SHEAR FLOW PAST A FLAT PLATE IN HYDROMAGNETICS

The problem of simple shear flow past a flat plate has been extended to the hydromagnetic case in which a viscous, electrically conducting, incompressible fluid flows past an electrically insulated flat plate with a magnetic field paral- lel to the plate. For simplicity all physical parameters are assumed constant. A series solution for the velocity field has been developed for small values of a magnetic parameter. The equations governing this flow field were integrated nu- merically It is found that the effect of the magnetic field is to diminish and increase respectively, the first and second order contributions for the skin frlc- tion.

theory at solid boundaries, forms a uniformly valid leading term in the asymptotic expansion of the solution of the Navier-Stokes equations in ascending powers of viscosity for a wide class of problems, provided the solutions obtained do not break down.Attempts have been made to study higher order terms in the asymptotic expansion which take into account the second order effects related to vorticity of the main flow, the longitudinal and transverse curvature, and others.The im- pact of the second order effect is increasingly felt due to its considerable in- fluence on such important fluid phenomena as skin friction and heat transfer.
The effect of external vorticity was first pointed out by Ferri and Libby   Solutions of the two dimensional boundary layer equations for flow over a flat plate, including the boundary condition of finite vorticity at the edge of the boundary layer, have been presented by Li [2,3] and Ting [4] and others.These authors show that positive vortictiy increases skin friction.
The extension of this classical theory of simple shear flow past a semi- infinite flat plate to the hydromagnetic case has been attempted here.The physical phenomena investigated is the shear flow of a viscous, electrically conduct- ing, incompressible fluid past an electrically insulated flat plate in the pre- senceof a uniform field parallel to the plate.All physical parameters are as- sumed to be constants.We wish to study, specifically, the effect of the uniform magnetic field on the second order contribution to skin friction.

THE EOUATIONS OF THE PROBLEM:
The classical boundary layer equations are modified so as to apply to an electrically conducting fluid in the presence of a magnetic field.The simplifi- cation of the hydromagnetic equations appropriate to the present problem, given by Greenspan and Carrier [4]

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The novel feature in our problem is that the free stream has a constant vort- icity.We are particularly interested in the effect of the magnetic field on this shear as the field penetrates the boundary layer and the concommitant effect pro- duced on the skin friction.
Thus, the boundary conditions are u 0, v O, Hy 0 at y 0 (2.4) Here is the permeability of the medium, is the magnetic viscosity of the liquid, ' is the magnetic viscosity and the constant external vorticity.Now we rewrite the free stream velocity distribution as follows:  where is the familiar variable in the boundary layer studies; : is the parameter of external vorticity interpreted as the ratio of free stream vorticity and the average vortictiy in the boundary layer.
Introducing the stream function (x,y) and the magnetic stream function Thus we have two pairs of coupled equations.The first pair of coupled non- linear equations describe the first order flow and magnetic fields.The second pair of coupled linear equations represent the flow and magnetic fields of second order arising due to external vorticity.
--2 2 Here s H o / u o is the ratio of the magnetic and kinetic energies and v/q' is the ratio of the kinematic viscosity to the magnetic viscosity.
The condition u 0 v at y 0 (2.17 Since on the plate, we have A ;P ---0 (2.23) A(x, 0) constant. (2.24) Choosing the llne of force at n 0 in the external flow to be A(x,0) constant, (2.25) and, since this llne passes through the origin, we have A( (2.26) This leads to the conditions h(0) 0 hl(0) Hence, the boundary conditions of the first order are (2.27) (

2.31)
There is mathematical difficulty involved in solving the coupled, non-llnear or- dlnary differential equations for the velocity and the magnetic field with two parameters present.Thus, we attempt a series solution for small s and the re- suiting equations are solved on an IBM 1620 electronic digital computer.

SERIES SOLUTION FOR SMALL s
We look for solution, of the form for small values of the magnetic parameter s.
Substituting (3.1) to (3.3) in the equation (2.15) to (2.17), we obtain for the first order flow and magnetic field, the equations fc0 + f00f00 0 (3.5) fOl + foofol + foofol hooho0 0 h00 -e(hoofo0 hoofo0) 0 h01-e(f00h01 f00h01 f0h00 + h00f01) 0 Now, the non-linear equation uncouples from others and reduces to the Prandtl- Blasius problem.The equations (3.5) and (3.9), together with their boundary con- ditions, constitute the vorticity interaction problem discussed by Li ((3) for the field free case.For simplicity, only equations pertaining to the first power of the magnetic parameter have been taken for investigation.All these equations have been integrated by a Runge-Kutta fourth-order process due to Gill (4).The essential results are given in the following tables.The skin friction at the plat, given as the viscous force per unit area acting at the plate, is given by (5.) r ---y=O (5.2) The first factor within the brackets is the hydromagnetic skin friction for irrotational flow.The second factor is the contribution to the skin friction by interaction with external vorticity and is a measure of the kinematic effect of external vorticity at the surface.The two factors are given for various values of the magnetic parameter in the following table: The effect of a magnetic field on simple shear flow of a viscous, electrically canductlng, incompressible liquid past an electrically insulated seml-lnflnlte flat plate has Seen considered.A series solution in powers of a magnetic parameter has been attempted.The equations governing the flow and the magnetic el have been integrated numerically on a digital computer.
To a first approximation, it is clear that the effect of the magnetic field on the boundary layer is to reduce skin friction.The correction to the first order skin friction for small values of the magnetic parameter s is given by (i .5806 s. The magnetic field tends to +/-ncrease the shear at the surface over that due to external vortlclty.For the fi'eld free case, the kinematic effect of the shear associated with external vortlcity at the surface is 80% of that just outside the boundary layer.There 's a correction of (i + .3988s) to this part of the shear for small values of the magnet'c parameter.
Thus, it appears that a uniform magnetic field parallel to the plate tends to increase the second order contriSution, while reducing the first order con- trlbutlon to the value of skin friction.It is easily noted that the first order term is affected more the the second order term.The above analysis is valid only for small values of s, and the results are given in tabular form.It means topics where a satellite, a rocket, or any kind of man-made object is travelling in space governed by the gravitational forces of celestial bodies and/or forces generated by propulsion systems that are available in those objects.Many topics are related to orbit determination, propagation, and orbital maneuvers related to those spacecrafts.Several other topics that are related to this subject are numerical methods, nonlinear dynamics, chaos, and control.
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Table 6 .
Second order velocity and magnetic field functions.Space dynamics is a very general title that can accommodate a long list of activities.This kind of research started with the study of the motion of the stars and the planets back to the origin of astronomy, and nowadays it has a large list of topics.It is possible to make a division in two main categories: astronomy and astrodynamics.By astronomy, we can relate topics that deal with the motion of the planets, natural satellites, comets, and so forth.Many important topics of research nowadays are related to those subjects.By astrodynamics, we mean topics related to spaceflight dynamics.