SINGULARITY METHODS FOR MAGNETOHYDRODYNAMICS

Singular solutions for linearized MHD equations based on Oseen alproximati(,r.s have beer obtained such as Oseenslet. Oseenrotlet. mass source, etc. Cy suitably distributing these singular solutions along the axes of symmetry of an axially sy..metric bcdies, we derive the approx;mate values for the velocity fields. he force ar.d the momentum for the case of translational and rctational motions of such bodies in a steady flow of an ircompressble viscous and magnetized fluid.

The motion of a body in a steady flow of an incompressiblE, viscous and mau.etizeo fluid is gover,ed by a set of nonlinear equations known as magnetohydro- dynamic (MND) equations.Exact solutions for these equations have bee.obtained only for a few very specific problems.However, or many applications these equations can be lnearized ;y using two ir.earization schemes known as Oseen and Stokes approxima- tiens [1.2].
A method of singularities has been developed recently to solve various boundary value preblems in matenoical Fhysics dispi.essuch as potential theory, scattering thecry [5], h,rodynami(.[6.7], en elasticity [8].Our aim n this paper is to extend this metho to solve some boundary value ,roblems i MD. using Oseen's appro:<:ration of the V, HD equatlon..In Section 2 we present the mathl,,atical formu'dtion of the Eo:uations oi the basis of this aPF, Yf.,ximation.In Sectiop 3, we pre:.e:t the fundomeptal solution (singularity) of Oseens equatior, s and cclstruct other sipoulor;ties reded in our analysis, including Ocens rotlet, Oseens doubl., (!seens stveslet.In the last two sections we so le two typ(:s ef motion problems for o>-:!Iv symmF..ic bedes by s11itably distributin 9 singularities about their axes of symmet,',;-First, ti.E steady rotation of the bocies about their lonstudinal a><s; ttccnd, the uF,form Lranslational ruction of those bodies in the direction of their axi-L, symmetry.C.(r, iguratior.. of interest in this tudy are pro!te and ob]t _pi:erinds and *Peir limit ipg cases including the sphere, the circular disc, and the slende' body.For these probler,;, we derive formulae for the velocity fields along with tF.plysical qt:cr,tities, the drag and the force.
ll nen-dimensicnal equations governing the steady flow of d: incompressible, visccus, electrically copducting fluid are where R aU/u is the Reynolds number, R m OeaU is the magnetic Reynolds number, o 1/2 and M PeHo a () is the Hartmann number.
TI:e vectors , and are the velocity field, the electric current density, he magnetic field strength, and the electricl field strength, respectively.
The cohstants p, , u, u e, and c are the fluid density, pressure, kinematic viscos,ty, magnetic permeability, and the electric conductivity.The constants U, d, an H aFe the typical velocity, characteristic length, and the uniform o magnetic field.
It is assumed that the magnetic field is orieted in the e dection i.e., along the x-axis.Furthermore, for the steady rotation problem the typical velocity is U a q where R is the uniform angular velocity, and for the tar.slational motion U is the velociSj' of the uniform flow in the e x direction.
The O, seeFs apprcximation replaces the convective (non-linear) terms in equations (2.]) b) convectiop due to the uniform velocity and uniform magnetic fields at iT, finity.Furthermore, bcause of t.e symmetry conditions, the electric field E is taken to t.e zero.So writing the velocity field and the magnetic field H as and H e + H' x in equations (2.1), neglecting the quadratic terms, and dropping the primes, we obtain the f(jllowing lir, ear sysi-e:, [2]" . , . (2.4) ui 2 R + vP r C'. ( ,,x EKe E and R_ are the roots o. the equatior, Z M 2 R {R + Rm) R 'P'm M2 O, and P p + -H-e (2.6) m O,.,eer,s aplro, imation is volid for Rr, -<I apd R/M 2 <<1, hat is, when the magnetic ieid oni;,ates over the inertia f(,rces; however, it is valid for small and large P, al' Llqdnn number. in th presence of : solid, the boundary conditions associated with the above s,'.tem of equations, in 6ddition to the no-slip copdition ( O on the surface), are H C inside and c, the solid since it is ir.sulated, and all the perturbations must vanish at iniinity.Two ir.,por*ant special cases emerqe from the above system of equations" FLsly hn Pi R 0 equations (2.4) and (2.5) reduce to Stokes flow of non-conducting fluids.Secondly, the case IP, R # O) gives the equations of Oseens flow.where g forcing function having some singular behavier in an infinite medium, are called fundamental solutions.The primary fundamental solution is called the Cseenslet .pd it correspcnds to d forcing function 8.-6() where a is a constant ector, and 6() s the three dimensional delta function.

