H.K. Moffatt- On fluid flow induced by a rotating magnetic field

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  • 8/3/2019 H.K. Moffatt- On fluid flow induced by a rotating magnetic field


    J . Fluid Mech. (1965), vol. 22 , pavt 3, p p . 521-528Printed in Great Britain

    52 1

    On fluid flow induced by a rotating magnetic fieldBy H. K. MOFFATTTrinity College, Cambridge

    (Received 12 November 1964)The interior of an insulating cylindrical container is supposed filled with anincompressible, electrically conducting, viscous fluid. An externally appliedmagnetic field is caused to rotate uniformly about an axis parallel to the cylindergenerators (by applying two alternating components out of phase a t right angles).Induced currents in the fluid give rise to a Lorentz force which drives a velocityfield, which in general may have a steady and a fluctuating component. Theparticular case of a circular cylindrical container in a transverse magnetic fieldis studied in detail. Under certain reasonable assumptions, the resulting flow isshown to have only the steady component, and the distribution of this componentis determined. Some conjectures are offered about the stability of this flowand about the corresponding flows in cavities of general shape.


    1. IntroductionProgress in magnetohydrodynamics has been greatly hindered by the diffi-

    culties of achieving high magnetic Reynolds numbers in the laboratory. Thisnumber R, is a dimensionless measure of the conductivity of the fluid and isdefined bywhere L and V are typical length and velocity scales of the moving fluid, and pand r~ its permeability and conductivity, respectively. The condition R, B 1is usually satisfied in astrophysical or geophysical problems on account of thelarge length scale involved, and many problems have been intensively studiedby theoreticians for the limiting situation of infinite conductivity (or R, = CO).Laboratory experiments with mercury or liquid sodium are usually performedat R, < 1 , or at the highest speeds and largest length scales practicable, atR, M 1. Consequently, it has not yet been possible to observe in the laboratorymany of the effects predicted by theoreticians for the case R, 9 1, in particularthose effects associated with the freezing of the magnetic lines of force in thefluid.The velocity V appearing in (1 .1 ) must be interpreted as the relative velocitybetween the fluid and the magnetic field. It is difficult to obtain a high value of Vin the laboratory simply by moving fluid relative to a stationary magnetic field.

    R, = ~ T ~ c T L V , (1 .1 )


  • 8/3/2019 H.K. Moffatt- On fluid flow induced by a rotating magnetic field


    522 H . K . MoffattA field of high frequency w gives rise to a high relative velocity, V = wL , and thissuggests that high effective values of R,( = 477,uuL2w)may in fact be accessibleby quite easy means. This is no more than a restatement of the well-known factthat a conductor behaves almost like a perfect conductor when immersed in ahigh frequency magnetic field. Henceforth we shall assume

    R, = 4 7 r p ~ ~ L ~ w. ( 1 . 2 )If the fluid is contained inside an insulating container immersed in such a high

    frequency rotating field, then skin current flow, effectively excluding the fieldfrom the interior of the fluid. I n the language of fluids, this skin is a magneticboundary layer. Within this layer the curl of the Lorentz force generates vor-ticity, and a motion is transmitted to the core of fluid. The characteristic velocityof this fluid motion, U say, may be several orders of magnitude smaller thanwL . Indeed U will decrease as B, decreases so that circumstances certainly existfor which the magnetic Reynolds number defined in terms of U ,

    2, = 4Tf,LULU, ( 1 . 3 )R, B 1 , 8, < 1, ( 1 . 4 )

    is small compared with unity. This combination of conditions,is the basis of the work described here; the latter condition may be expressedin terms of the amplitude of the applied magnetic field, when the resulting fluidmotion is determined (see (2.21)and (2.22) below).

