David W. Taylor Naval Ship Research and Development CenterBethesda, MD 20084-5000

C) DTRC-88/017 May 1988

Propulsion and Auxiliary Systems DepartmentI" Research and Development Report0)

o Transient Magnetohydrodynamic Liquid-Metal Flows in a Rectangular Channelwith a Moving Conducting WallbyFrederick J. YoungScientific Division of the Frontier Timber Co.Bradford, Pennsylvaniaand

.2Samuel H. Brown and Neal A. SondergaardDavid Taylor Research Center

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63224C I IDN50752611. TITLE (Include Security Class fication) 61152N ZROOOOl ZR0230201 DN570523TRANSIENT MAGNETOHYDRODYNAMIC LIQUID-METAL FLOWS IN A RECTANGULAR CHANNEL WITH A MOVINGCONDUCTING WALL.

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This work was performed in conjunctio with the Scientific Division of the FrontierTimber Co., Bradford. PA 16701-3718./

17. COSATI CODES 18. SUUT TERMS (Continue op reverse if necessary and identify by block number)FIELD GROUP SUB-GROUP Curr'ent Collectori Rectangular Channel

Magnetohydrodynamics ' Transient Channel Flows ...

Liquid Metal Flows19. AUST (Continue on reverse if necessary and identify by block number)

The magnetohydrodynamic equations for temporal transients have been formulated andsolved for a liquid metal flowing in a rectangular channel. The rectangular channel hasa perfectly conducting moving top wall and a perfectly conducting stationary bottom wallin the presence of an applied external magnetic field aligned perpendicular to theconducting walls. The side walls of the channel are stationary insulators. Calculationsshow that the temporal transients of the fluid velocity and induced magnetic intensitycomprise two exponentially decaying parts. One fast transient is believed to be associatedwith the propagation of Alvgn waves and the other a slow transient being the result ofviscous and electrical diffusion. Curves of the transients are presented at several

stations in the channel.

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CONTENTS

PageA bstract ........................................................... IAdministrative Information .......................................... 1Introduction ....................................................... 1General Transient Magnetohydrodynamic Equations .................... 2

Rectangular Channel With Moving Wall ............................... 3Boundary Conditions ............................................. 4M athem atical Solution ............................................ 5

Lim iting Case Solutions ........................................... 1Results and Discussion .............................................. 12

T ransients ....................................................... 12

C onclusion ......................................................... 13Acknowledgm ents .................................................. 13

R eferences ......................................................... 19

FIGURES

1. Rectangular channel cross section .................................. 14

2. Velocity transients at z = 0 and y = 0.25, 0.5 and0.75 for k = 1 and Hartmann number of 5 ......................... 14

3. Magnetic intensity transients at z = 0, y = 0.25, 0.5 andfor k = 1 and a Hartmann number of 5 ........................... 15

4. Velocity transients at z = 0.25, y = 0.25, 0.5 and 0.75for k = 1 and a Hartmann number of 5 ........................... 15

5. Magnetic intensity transients at z = 0.25, y = 0.25, 0.5and 0.75 for k = I and a Hartmann number of 5 ................... 16

6. Velocity transients at z = 0.5, y = 0.25, 0.5 and 0.75for k = I and a Hartmann number of 5 ........................... 16

7. Magnetic intensity transients at z = 0.5, y = 0.25, 0.5and 0.75 for k = 1 and a Hartmann number of 5 ................... 17

8. Velocity transients at z = 0.75, y = 0.25, 0.5 and 0.75_ for k = 1 and a Hartmann number of 5 ........................... 17

,,. 9. Magnetic intensity transients at z = 0.75, y = 0.25, 0.5and 0.75 for k = 1 and a Hartmann number of 5 ................... 18

'Ini

9 NOMENCLATURE

B0 Orientation of the dimensional externally applied, constant magnetici field

C1, C2 Fourier coefficients for transient fluid velocity in channel

GDimensionless pressure gradient in x* direction in rectangular channely0

2 a P*i tAfU 0 a x*

H Dimensionless induced magnetic field in x* direction in channelH*/Uo oV'-

H* Dimensional induced magnetic field in x* direction in rectangularchannel

h(y,z,t) Mathematical solution for the dimensionless induced magnetic fieldtransient. The Fourier series solution is represented by

*00 00

h(y,z,t) 1 1 Hmn(t)cos(mry) cos(nirz/(2k))m=O,1,2.. n= 1,3..

-PteP2t

where Hmn(t) = aC, e- P t + PC2 e and

P1 = [a(1 + Pm) + (1 - Pm)2a2 - 4Pm(Mmir) 21/(2Pm)

P2 = [a+(Mmir) 2]/(PiPm)

HSS(y,z) Mathematical solution for the dimensionless induced magnetic fieldfor the steady state problem. The Fourier series is represented by

00

HSS(y,z) = I Hn(y) cos(nitz/(2k))1,3,5..

* where Hn(y) = [-g!(el +e 2)/dI +g 1(e3 +e 4)/d2]F

F = (4 sin (nn/2)/((gI + g2)nTr)

1 g = (M+/M 2 +n 2nr2/k 2 )/2

g2 = (-M+V'M2 +n 2n 2/k 2 )/2

k Dimensionless aspect ratio

_ M Hartmann number a dimensionless quantity y0B 0 Jo/' f

iv

0S

Pm Magnetic Prandtl number a dimensionless quantity opouf/Q

u(y,z,t) Mathematical solution for the dimensional fluid velocity transient.The Fourier series is represented by

00 00u(y,z,t) = I I- Umn(tOsin(mnry) cos (nnrz/2k)

m=0,1,2.. n= 1,3..

where Umn(t) = Ci e - Pit + C2e - P2t and

P1 = [a(1 + Pm) + (1 - pm)2 a2 - 4Pm(MmT[) 2 /(2Pm)

P2 = [a+(MmTt)2]/(P1Pm)

USS(y,z) Mathematical solution for the dimensionless fluid velocity for thesteady state Rayleigh problem. The Fourier series solution isrepresented by

00

- USS(y,z) = I Un(Y) cos. n =1, 3,5.. 2

where Un(y ) =[g2 (el -e 2)/dl +gl(e 3 -e 4)/d 2 ]F

F = (4 sin (nrT/2))/((g I + g2)nr)Aooession For

= (M + V/M 2 + n2nr2 /k 2 )/2 NTIS GRA&IDTIC TAB

92 =( -M+/M 2 + n2l2/k 2 )/2 Unannounced ElTustifteationi

t* dimensional timeBy

U* dimensional fluid velocity in x* direction Distribution/Availability Codes

U0 dimensional velocity of top perfectly conducting wall Avai and/orDist Special

x* ,y*,z* dimensional cartesian coordinates

xy,z dimensionless cartesian coordinates y*/y 0 , z*/yo ___

Yo dimensional height of rectangular channel

2zo dimensional width of rectangular channel

SC,'. * all dimensional variables are denoted with a superscript asterisk

which is contrary to previous work

0v

' C:11.11

o Electrical conductivity

MAf Viscosity

aC1 ,aC 2 Fourier coefficients for transient induced magnetic field

•O o Permeability of a vacuum

or r

-

S ,

N)vi

S.

7"11

ABSTRACT

The magnetohydrodynamic equations for temporaltransients have been formulated and solved for a liquidmetal flowing in a rectangular channel. The rectangularchannel has a perfectly conducting moving top wall and aperfectly conducting stationary bottom wall in the presenceof an applied external magnetic field aligned perpendicularto the conducting walls. The side walls of the channel arestationary insulators. Calculations show that the temporaltransients of the fluid velocity and induced magneticintensity comprise two exponentially decaying parts. One fasttransient is believed to be associated with the propagation ofAlvin waves and the other a slow transient being the resultof viscous and electrical diffusion. Curves of the transientsare presented at several stations in the channel.

ADMINISTRATIVE INFORMATION

This work was a cooperative effort between the David Taylor ResearchCenter and the Scientific Division of Frontier Timber Co., Bradford, Pa. It was

* performed under Program Element 63224C, Work Unit 1-2712-501, ProjectTitle: SDI Pulse Power Key Technology, Responsible Individual P. Filios. Thework was also partially supported by the DTRC Independent Research Program,Director of Naval Research, OCNR10, and administered by the ResearchDirector, DTRCOI 13 under Program Element 61152N, Project NumberZR00001, Task Area ZR0230201, Work Unit 1-2712-125, Project Title:Orientation Effects in Liquid-Metal Collectors.

INTRODUCTION

The use of liquid metals for current collectors in homopolar motors andgenerators has led to the design of machines of superior performance. Thesteady state power losses have been studied in a model of a current collectorcomprising a liquid-metal rectangular channel with an applied external magneticfield and boundary conditions containing a combination of moving and fixedinsulating and perfectly conducting walls.1 The top moving wall and stationarybottom wall are perfect conductors. The side walls are insulators. The appliedexternal magnetic field is perpendicular to the conducting wails.

In some applications of homopolar generators it becomes necessary notonly to start and stop the machines but also to operate them under oscillatingconditions. This could be the case in an application where a homopolargenerator behaves as an extremely high energy capacitor. Therefore, one isinterested in examining energy losses caused by these non-steady statephenomena.

Studies of non-steady state Couette flow in an external magnetic field areseldom found in the literature. There have been some papers which augmentedthe classical Rayleigh problem 2 with an external magnetic field in an electricallyconducting medium. 3"8 The presence of Alvin waves and slowly propagatingwaves was observed in detailed studies of the Rayleigh problem. Theorypredicted the possible formation of switch-on shocks and current sheets to be

0I

present in the Alvin wave front under certain conditions. Currently, it wouldappear that a complete theoretical analysis and explanation of the variousphenomena comprising the Rayleigh problem is lacking. This is partly caused bya waning interest in the field of magnetohydrodynamics in the past decades andthe complexity of the mathematical solutions in these magnetohydrodynamicproblems.

Our objective is to present a transient two-dimensional solution and some'. numerical results for one particular geometry of finite excent, shown in Fig. 1.

The solution enables the investigation of the influence of the sidewalls on thevelocity and the induced magnetic field that affects the joulean and viscousdissipation. The geometry chosen roughly corresponds to an axisymmetric liquidmetal current collector having channel dimensions that are small compared tothe radius of curvature of the current collector. The moving and fixedconducting walls correspond to the rotor and the stator, respectively. Followingearlier work, 1 the orientation of the applied magnetic field or the appliedinduction, B0 (see Fig. 1) is chosen to generate the maximum eddy current. Inaddition, current parallel to the applied magnetic field produces noelectromagnetic body force and need not be considered. Analytical expressionsare derived for the induced magnetic field and the fluid velocity in terms of

s, their spatial and temporal variations. The parameters of variation are applied* magnetic induction, fluid viscosity, density and electrical conductivity, and the

channel aspect ratio. Numerical results showing the buildup of the velocity andinduced magnetic field transients are presented for several cases.

GENERAL TRANSIENT MAGNETOHYDRODYNAMIC EQUATIONSConsider the case of laminar flow of an incompressible, uniformly

conducting fluid of density Q, viscosity pf, and conductivity o flowing in arectangular duct of width 2zo and height of y0 with a uniform external magneticinduction, B0 applied in the y direction as shown in Fig. 1. The flow is drivenby the sudden movement of the top conductor. This conductor, initially at rest,is assumed to attain a constant velocity at the onset of the transient calculationand maintains that velocity. It is assumed that the pressure gradient in the fluidis zero to simplify the solutions. Such would not be the case if there were anyobstructions in the channel that limited the net flow of fluid. It is assumed thatno secondary flow is generated during the transient, and there is no variation in

* the duct cross section or distortion of the applied induction in accordance withthe assumption of a fluid velocity and an induced magnetic field intensity in thex direction only. The equation for the fluid velocity in the x direction is givenby

Q aU* a2U * a2U * H* aP*at* =f y + f z + 0 3 (1)

and the induced magnetic intensity in the x direction is given by the magneticdiffusion equation as

aH* a2H* a 2H * aU*CIAO t* y, + -* 2 + oB0 y (2)

2

' % X

0k -

[ * r , ~ .

where all dimensional variables are denoted with a superscript asterisk, contraryto earlier work. 1,8 Equation (1) is the two-dimensional Navier-Stokes equationincluding an electromagnetic body force and Eq. (2) is the magnetic diffusionequation obtained by combining both of Maxwell's curl equations and theelectrical constitutive equation for a moving conducting medium. All equationsare in the SI or RMKSA system of units.

RECTANGULAR CHANNEL WITH MOVING WALLThe problem solved in this paper considers a rectangular channel (see

Fig. 1) filled with a liquid metal in a uniform magnetic field, B0. The top wall is,- assumed to be perfectly conducting and is initially at rest. At time t* = 0 its

velocity jumps in a unit step to U0, where it remains constant. The bottom wallis stationary and is also an ideal conductor while the side walls parallel to theapplied external magnetic induction field are stationary electrical insulators. Tomake the handling of the equations easier we define a number of convenientand physically meaningful dimensionless quantities. The dimensionless quantitiesare not denoted with asterisks as in previous work and are defined as

I%. -

U = U*/U 0 , y = Y*/Y 0 , z = z*/y 0 , t - t*/(y 2Qmf),

H = H*/U\/ of) and k = z0/y 0 , the Hartmann number

M = y 0 B 0 V';f_, G = [yo2 /(MfUo)10P*/8x*

the magnetic Prandtl number is given by Pm = cJaolf/Q, where Io is thepermeability of a vacuum. Even though the Hartmann number was undefi-ied inthe magnetohydrodynamic Raleigh problem (because y0 was unbounded), it isthe most important parameter in this problem. It is a measure of the ratio ofponderomotive force to the viscous force (in this interpretation it is assumed theponderomotive force is of the order of U0 , certainly true for the electricalconductivities encountered in liquid metal current collectors). The magneticPrandtl number is small in this problem but cannot be neglected entirely withoutcausing great inaccuracy in the solutions. It can be discarded only in directcomparison with other larger quantities. The magnetic Prandtl number is ameasure of the ratio of vorticity diffusion to magnetic diffusion. For liquid NaKthe magnetic Prandtl number is of the order of about 2 x 10-6. If liquid metal