dt (3.3)
The velocity and p[essure for the Oseenslet are Tki where r lI and i li lhe r,c.t force experienced by a control volume containing the Ossenselt is given by 8 (.5) l,.e ]inearlty of Oseep's equation nplies that derivatives of any order of the Cseenslet iF an arbit'E, ry fixed direction is again a solution of (3.1), ith a forcer.9function Faving the same derivatve of .Thes derivatives can be obtained easily f,y conslderir, g the Taylc.rsries expansions of the velocity and the pressure of the Oseenslet abcut a fixed point # O, that s, with a similar expa,sion for the pressure pCS (_,e) The first term in (" ,) is tt, Oeenslet itself.The second term is the "OseensHoublet" and the third one is the ..(.CnSQuadropole".he Oseensdeublet is oiven by eki (.9) The Oeensdoublet can be written as a s,m cf antisyetric and symaetric (witt respect to intercancing and ) team, s, respectively calle "Oseensrotlet" and "Cseensstresslet" as in hydrodynamics Stokes flow [7].The velocity and the pressure of the Oseensrotlet Ere give by ki(x-r) e T (:x) ( IO) orti (,y) v x r p?r (,) 0 (3 11)   th the corresponding (orcing functien -gor -4 v x () y (3.12) The net torque exerted by an Oseens rotlet enclosed by a control volume on the surrounding fl,id is -4, y (3.3)The velocity vector, the pressure, and the forcing furction of the Oseens stresslet are eki (x-r) -SS(x,,)u hue to the symmetry property this singularity contributes neither a net force nor a r,omentum to the surrounding medium.
Another sir.oularity which is useful in the present study i. called "mass source".Its velocity, pressure and forcing functions are pmS () Solutic,,s (,f variou'-boundary vaue problems in MHD irvolving the motion of xily syiT,etric bodies can be obtained by superposition cf flows due to a suitable distrit, utlon uf some of these singular solutions along tt' axis of symmetry of the b,c).This wl! be demonstrated in zl following sections.
Formulae for the torques on the rotating sphere and rotating circular disc about its ('iameer have been obtained previously by severa authors [9,2] using different techniques, up to the third order.Those results are special cases of (4.26) and (4.27) whe ak bk.
M/2 while the fourth order term appear to be new 5.
In this sectlon the pYolate spherold (4.1) is assumed to have a unifurm velocity U orected along its a>is o symmetry.In this case the velocity will be ob;ained by eI;loylni a i,,e distribution1 of Oseenslets in the e directicr with strength x t(x): and a line distribution of mass sources with strength h( , between the foci of tile spleriod.Thus the solutior will have the followir, c. functional expression c i(;-) U x + f(t) iis(-, x dt -c c f h(t p.ns(_) dt (5 I)   -c On the sbr;a(.e of the sheroid the no-slip cGndition gives the follewing integral equation ,or f(t) and h(t).
The first system is v2j p 0 v. u which describes the steady Stokes flow, thus all the previous results in Section 4 and 5 reduce to that of CHWANG and WU [6,7], by putting k.O. Secondly, the system associated with the root R is while the second order tern appears to be new.
(4.6}is the same integral equation which appears in the rotational motion of pro'ate spheroids in Stokes flow.Thus its solution is Equation