    The significance of the condition8, < 1 (which would in fact be hard to violatein the laboratory) is that the magnetic field distribution is unaffected by themotion of the fluid, and may be calculated as though th e ju id were a solid conductor(It s of course far from uniform being zero in the core of the conductor; the dis-tortion of the field is associated with the condition R, & 1 . ) This is a strikingsimplification since the determination of the magnetic field everywhere is now alinear problem of well-known type (Landau & Lifshitz 1960). There remains,however, the formidable problem of solving the Navier-Stokes equations for thefluid motion, under a known distribution of Lorentz forces in the magneticboundary layer. The type of motion t hat is produced will be largely governed ingeneral by the magnitude of the Reynolds number R = UL / v ,where v is thekinematic viscosity of the fluid. The particular case of a circular cylinder in arotating transverse field is greatly simplified by the fact that the streamlinesturn out to be purely circular, and the inertia forces can be balanced by a radialpressure gradient; only in this case is the Reynolds number irrelevant.2. Circular cylinder in a rotating transverse field

    The magnetic field distributionSuppose that the conducting fluid is contained inside the insulating boundary

  • 8/3/2019 H.K. Moffatt- On fluid flow induced by a rotating magnetic field


    Plow induced by rotating magnetic field 523The field and resulting flow are independent of x and are confined to the ( r , @ )plane.It is convenient to introduce the magnetic stream function $B, defined by

    where from ( 2 . 1 ) ) $rg Bor mei(e-wt) as r + CO. (2 .3)(2.4)= [Bor+ g ( r ) ] i(e-wt) ( r. > a),

    We assume that $B = f(r) i(e-wt)

    the imaginary part being understood. For r > a, A B = 0 so that V211. = 0, .e.(V2- / r 2 ) ( r ) = 0,

    i.e. g ( r ) = C / r (C = const.), (2.5)aB/at = hV2B, (2 .6)

    (the solution linear in r being rejected by virtue of (2 .3 ) ) .For r < a,

    (neglecting the effect of the motion of the fluid by virtue of 8, < l ) ,whereh = (47r,ucr)-l is the magnetic diffusivity of the fluid, i.e.

    or, using

    with general solution

    where k = (w / 2h ) * ,The part involving Ydzgives an infinite magnetic field a tr = 0 and must be rejected, i.e. B = 0.The constants A and C are determined by the conditions that both B, and

    Be are continuous across r = a (the possibility of surface currents being dis-missed since cr is finite). These conditions give

    f = Ar J e [ k (1+i) ]+BrY,[k( 1+ ) ] , (2.8)I

    where p = k ( l+ )a.This completes the determination of the magnetic fielddistribution.

    Now R, = 2 ( k ~ ) ~ ,o tha t for R,,9 1 the asymptotic formulae for the Besselfunctions for large kr and p may be employed, e.g.

    ~ ~ ( p )2/np)*os [ p- -t n - n], J:& N - 8 / n p ) * in [ p-2-tn -an],1

  • 8/3/2019 H.K. Moffatt- On fluid flow induced by a rotating magnetic field


    524 H . K. MoflattThis form of the solution may alternatively be determined by first anticipatingtha t for R, > 1 the solution will have a boundary layer character, and that theterms r-l ajar (f i r ) and f i r3 in (2.7) will be negligible compared with a2/8r2 f i rwithin the layer, but there is then a slight difficulty in eliminating the termcorresponding to the Yd2-term hat appears in the full general solution.

    t wfFIUURE . Lines of force ~B = const., when a magnetic field rotates about

    a conducting cylinder.The lines of force, $B = const., are sketched in figure 1 . The first part of the

    solution (2.10), B,(r - 2 / r ) in (8-w t ) , is the same as the stream function forpotential flow past a cylinder; the remaining part of the solution (for ka > 1 )is a small perturbation of this representing the penetration of the lines of forceinto the cylinder. The lines of force which start within a distance O(k - l ) of theaxis 8= 8- wt = 0 intersect the cylinder, intersecting fmt at 8= in and8= 27r. Those lines of force that intersect the cylinder surface within a distanceO(k-1) of the points 8 = $7r, f 7 r penetrate to near the centre of the cylinder, butthe magnetic field is exponentially small inside the skin. The tendency forthe conductor to impede the motion of the lines of force across it is clearly marked.The skin thickness isThe whole pattern of course rotates with angular velocity U . The lines of force infigure 1 have been sketched for a moderate value of R, in order to indicate thepattern. As R, + 00, the pattern inside the cylinder shrinks into the skin, thedotted line of force being singular in the sense that it is the only line of forcewhich penetrates to the centre in this limit.

    6 = O(k - l ) = O(aR;*). (2.11

    T h e velocity distributionThe electric current in the fluid j is purely in the axial direction, and from47rpj = V A B, is given bymaking the thin layer approximation already employed above. Hence from(2.10), 47rpj = 23B, k e-lc(a+ cos 6, (2.13

    4npj = - V2$, M ( a2p r2 ) B, (2.12

  • 8/3/2019 H.K. Moffatt- On fluid flow induced by a rotating magnetic field


    Flow induced by rotating magneticJield 525where k s a unit vector in the z-direction. Using (2 .10) and (2 .1 2 ) , this gives

    V A (jA B ) = +Bi Ic/n,ur [- in C(sin g- os5 )- os COS 5+sin 5 )] -2k(a-T)= - Bi / n ,ur )e-2k(a-r). (2 .15)

    The rather surprising disappearance of the dependence on (0-ot) s due to thefact that j nd $* are exactly out of phase at every point. (This is certainlyuntrue if R, is not large, in which case the flow would have a steady and afluctuating component.)

    Since the rate of production of vorticity in the skin depends only on r , thestream function $ of the resulting velocity field likewise depends only on r ,and the equation determining $ (again using the thin layer approximation toreplace r by a n the denominator of (2 .1 5 ) ) is

    vV4 y? = (BEk/n ,upa) -2k(a+),where p is the fluid density. The particular integral is

    = ( B ~ / 1 6 n , u p v a k 3 )-2wa-r),

    (2 .16)

    and the complementary function is$, = A'r2 +B'log r.

    The combination satisfying a$/ar = 0 on r = a and a$pr finite at r = 0 is

    The velocity distribution is purely in the &direction and is(2 .17)

    (2.18)In the core region, defined by k ( a- )% 1, the velocity is the rigid body rotation

    u0- i r (2 .19)M2

    8n,upva2k2- RLBi --Q =ith angular velocity (2 .20)where M = ( v / p v ) *B o a (2 .21)is the Hartmann number as usually defined.

    The relation between Rm efined in (1 .3 ) and R, is(2 .2 2 )

    and the conditions (1.4)are therefore compatible asR, + 00 keepingM constant.Possible orders of magnitude for mercury are indicated in table 1. Note that

  • 8/3/2019 H.K. Moffatt- On fluid flow induced by a rotating magnetic field


    526 H . K. Moffattobservable in an experiment. The Reynolds number of the flow, though notrelevant for this case of circular cross-section, is given as an indication of theimportance of inertia forces for the flow inside other cross-sections.

    3. The stability of th e flow inside a circular cylinderThe flow described above has the property that the circulation decreases out-

    wards within the layer between the rotating core and the cylinder which is a trest. Such a flow is prone to instabilities (Rayleigh 1916), and one might expectinstabilities of the Taylor vortex type (Taylor 1923) when the rotation in thecore exceeds some critical value. At high rotation rates, there would seem to bethe possibility of a steadily rotating stable core bounded by a turbulent magneticboundary layer.

    a w R m M s2 2 m R10 106 103 1 10-1 10-3 1031 106 102 1 10+2 10-2 104

    cm sec-l sec-'10 1 0 6 104 1 10-2 10-4 102

    TABLE . Possible orders of magnitude in mercury

    If a small perturbation is superposed on the steady flow determined above,the linearized equations describing the evolution of the perturbation are notinfluenced at all by the Lorentz force, whose only role is to determine the un-perturbed state. The dimensionless number which appears in these equations,and which determines the onset of instability is the Taylor number,

    T = ( c z ! ~ ~ ~ ~ / v ~ )( u / v ~ )LW~GJ/R:)~c z / R ~ ) ~ ,i.e. T = (A/v)' M4R,H. (3 .1Taylor (1923) found that the circulating flow between cylinders of nearly equalradii when the inner one rotates with angular velocity !2 is stable or unstableaccording as T: 1709. The situation studied here is different only in that thevelocity profile in the layer (2.18) and the inner boundary condition on thevelocity perturbations are different; one might expect these differences to changethe critical Taylor number by a t most a factor of order unity. Hence it is probablysafe to say that the criterion for stability of the flow studied here is

    ( A / v ) ~4R;4 < 103a, (3.2where ct is some number of order unity. For mercury, h/v E 105,and the values

  • 8/3/2019 H.K. Moffatt- On fluid flow induced by a rotating magnetic field


    Flow induced by rotating magnetic field 5274. The flow inside non-circular containers

    Consider now the case of a cavity which is cylindrical but not circular. Thisis equivalent to allowinga variation in the curvature ~ ( 6 )f the cylinder surface,where 6 is the distance along the arc of the surface. It seems likely th at theflow will be described by an equation like (2 . 1 6 ) , allowing for inertia forces,and for the variation of curvature of the surface, viz.

    where 7 is measured normally in from the surface. Built into this equation is theassumption, certainly valid for the circular case, that the only effect of the rotatingmagnetic field is to generate vorticity in the magnetic boundary layer at a rateconstant in time, but proportional to the local curvature of the surface. Theremay in addition be a time dependent component of the flow of frequency 2w ,but for large w one would expect this component to be of small amplitude (dueto large viscous damping) and irrelevant to the determination of the mean flow.

    If the Reynolds number R, ased on the skin thickness E - l is small, the velocitycomponent w in the 6 direction a t the inner edge of the skin may be determinedapproximately from (4 .1 ) in the form

    the condition for the consistency of this derivation isR,= ( w / E v ) = ( N 2 / R i )h / ~ ) , (4 -3 )

    and if this is satisfied, the problem reduces to that of determining the flow whenthe tangential velocity a t the inner edge of the skin is prescribed by (4 -2 ) . Thisproblem further subdivides according as the Reynolds number based on the over-all dimension a = K - ~ ,

    is large or small. If large, then there is a further boundary layer with closedstreamlines of thickness aR-8, and inside this a steady flow of uniform vorticity(Batchelor 1956) . If small, then the non-linear terms in (4 . 1 ) are negligibleeverywhere, and the stream-function satisfies the biharmonic equation in thecore.

    If, on the other hand, R, s large, but small compared with R,, then a conven-tional boundary layer analysis would be needed t o determine the solution of (4 . 1 )through the skin. In this case R is necessarily large, and the flow in the core isof the uniform vorticity type.The rate of production of vorticity in the skin is greatest where the curvature

    R = vajv z R, A , (4.4)

  • 8/3/2019 H.K. Moffatt- On fluid flow induced by a rotating magnetic field


    R E F E R E N C E SBATCHELOR,. K. 1956 J. Fluid Mech. 1, 177.LANDAU, . D. & LIFSHITZ, . M. 1960 Electrodynamics of continuoue media, Chap. VII.RAYLEIUH, ORD1916 Scientific Papers, vol. 6, . 447. Cambridge University Press.TAYLOR,. I. 1923 Phil. T r a m . A, 223, 289.

    Pergamon Press.

  • 8/3/2019 H.K. Moffatt- On fluid flow induced by a rotating magnetic field


    On luid flow induced by a rotating magnetic fieldBy H , K . MOFFATT

    J . Fluid Mech. vol. 22, 1965, pp. 621-528t

    am grateful to Dr E. Dahlberg who has recently (Dahlberg 1972)drawn attens . Ito the following points in the above paper which roquirc amendment,(i) There is an error in equation (2.7)which affects equations (2.8) aitd (2.9)t none of the subsequent equations or discussion. The operator V 2 - 2 / ~ 2 inshould be replaced by V2- / r 2 ,and the order of the Bessel functioiis inand (8.9) should be 1and not J2 (see footnote to Siieyd 1971, p. 818).(iiJ The sketch of the linesof force (figure1) suggestsa slight transverse dis-from the cylinder; this :Y spurious.(iii) The parenthetical statement after equation (2.15) This is certabdy

    if R,fiis not Iarge, in which case the flow would havc a steady and acomponent) should be deleted.(iv) The complementary function given after (2 .10) should BeP $2 = Ar2+B10g 4- Cr210g r+D

    of 9% Ar2+B logr,d the word finite in the following line should be replaced by regular.correction does not affect any of the subsequent equations or conolusions.(v) The statement in $ 3 hat the linearized equations dcscribing the evolu-

    of the pcrturbation are not influenccd at all by tho Loroatz force shouldbeby the phrase provided the Hartniann riurnbcr is not large. Thequalification is in fact required to justify the approach adopted in doter-the mean flow.(vi) The plume For mercury, A /v x 106 hould be replaced by For mer-at room temperature, A/v x 6 x 10s; he interpretation of the orders of

    giveii in table 1 requires corresponding minor adjustment.REFERENCES- DAELBERQ,. 1972 AB Atonzenergi,Studmik, Sweden, Rep. no. -447.

    SNEYD, . 1971 J . Fluid Mech. 49, 817.