MAGNETOHYDRODYNAMIC CHANNEL FLOW AND VARIATIONAL PRINCIPLES

ADSTRACT. This paper deals with magnetohydrodynamic channel flow problems. Attention is given to a variational principl% where the boundary conditions are incorporated via a suitable functional which is stationary at the solution of the given problem; the trial functions used for the approximate solution need not satisfy any of the given boundary conditions.


I. INTRODUCTION.
In a number of papers variational principles have been derived for the solution of linear magnetohydrodynamic channel flow of various types.Tani [I] has presented a variational principle for a problem involving the Hall effect.Smith [2] has deriveda functional giving bounds on the flow rate.Sloan [3] has derived a variational principle which can be used to obtain sequences which converge to the real solution.More recently a variational principle has been derived by Barret [4] to the problem of viscous, incom- pressible, electrically conducting fluid flows steadily in a rectangular channel with a uniform transverse magnetic field parallel to one pair of walls, opposite pairs of walls being either insulators or perfect conductors.The equation for the nondimensional axial velocity V, and the nondimensional induced magnetic field B, are coupled and may be written as where M is the Hartmann number.On the boundary, V is always zero and B is zero if the walls are insulators and the normal derivative of B is zero if the walls are perfect conductors.
Barrett and the above mentioned authors assume that the trial functions satisfy the same boundary conditions as the real field.Moreover, attempting to solve the problem of non-conducting walls, Barrett has restricted the choice of trial functions even more by reguiring that one of the governing equations be exactly satisfied.In this paper we develope a variational principle in which the given boundary conditions of the problem A.A. EL-HAJJ are imposed implicitly via suitable terms in the functional and hence the expansion set of functions need not satisfy any of them.

THE VARIATIONAL PRINCIPLE.
The variational techrque derived in this sction, applicable to a wide range of linear non-seif-adjoint problems, is used to derive a stationary principle for a magnet- ohydrodynamic channel flow In the particular case of (I.I) and (1.2) subject to the boundary conditlons V=O and B=O a variational principle is derived by first introduc- ing the folloing adjoint problem of(l.l) and (1.2): where U and C are zero on the boundary.The functional delends on four variables: , , , and T and is THEOR I.The solution to (I.I), (1.2), (2.1) and (2.2) renders the following functio- nal stationary at the solution points V, B B, m U and T C J(, 8, , T) {-V" Vv-VT.VB+MB/x+MTv/x+u+}dS where 8s is the boundary of the region, Vdenotes differentiation along the normal and ds is a boundary element.
PROF.Let V and B be the solution of (1.1) and (1.2); U and C the solution of (2.1) and (2.2); ,, , 6 and arbitrary functions.
NOW, dF1 Is-VU" 7i dS I s VU 4 IsV nds (2.5)   and the fact that The f.rst .ntegral in (2.7) is equal to zero by (2.1) and it is obvious that the Iane Integrals cancel one another.Hence, dF,(0)/de --O.
In much the same way, it can easily be shown that dFi(O)/de 0 for i 2, 3, 4. Hence.
the functional J is stationary at the solutions of the given problem and its adjoint.
The variational pr+/-nciple derived in the privous section will now be used to obtain the defining equations for the approximate solution to the problem of channel flow with non-conducting walls.
We +/-ntroduce appropriate basis functions--typically, a set of product orthogonal poly- nom'ials and a global expansion of the solution is made: Inserting these expansions into (2.3) and finding the stationary value of the functional leads to he symmetric block matrix equation for the coefficients a_(k), k= An important feature of our technique is that the basis functions hi(x0y) are not required to satisfy any of the boundary conditions of the problem; these conditions are imposed mplicitly by the functional, and are satisfied exactly only at the solution point.We only require, for stability reasons see Mikhiln [5]), that the set of basis functions be a set of product orthogonal polynomials.