity field vector and the magnetic field vector are mutually perpendicttlar in the flow region.From this definition, we have where k (0.0.I) and f is a scalar fimctiou.Using (2.15) in the system of equations (2.8)  o.f of ('2.22) 0!1 3. Equations in the Hodograph Plane.Letting the flow vm'iables u u(x,.!l), v v(z.!1) to bc such that.in the regi, m ,ff fl,w under cmsideration, the .lacbianwe may consider x. y as fimctions of u. v By means : :( ,,.v).O9 0(.u) 0(0.v) yO(O.j) I 0.. sbZ;;,i 0,,.
Equations in Legendre transform function and j(u.v).
Once a solution L(u. v )./(u.v is found, for which J evaluated from (3.21) satisfies 0 < IJI < .the solution for the velocity components are obtained by solving equations {3.13) sinndtaneously.Having obtained the velocity components u u( , y ).v v{ x, y ), we obtain f( , y in the physical plane fl'om the solution for ](u, v in the h<dograph plane. Using V(z.y) and f(z.y in 2.6 ). (2.7). (2.9) and (2.10), we deternfine other flow variables in the physical plane.
Therefore.L*(q) D,.-oln q + Da is not the Legendre transform function of the stream- function of the flow.

1 .
v axe the components ,,f tile velocity fieht V. HI.H2 e the component of the magnetic field H. p the pressure.( the constant fluid density, p the c, mstant c,cient of scosity d p* the constant magnetic permeability.Here K is arbitrary constant of integration obtned from the diffusion equation curl[VxH-p.alcurlH] =0Introducing the functions o-N.h-5<IV +v where to the above system of equations, we obtain the following system: