H.K. Moffatt and J. Toomre- The annihilation of a two-dimensional jet by a transverse magnetic field

8/3/2019 H.K. Moffatt and J. Toomre- The annihilation of a two-dimensional jet by a transverse magnetic field

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Q.4 72.J . Fluid Mech. (1967), vol . 30, part 1 , p p . 65-82Printed i n Great Britain

65

The annihilation of a two-dimensional jet by atransverse magnetic field

By H. K. MOFFATT A N D J. TOOMREDepartment of Applied Mathematics and Theoretical Physics,Silver Street, Cambridge

(Received 4 February 1967)The effect of an applied transverse magnetic field on the development of a two-dimensionaljet of incompressible fluid is examined. The jet is prescribed in termsof its mass flux pQo and its lateral scale d at an initial section x = 0 . The threedimensionless numbers characterizing the problem are a Reynolds numberR = Qo/v ,a magnetic Reynolds number R , = f ig&, ,and a magnetic interactionparameter N = uBtd2/pQ,,where p represents density, r~ conductivity, ,U per-meability and B, applied field strength, and i t is awumed that

R,g 1 , R > , N < 1.It is shown that when M2 = RN B 1 , an inviscid treatment is appropriate,and that the effect of the magnetic field is then to destroy the jet momentum

within a distance of order N - l in the downstream direction. A general solutionfor inviscid development is obtained, and it is shown that a large class of velocityprofiles (though not all of them) are self-preserving.When M2 4 , it is shown that the viscous similarity solution obtained byMoreau (1963a,6) is relevant. This solution is re-derived and re-interpreted;i t implies that the jet momentum is destroyed within a distance of order RfN-fin the downstream direction.

Some further aspectsof the jet annihilation problem are qualitatively discussedin 0 4, iz. the nature of the overall flow field, the effect of the presence of distantboundaries, the effect of increasingR , to order unity and greater, and the effect ofoblique injection. Finally the development of a jet of conducting fluid into a non-conducting environment is considered; in this case the jet is not stopped by themagnetic field unless a return path outside the fluid for the induced current isavailable.

n

1. IntroductionThe effect of a transverse magnetic field on the development of a two-dimen-sional submerged je t of conducting fluidbas been examined by Moreau (1963a,6 ) .He obtained a similarity solution of the equations of magnetohydrodynamics(in boundary-layer approximation) representing a balance between inertia,magnetic and viscous forces. The solution revealed that the magnetic field tendsto destroy the momentum of the jet and to cause it to diverge at a certain finite

5 Fluid Meoh. 30

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6 6 H . K . Moflatt and J .Toomredistance downstream from its point of origin. The same solution has beenobtained by Tsinober & Shcherbinin (1965).The solution is of great interest in that it is one of the very few in which fullynon-linear effects compete with magnetic and viscous effects to establish thepattern of flow. However, the solution has certain limitations which have not asyet been fully appreciated. Like any similarity solution, the Moreau solutionmust be understood as being asymptotically valid at a large distance from themomentum source. However, the fact that the jet diverges at a certain finitedistance downstream from the source means that there may not be sufficientspace for the jet to settle down to its asymptotic form before it diverges. It willemerge from the analysis of this paper that the condition for the flow to settledown to Moreaus similarity form before the jet diverges is that the Hartmannnumber M = ( R N ) * should be small. If M 9 1 then the Moreau solution isirrelevant, and an alternative description of the flow is required. It is the purposeof this paper to provide such a description. The analysis applies in particular tothe inviscid limit U-+ 0 (keeping all other parameters fixed).

Consider for definiteness the situation depicted in figure 1a (dimensionalvari-ables are distinguished by an asterisk). Incompressible conducting fluid fills thespace x* > 0 , ly*l < yod, z*( < x o d , and a uniform magnetic field B, = (0 ,B,, 0 )is externally maintained. Fluid is steadily introduced across the boundary x* = 0with x*-component of velocity U(y*/d); he y*-component of velocity on x* = 0will for the moment remain unspecified. It will be supposed that U(y*/d) issummetrical about y* = 0 , has a single maximum U at y* = 0 , and has a finiteflux,t CO2Q0 = Udy* = 2U,d say, (1 )

-02

where d is the lateral scale of U(y*/d).We suppose, moreover, that y,, z, 9 1 ; infact we shall be interested in the nature of the flow at fixed values of the dimen-sionless variables y = y*/d, x = x* /d in the limit y,, zo-+ CO. It is to be expectedthat the (dimensionless)velocity fieldu(x) ecomes two-dimensional n the limitz,, -+ CO , i.e. that = ( N X , Y ) , V ( X , Y ) ) = ( w a y , -a@/ax), (2)whereQ, @(x, ) is the stream-function of the flow.It is reasonable to suppose that@(x,y ) is insensitive to the nature of the boundaries y = & y,, x = f ,, e.g. as towhether they are porous or impermeable, electrically conducting or insulating,etc. It will appear below, however, that the pressure distribution is to some extentaffected by the conditions on these distant boundaries. Possible effects due tofinite values of y, will be considered in 5 4.

The prescribed condition on the boundary x = 0 is perhaps somewhat artificial,and requires comment. It may be imagined that the boundary x = 0 is a rigidsheet of material of variable porosity proportional to U(y); if fluid is continuallysupplied to the region x < 0 at a pressure higher than the pressure a t x = 0+ ,then the steady velocity profile U(y)can (in principle) be maintained. In practice,the simplest possibility is that the fluid is supplied through a slit Iy*I < d, in the

t The momentum flux defined in (10)below is then also necessarily finite.


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The annihilation of a two-dimensional jet 67plane wall z= 0 (figure l b ) . In this case, viscous forces may affect the velocityprofile U(y) , although perhaps the most reasonable possibility is that the profilein the slit is an undeveloped ' op-hat ' profile U(y) = const; this possibility is allthe more reasonable if the Hartmann number based on d is large, since the well-known Hartmann profile for channel flow then has the same top-hat structure;(d = d, in this ca8e).

t;" t Bo

( b )FIGURE. Symmetrical injection of a two-dimensional jet into a regionpermeated by a uniform transverse magnetic field.

It will be assumed thatR, = ~ I J Q O 4 1, (3 )

where IJ is the conductivity and ,U the permeability of the fluid, and distortion ofthe magnetic field will be neglected (Shercliff 1965, 0 3.8). (Qualitative effects ofincreasingR, will be considered in 0 4 .3 . )

The governing equations are thenpu*.V*u*= - V * p * + j * ~ B ~ + q V * ~ u * , (4)

( 5 )(6)

j* = a(E* U *A Bo),v*.u*= v*.j*= 0.Moreover, under steady conditions, the electric field E* is derivable from apotential, E* = -V*$*, and equations ( 5 ) and (6) together imply that

V*2$* = V*.(U*AB,) -B,.(V*AU*). (7 )5 -2


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68 H . K . Moffat and J . ToomreHence in the two-dimensional region of the flow (i.e. everywhere except nearthe distant boundaries z = zo), V*2#* = 0. The electric field in the two-dimensional region is therefore unaffected by the flow in that region and isdetermined only by the conditions near the distant boundaries z = & zo. Since thefield E* is independent of z far from these boundaries, we must have

aE:lax* = aEfiaz* = 0, aE:lay* = aE,*laz* = 0,so that E,*= const. The value of E: is determined by the electrical properties ofthe boundaries z = +_ zo , y = +_ yo (see the last paragraph of this section).

In terms of the dimensionless stream function $(x, ) , the curl of (4 ) reduces tothe dimensionless form

where R = Q0lv, N = aBid2/pQo, (9 )the Reynolds number and magnetic interaction parameter respectively.momentum flux at x = 0,

If the flow in the region x > 0 is t o have the character of a jet, then clearly theLa2F0 =I U2 d y = k Q i d - l , say, (10)-La

(wherek s a constant of order unity characterizing the shape of the initial profile)must be sufficiently large for the inertia force to dominate (over a considerablepart of the flow field) over both the viscous force and the magnetic drag,represented by the term N a2$lay2 in (8).It will therefore be assumedt that

N < l , R 3 1 . (11)The conditions (3 ) and (11) are realistic for mercury jets in transverse fields(Moreau 1966cc,b ) . Under these conditions, it is to be expected that the jet willhave a long thin character, a t any rate for some distance downstream, and theusual boundary-layer approximation, V2 M a2/ay2is legitimate; it may of coursetranspire that the approximation breaks down in certain regions of the flowfield. Equation (8) then takes the approximate form

which may be integrated once with respect to y to givea$ a2$ a$az$ a$ a3$ay a x a y ax ay2 ay ay3 *- N - - R - - I -

The arbitrary function of x which appears on integrating has been set equal tozero, to conform with the conditions that U ( = a$.jay) and the derivatives aulay,a2u/ay2 hould vanish as y -+ +_ CO.$

Some comments on the nature of the flow if N 1 are given in $4.2.$ A sufficient condition would be U = o(ly1-l) as Iy /+ CO ; this condition is satisfied bythe initial profile U(y), and the subsequent analysis confirms th at i t is invariably satisfiedby U@, Y) (x> 0).


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Th e annihilation of a two-dimensional je t 69Equation (1 3 ) is simply the x-component of equation (4) n the boundary-

layer approximation, and i t isevident that the vanishing of the arbitrary functionof x outside the jet region is equivalent to the condition

ap*px* = - Bo, (14)which follows likewise from (5)and (6)with U = 0; if E: + 0, a pressure gradientap*lax* is established to counterbalance the j*ABo force outside the jet. Asmentioned above, the presence of an electric field has no effect on the velocityfield, but it does affect the pressure distribution.

In terms of the velocity distribution U , equation (13) takes the form(U. ) U = -NU+R-l a2u/ay2. (15)

It is easy to see that this implies that the jet momentum is destroyed by themagnetic field. For, integrating from y = -CO to y = f oo we get

whereP nd Q are the (non-dimensionalized)momentum and flux at the section x.Thus, as long as Q > 0, F decreases monotonically with increasing x. Thisdecrease is directly due to the magnetic drag experienced by each fluid particle;it is also affected indirectly by viscous forces, in th at they can (through viscousentrainment) locally increase the value of Q and so accelerate the decay of F(see 5 3). Incompressibility of course requires that the destruction of momentummust be accompanied by a spreading of the jet in the lateral directions; thisspreading is symmetrical due to the assumed symmetry of conditions at x = 0.At some stage, the assumption ajay % ajax must break down; however, providedATis sufficiently small, equation (16) suggests that the jet will retain i ts jet-likecharacter for some considerable distance downstream.

The destruction of jet momentum raises the question of how the over-allmomentum balance of the fluid and its boundaries and any external electriccircuits is maintained. A detailed description would require a consideration of theprecise effects of the boundaries y = f o, z = f o , with the attendant three-dimensional difficulties. However, the essential nature of the balance may beunderstood from t he following simple arguments. The jet momentum isdestroyedessentially because a current j , = PUB,, s induced in the jet region, and theassociated Lorentz force - aBi U is retarding. The current circuit must howeverbe completed in some manner that is determined by the distant boundary condi-tions. The return path for the current may be either through the fluid outside theje t region (as must happen, for example, if all the distant boundaries are insu-lating), or through the boundaries if they are perfectly conducting. In the formercase, there must exist an electric field E: and an adverse pressure gradient givenby (14); the momentum balance is then provided by the pressure distributionthat must be applied on the plane x = 0 to maintain the flow. In the latter caseth e integrated Lorentz force in the jet is balanced by a net Lorentz force acting


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70 H . K . Moffatt and J.Toomreon the boundaries that conduct the return current. I f the boundaries have finiteconductivity then the return current flows partly through the fluid and partlythrough the boundaries.2. Behaviour of an inviscid jet ( R N B 1)

In the limit v -+ 0, or R + CO, equation (15)becomes, in Lagrangian form,D u / D t = (U. ) u = -Nu. ( 1 7 )

(Itwill appear in retrospect that the necessary condition for the neglect of viscousforces is R N B 1 .) The solution is, evidently,

(18 )(xo, ) = u(xo,o ) -N(t-to),for the fluid particle which was a t xo t time to. The x co-ordinate of this fluidparticle a t time t is determined by

Dx/Dt = u(x0,) = u(xo,o) -N(t-lo),so thatHence, as t+ CO, a fluid particle initially on x = 0 asymptotically approaches theline

x - o = N-Iu(x0 , o ) 1- -N(1-4)). (19)x = N-lu(0, yo). (207)

Since, by definition, $ is constant on streamlines, it follows thatW A Y ) = $(O,yo);

N x , ) = %I($) Nx,hence (18) and (19 ) may be written in the Eulerian formwhere u0($)= u(0,o). Since

= agiay, so that $ =logay ,uo($)may be regarded as known in principle (in its range of definition 111.1 < 1)if u(0, ) = U(y) is known. Then from (21 ) ,

and this implicitly determines $(x, y).meaning of the above solution clear.dimensional form.

The following examples of possible initial profiles will help to make the(i) First, suppose that the jet has a top-hat profile at z= 0, i.e. in non-

t The analysis is invalid if the initial profle has more than one maximum; for in thiscase there exist points (0, yl) and (0, y2) with 0 c y1 c y, for which u(0,y2) > u(0,yl),and according to (20) the streamlines through the points (0, yl) nd (0, y2)would intersect.Lateral pressure gradients, represented by the term ap/ay, must prevent this unphysicalbehaviour; in other words, the boundary-layer assumption a/ay + a/ax is necessarilyviolated.


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The annihilation of a two-dimensional jetThen, for y > 0, on x = 0,

@ = {; :;;:;:IHence %I($)1 ( I$ \< 11,and from (21) , a $ P Y = +,y) = 1-Nx ( I $ / < 11,i.e.and of course, $(x, - ) = - $(x,y).

7 1

(25)

J = 1IIx=O

FIGURE. Annihilation of an inviscid jet by a transverse magnetic field. The flow issymmetric about y = 0, and only the regiony > 0 is sketched: ( a ) op-hat profile a t 2 = 0;( b ) secha profile at z = 0. Boundary-layer theory breaks down between the broken linesin both cases.The flow is sketched in figure 2 a . The flow in the jet region has the same

streamlines as the inviscid flow near a stagnation point on a rigid boundary.Outside the jet, i.e. for IyI > (1-Nx)-l , the fluid is at rest. The magnetic field hasan effect similar to that of a rigid boundary placed at x = N - I . It would bemisleading, however, to press the analogy too far; the streamlines within the jetare identical with the streamlines in a stagnation point flow, but they are notidentical with the streamlines of the flow in an inviscid jet impinging normallyon a rigid wall.

The transverse velocity v = - $/ax is given, from (26) , by


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7 2 H . K . Moffatt and J . ToomreTwo observations may be made. First, within the limits of an inviscid analysis,v cannot be prescribed arbitrarily (in addition to U) on x = 0 ; t is determined bythe solution ( 2 7 ) . If the fluid is constrained to have a distribution of v(0 ,y)inconsistent with ( 2 7 ) ) e.g. by the use of guide vanes a t the slit) then presumablythere must be a viscous boundary-layer on the plane x = 0. If the naturalcondition w(0, ) = 0 is imposed, then this layer is weak when N < 1.

Secondly, it is clear that the boundary-layer approximation (a/ax< a/ay), onwhich the solution ( 2 6 ) is based, breaks down at values of x where v becomescomparable in magnitude with U. The maximum value of v at a section x isN(1-Nx)-l, and this is of the same order of magnitude as u(x,0 )= 1-Nx whenx = N-1- 6 where 6 = O(N-3). The above analysis is therefore valid only for

l - N x 9 Nt; ( 2 8 )in particular, it tells us nothing about the nature of the flow for x 2N-l. Thecondition N < 1 ma,y now be seen to be an essential prerequisite for the use ofboundary-layer methods.

The breakdown of boundary-layer theory a t 1-Nx = O(N4) s associated withthe growing importance of pressure forces as this region is approached. Thepressure distribution that is implied by the solution ( 2 6 ) may be obtained byintegrating the y-component of (4), nd in non-dimensional form this gives,within the jet, p +Ex = - +N2[y2- 1-Nx)-2] +const.,where E = ( a B o d 3 / p Q g ) ,*a n d p = ( d 2 / p Q i ) p * .Hence

aplax+E = ~ y 1 -X)-3.The term on the right is neglected in the boundary-layer approximation. Itsactual effect is probably to cause the jet t o diverge a t a value of x a little less thanthe value x = N-1 suggested by boundary-layer theory.-=+ 1 Suppose now that the initial profile is

d

I u(0,y) = sech2y, @(O,y)= tanhy,so thatEquation ( 2 3 ) hen becomes uo($) = 1-p.iThe streamlines are sketched in figure 2 b . I n this case each streamline has adifferent asymptote, the streamline through (0 ,yo)having the asymptote

x = N-lsech2y,.The slower moving fluid near the outer edges of the jet is stopped a t a n earlierstage than the faster moving fluid at the centre. The flux 2Q = $(x, CO) - (x, - O)decreases continuously from 2 a t x = 0 to zero at x = N- I .

1c.(l--NX)*(1-Nx)-fr tanh-l


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The annihilation of a two-dimensionalj e t 73The maximum value of lvl at the section x is in this case of order N (1-Nx)-&,

and this becomes of the same order as U(%,0) = 1-Nxwhere 1-Nx = O ( N 8 ) .Boundary-layer theory therefore in this case breaks down at a distance O ( N - ) )short of the stopping plane x = N - l .

(iii) In the examples (i) and (ii) given above, the velocity profiles are self-preserving in the sense that in both cases it is possible to express $(x, ) in the

(33)formIn case (i),q = 1, 6 = (1 Nx)-l , and

$(x, ) = d x ) f 71, r = Y / 0 4I7 (I4< I ) ,

- 1 ( r < 1 1 ,f ( r )= 1 1 ( r > l L i (34)while in case (ii), q = (1 N x ) t , 6 = (1-Nx)-$,nd f ( r ) tanhr . It is easilyverified, on inspection of (23), that any initial profile of the form

uo($) = 1- I$ \ ( I$\ < 1,P 3 01, (35)is likewise self-preserving.?example, the initial profileNot every initial profile however behaves in this simple way. Consider, for

2 TY$ ( O , y) = - an-I-.u(0,y) = 1+ (7Ty/2)2 7r 2For this case, uo($)= c o s 2 ( n $ / 2 ) , nd integration of (23) leads to the stream-

(37)unction $(x,y) = 2 an-I ( ( l ~ ~ ) t a n h 2 ~ N x ( 1 - N x ) l i=Y7Twhich cannot be expressed in the form (33).

The neglect of viscous forces is justified only if they cause negligible change inthe je t profile over the distance N - l to the stopping plane. I n fact, viscous effectsdiffuse across the width of the velocity profile within a distance of order R down-stream from the initial plane x = 0; hence the inviscid analysis of this section isreasonable provided R $ N - l , i.e. provided

M 2 = R N = aBid2/pv 1. (38)If this condition is satisfied, then the jet is stopped long before viscous diffusionhas any significant effect.

3. The effect of viscous entrainment when M 2 = RN < 11.Then the magnetic force may be expected

to have negligible influence over a range x 5 O(R) ver which viscous diffusionthoroughly redistributes the initial distribution of momentum in the jet. I n therange in which x is large compared with R but still small enough for magneticforces to be negligible (see (51) below), the well-known similarity solution

t In fact, by substitution of ( 3 3 ) into ( 1 3 ) (with R = CO), it may be shown that (35)

Suppose now instead that M 2

is also a necessary condition for self-preserving flows.


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7 4 H . K . Mofla t t and J . To o m r e(Bickley 1 9 3 7 ; Goldstein 1 9 38 , $ 5 7 ) may be expected to apply. This solution isusually expressed in terms of the jet momentum 2F0 defined in (10).In terms ofthe dimensionless variables used in this paper, Bickley's solution may be written

9 k ( x+ x o ) k R 2 *@(" y, = [ 2 8 ]" tanh [ 8 ( x+ ,)2] y, ( 3 9 )where - o = O ( R ) s the 'virtual origin' of the jet. The exact value of xo dependson the initial profile on x = 0. The velocity profile corresponding to ( 3 9 ) s

It is a lucky coincidence that one of the self-preservingprofiles considered in theinviscid analysis in $ 2 has a similar 'sech2' dependence on the transverse co-ordinate y , It is fo r this reason that the solution given by Moreau ( 1 9 6 3 a ,b ) ispossible. (It may be remarked that there is no self-preserving solution for theproblem of the two-dimensional jet in an aligned field.)The following derivation of the relevant similarity solution differs from thatgiven by Moreau in certain respects. The simple Lagrangian form of the solutionwhen v = 0 suggests that we use ( x ,@) as independent variables instead of (x, )(the 'von Mises transformation').? Since

equation ( 1 5 ) akes the form

Let us look for a similarity solution of the form4 x 9 $1 = u l ( x ) f 71, 7 = @ / Q ( x ) * ( 4 2 )

Here u l ( x ) s the velocity on @ = 0 (i.e. on y = 0) so that f (0)= 1. Since @ variesfrom -Q to Q as y varies from -00 to 00, the range of 7 is - to 1 . Substitutionof ( 4 2 ) n ( 4 1 )gives

(43 )U; f - " ' 3 7 f f = - h l + R - l A ( f l f ) f .2Q Q 2Now suppose that f can be expanded in Taylor series

00f = 2 atnv2% (a0 = 1) ' ( 4 4 )n= Oconvergent in some neighbourhood of 7 = 0. The series contains only evenpowers of 7 since the velocity profile is symmetrical. Then

t A similar approach has been explored by Moreau (private communication).


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The annihilation of a two-dimensional etOn 7 = 0, equation (43)becomes

U; = - N +2a2u2,1RQ2;moreover, equating the coefficients of yzn in (43) gives

75

(46)

These equations are clearly independent, and since there are only two functionsul(x) and Q(x) at our disposal, we can satisfy only one of the equations (47) inaddition to (46).Hence the coefficientsa2, must vanish for n = 2,3, . .t and thenfrom (45), C, = 0 for n = 3,4, ...). Hence f = 1+a2r2 . Since the series (44)terminates, it is (trivially) convergent throughout the range 171 < 1, and theconditionf(1) = 0 implies ap= - 1, i.e.f ( r ) 1 r 2 . (48)As might be expected, this corresponds o the sech2 profile when (42) s expressedin terms of (x,y).Equation (47),with n = 1, now integrates to give

where xois a constant of integration, and (46) then integrates to give

where C is a further constant of integration.Comparing (50)with (40)shows that C = (3k2/32)* =O(l ) ) , nd tha t magneticforces are in fact negligible for (x+xo)< RiN-4, or, remembering that xo = O(R)and that M2 = RN < 1, forThe flux in the jet for x B xo is given from (49)by

x < RiN-2. (51)(52)x ) = (%) (CRW - N X ~ I + .

x = 0.545C%R4N-g = x,, aay, (53)

(54)

6 4This reaches a maximum at

and falls to zero a t the stopping plane,x = 1.24CPRiNd = 1.24CPM i N - l w 2 . 2 8 ~ ~X, say.

This is to be contrasted with the corresponding result x, x N-l valid in theinviscid limit M 9 1 (03). When M < 1, the result (54) clearly indicates theextent to which viscous forces can modify the annihilation process.t We must have a, * 0, since otherwise, equations (47) successively imply a4= 0,

aB= 0, ,and the condition f( ) = 0 cannot be satisfied.


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7 6 H . K . Moffatt and J . ToomreAgain, boundary-layer theory breaks down as the stopping plane x = x, is

approached. It may easily be verified that 12i(x,00)l = IdQ/dxl is small comparedwith ul (x )only if

when x/x,= 1 O(N*R-*) , pressure forces again modify the final stage ofannihilation, in a manner whose elucidation requires analysis of the full Navier-Stokes equations.

A sketch of the streamline pattern determined by the solution discussed in thissection has been given by Moreau (1963a,b ) and by Craya & Moreau (1964).

1- > N*R-*; ( 5 5 )

4. Speculations on further physical effects associated with jet4.1. The nature of theflow near and beyond the stopping plane

The fact that a jet is stopped by a transverse field raises the question of whathappens in the neighbourhood of the stopping plane x = x,. The jet is clearlysplit into two deflected jets alined in the 5 y directions, and centred approxi-mately on x = x,. The momentum in each of these deflected jets is of the sameorder as the momentum in the original jet a t the section a t which boundary-layertheory begins to break down, i.e. a t x = x,( 1-O(N-A) ) or the top-hat inviscid jet( M 2> 1)and at x = xs( -O ( N R - 4 ) ) or the viscous (Moreau) jet ( M 26 1).This

annihilation

I n either case, Fl 6 Fo,so th at the momentum in the deflected jet is small com-pared with the initial momentum in the primary jet.

In some respects, the strictly inviscid jet is simplest to visualize (figure 2).There is no further magnetic resistance to the flow after deflexion, and thedeflected jets can flow parallel t o the field B, without any complications associatedwith viscous entrainment.In the case of viscous deflexion however, it seems likely that the streamlinepattern should be qualitatively as indicated in figure 3. For y > x,, the flowpresumably settles down to a weak wall-jet in the y-direction with constant fluxQo. A t a sufficient distance, inertia forces must be negligible, and the flow iscontrolled by the balance of magnetic and viscous forces. The pattern is again ofcourse symmetrical about y = 0.

The eddies in figure 3 are generated by viscous entrainment, which is parti-cularly strong near the slit. The length scale in the y-direction, yc say, o fthe smalleddy may be crudely estimated by the following argument. The deflected jet isapproximately centred at x = x,. It is a low momentum jet for M 2< 1 (equation56),and at sufficient distance from the plane y = 0, inertia forces are certainlynegligible. The spread of the deflected jet is then described by the equation


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Th e annihilat ion of a two-dimensional j e t 77The relevant solutions depend on the similarity variable Mh(x-x,)/ya, i.e. thespread is parabolic. The jet interacts with the wall x = 0 when

y x y c = M x : = R I N B l . ( 5 8 )Note that yCIx,= R%N-*B 1 , (59)so that the eddy is extended in the y-direction.

I

FIGVRE. Qualitative sketch of the probable streamline pattern when M 2 Q 1.The eddies are generated by viscous entrainment.4.2. Th e eSfct of the presence of rigid boundaries at y = f o

The foregoing analysis is somewhat unrealistic in tha t the effect of the boundariesy = f oin the region x > 0 has been totally neglected.It is to be hoped that suchneglect is legitimate, at any rate in certain regions of the flow, when yo 9 1.However, i t is now possible to predict at least qualitatively what the effect ofrigid boundaries at y = & yowill be, even when yo is not large. There are essentiallytwo possibilities when N < 1, M2 9 1.

(i)N4 4 o l (figure4 a ) . n this case, an essentiallyinviscid jet emergesfrom theslit and begins to be influenced by the walls y = f o at a distance

xo x N-l(l- yol ) ( 6 0 )downstream. A Hartmann profile is established for x 9 xo.Weak viscous entrain-ment will generate corner eddies as indicated.

(ii)19 N J 9 y o l (figure4 b ) . In this case, the jet is stopped a t xo x N-l beforeit begins to interact with the walls y = f o. As N increases, the region of jet-typeflow shrinks in extent. Although the analysis does not permit us to increase Nto order unity and larger, it is not difficult to predict what happens in thisthird case.

(iii)N 1 (figure 4c) . Here the jet is extinguished as soon as it emerges fromthe slit, and there is a narrow region of adjustment from one Hartmann profile(for x < 0) to the other (forx > 0). This situation is related to that considered byHunt & Leibovich (1967) who studied Hartmann flow through a channel ofvarying cross-section under the conditions M 9 1 , N B 1. They were able to


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7 8 H. . Moffatt and J . Toomreanalyse the sudden adjustment of a Hartmann profile due to a sudden change ofgradient of the channel walls-in which case the thickness of the adjustmentregion is O(N-i)-but were unable to analyse the type of situation comideredhere in which the cross-sectionalarea suddenly changes. In the region of adjust-ment, V2 w a2/ax2, nd consideration of the inertia-magnetic balance jn (8)againsuggests that the region of adjustment has thickness O(N-4) as indicated infigure (4c) .

Bo

FIGURE. Qualitative sketch of the expansion of Hartmann flow with Ma & 1 a ta sudden increase in channel width.In the case M 2< 1, the width of the Moreau jet, from (50)and (52) is

and this becomes comparable with yo when6Rg(CR*-$Nx*) w -xQ,

i.e. atThere are three further possibilities, each giving rise to a distinctive patternof flow:

(iv)yo


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The annihilation of a two-dimensional e t 79(vi) yoB M -l (R3 / N)* ,or equivalently yo> yc; in this case, the deflected jets

interact with the wall z= 0 before they meet the walls y = & yo (cf. figure 3) .A situation similar to tha t depicted in figure4 has been studied experimentallyby Moreau (1966a,b) . The values of the parameters R and yo (in the notation ofthe present paper) were 2 x 103 and 25 respectively. The Hartmann number Mvaried in the range 1 5 M 5 3. Qualitatively, the jet annihilation effect wasdemonstrated. It is not possible to make quantitative comparison between thetheory, justifiable only when M 2< 1or M 2 B 1 , and the experiments for which

y = o ( a ) R, 4 1 insulating wall

y = o ( c ) R, * , insulating wall

y = O ( b ) R , 4 1, perfectly conducting wall

- -__J = O

( d ) R,!,S= 1, perfectly conducting wallFIGURE. The lines of force of the magnetic field when R, < 1 and when R, & 1. In (a)and (c ) the wall y = yo s insulating, and the current returns through the fluid outside the jet.In ( b ) and ( d ) , he wall y = y,, is perfectly conducting, and i t conducts the return current.M2 = O (1 ) . However, one feature of the experiments is a little unexpected. Theextent A B (figure 4 a ) of the region of closed streamlines should vary approxi-mately as N -l when M 2$ 1 and as N-f when M 2 Q 1. The three streatmlinepatterns inferred by Moreau from probe measurements suggest a dependencenearer to N-4. Some further investigation over a much wider range of valuesof M would clearly be valuable.

4 .3 . The effectof increasing R ,The lines of force of the total magnetic field (i.e. including the perturbation due tothe currents in the fluid) are sketched in figures5a nd 5 b in the two cases (a) nwhich the boundaries y = k yoare insulating and ( b ) n which they are perfectlyconducting electrodes. In case (a), he current Jinduced in the jet returns throughthe fluid outside the jet, and in case ( b ) t returnsin current sheets on the electrodesy = f o.In either case the maximum value of lByl/Bo s O(R,), nd this has sofar been assumed small.


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80 H . K . Moffatt and J . ToomreIf we now allowR, to increase to order unity and larger, ~ B ~ ~ / B oill certainly

increase (although not necessarily in simple proportion toR,),and it seems likelythat the lines of force in the two cases will then be as sketched in figures ( 5c ) and( 5 4 . The induced component B, of the field does not affect the streamwise com-ponent of the jet, although it must affect the entrainment; it seems likely that theentrainment will be along the magnetic lines of force, i.e. that U and B will beparallel outside the jet (and of course outside boundary layers on any solidboundaries).The important result that a transverse field annihilates a jet cannot be affectedby increasingR,, although the 'stopping distance ' x, may now depend on R, aswell as on N (and R also in the viscous case). Again for x % x,, the flow willasymptotically approach a Hartmann profile.

FIUURE. Oblique injection of an inviscid jet into a transverse field. The parameter N isdefined in terms of the %-componentof velocity at the slit.4.4.DeJEexionof a je t oblique to the applied field

So far it has been assumed that conditions are symmetrical about the planey = 0. However, if the jet, on emerging from the slit IyI < 1, x = 0, has a netmomentum in (say) the positive y-direction, then this momentum is conserved,and the jet may be almost totally deflected in this direction (figure 6a). Thisbehaviour resembles that of a jet directed into a transverse wind, although inthat case, the jet penetrates an infinite distance in the x-direction (on inviscidanalysis) (Taylor 1954), whereas in the magnetic problem considered here, itpenetrates the same finite distance x, as for symmetrical injection.If the initial transverse momentum is small, then the jet may be still deflectedinto two jets, but the deflected jet in the negative y-direction will have a smallermomentum than that in the positive y-direction (figure 6 b ) .


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The annihilation of a two-dimensional je t 81It is evident that the solution of the inviscid form of equation (14),viz.

subject to the condition that a$lay is prescribed on x = 0, is not unique, butphysical considerations suggest that it can be rendered unique if the initialtransverse momentumis also prescribed.

4.5. Free j e t in a transverse JieldThe behaviour of a free jet of conducting fluid such as mercury into a non-conducting environment such as air in the presence of a transverse field is verydifferent. Suppose again that the jet is approximately two-dimensional, that itemerges from a slit on the wall x = 0, and that i t is bounded by the surfacesy = If:yo(x).Suppose also that conditions at z = f o are such tha t no net currentcan flow in the z-direction, e.g. the jet might be bounded at x = f o by the samenon-conducting medium (e.g. air) that boundsit at y = +_ yo(x).Then any currentj, n the jet, must return through the jet, i.e. the total current

yt= -jYmg)=o dY (64)

must vanish. This condition determines E,. Moreover, since the medium forIyI > yo(x)is non-conducting, there need be no pressure gradient, since there isno electromagnetic force in this region, i.e. ap/ax = 0. The (dimensionless)expression for the current becomes simplyd

j , = N ( U - & ) , ( 6 5 )where Z(x) (= &/yo(x))s the average velocity at the section x, and the inviscidequation of motion is

( U E + V $ ) = - N ( u - E ) .Clearly if U = E , i.e. if U is constant across the jet cross-section, then there is noLorentz force, and the inviscid jet is unaffected by the presence of the magneticfield. I f U $: U at any section, then the Lorentz force is such as to accelerate thefluid where U < E and to decelerate it where U > Z,.e. i t tends to smooth outvariationin the velocity across the jet. The Lagrangian solution of (66),analogousto (18), hows that differences n velocityU - at any sectionx = x, are eliminatedin a (dimensional)distance of order p ( u-U)/gBEdownstream of xo.Viscous forcesagain have negligible effect if the Hartmann number based on the slit width islarge. R E F E R E N C E SBICKLEY,W. G. 1937 The plane jet. Phil. Mug. 7, 23, 727-31.CRAYA, A. & MOREAU,R. 1964 Jets libres et conf in6s en fluides de forte diffusivitemagn6tiques. Proc. XI t h Inter. Congr. Appl . Mech., Munich (ed. Gortler),pp. 696-706.GOLDSTEIN,. 1938 Modern. Developments in Fluid Dynamics, vol. I. Oxford UniversityPress.

6 Fluid Meoh. 30


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82 H . K . Mo#att and J . ToomreHUNT,. C. R. & LEIBOVICH,S. 1967 Magnetohydrodynamic flow in channels of variablecross-section with strong transverse magnetic field. J . Fluid Mech. 28, 241-60.MOREAU,R. 1963u, b Jet libre plan, laminaire, dun fluide incompressible en pr6sencedun champ megnetique transversal. C. r . hebd. Sdanc. Acad. Sci., Paris, 256, 2294-98,MOREAU,R. 1966u, b Gtudes experimentale de 16volution des jets confines en presencedun champ magnetique transversal. C. r . hebd. Sdanc. Acad. hi . ,Paris, 262, 259-62,304-7.SHERCLIFF,. A. 1965 A Textbook of Magnetohydrodynamics. Oxford: Pergamon Press.TAYLOR, . I. 1954 The use of a vertical air jet as a windscreen. Collected Papers, 111,TSINOBER,.B.& SHCHERBININ,. V. 1965 Two-dimensional magnetohydrodynamic

4849-53.

637-540.jets. Mugnetohydrodynarmics,3, 21-9.

H.K. Moffatt and J. Toomre- The annihilation of a two-dimensional jet by a transverse magnetic field

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