*electrical superconductors were available, the aagnetic Prandtl number would bevery large and could cause interesting effects. To make the solutions muchsimpler we assume G = 0. Then Eqs. (1) and (2) become

a 2U/ay 2 + a 2U/z 2 + MaH/ay = aU/at (3)

a 2H/ay 2 + a2H/az 2 + MaU/ay = Pm aH/at (4)

Here we note that the magnetic Prandtl number provides a coupling between theinduced magnetic intensity and the fluid velocity that does not exist in thesteady problem treated earlier. 1 Equations (3) and (4) must be solved

- simultaneously for U(y,z,t) and H(y,z,t) by any method which satisfies the

03

Nv~~' .-

boundary and initial conditions while satisfying Eqs. (3) and (4). It is temptingto consider the use of the Laplace transform to eliminate time as a variable untilthe difficulty of the inversion of the transform is considered. Instead of usingthe Laplace transform method for solution of the partial differential equations,we have decided to seek a solution in terms of exponentially diminishing timefunctions. To do that we let

U(y,z,t) = Uss(y,z) + u(y,z,t) (5)

H(y,z,t) = Hss(y,z) + h(y,z,t) (6)

where U,, and Hss are the solutions to the steady flow problem and u(y,z,t) andh(y,z,t) are transients that eventually fade. When Eqs. (5) and (6) are substitutedinto Eqs. (3) and (4) there results for the transients u and h

a 2u/ay2 + a2u/aZ2 + Mah/ay = au/at (7)

a 2h/ay2 + a 2h/az2 + Mau/ay = Pm ah/at (8)

* BOUNDARY CONDITIONS

The no slip condition applies to the total solution for the velocity at all ofthe boundaries. In that case

U(y = 1,z,t) = 1 implying that u(y = 1,z,t) = 0

U(y = O,z,t) = 0 implying that u(y =0,z,t) = 0

U(y,z = _ k,t) = 0 implying that u(y,z = ± k,t) = 0

Because there must be zero electric field in the rest frame of .he perfect

conductors for finite current flow, the normal derivative of the magneticintensity is zero at the top and bottom perfect conductors. Currents generated in

V the top conductor and the fluid move in a loop (in the y-z plane) bounded bythe conductors and the side insulating walls. All the current generated byconductors moving in the applied external induction field returns primarily in

* the bottom perfect conductor. Thus by Ampere's law the induced magneticintensity at the insulating walls is zero. In that case

alH/ayJlY= = 0 implying that ah/aylYl = 0

aH/ayy~ = 0 implying that ah/ayjy= 0 0

for all values of z and t, and

H(y,z= ±k,t) = 0 implying that h(y,z= ±k,t) 0

.- - 4

%

I-These conditions hold only for the open circuit case with no net current betweenconductors.

Initially the total velocity and induced magnetic intensity are zero, whichmeans the transient velocity and the transient induced magnetic intensity must bethe negative of the steady solutions. In that case

u(y,z,t =0) = - Uss(y,z) and

h(y,z,t=O) = -Hss(y,z)

MATHEMATICAL SOLUTION

To automatically satisfy all of the boundary conditions we choosesolutions of the form

u(y,z,t) = I Umn(t)sin(mty) cos(nirz/(2k))m=0,1,2.. n= 1,3..

00 00

h(y,z,t) = I I Hmn(t)cos(miTy) cos(nnz/(2k))m=0,1,2.. n= 1,3..

To satisfy the partial differential equations governing the solutions for u(y,z,t)and H(y,z,t), the above assumed solutions are substituted into Eqs. (7) and (8).There results a pair of coupled time dependent differential equations in timegiven by

dUmn/dt + r 2[m2 + (n/(2k))2]Umn = -MmrHmn (11)

PmdHmn/dt + i 2[m2 + (n/(2k))2]Hmn = MmrUmn (12)

Equations (11) and (12) are combined to yield a pair of second order timedependent differential equations for Umn and Hmn. They are given by

,-md2/dt 2 + (1 +pm)[(mn) 2 + (nn/(2k))2]d/dt +

(Mmn) 2 + [(mTI) 2 + (nit/(2k))2]2 Hm n 0 (13)

We assume the solution to this equation is Umn(t) = C e-Pt whence P must befound from the roots to the equation

PmP 2 - (I+Pm)aP + (MmR) 2 + a2 = 0 (14)

5

r0 i

where a = ir2[m 2 +(n/(2k)) 2]. Because the two solutions of Eq. (14) may differ

greatly in magnitude we choose them to be of the form8

P1 [a(l + Pm) + (1 - pM) 2a2 - 4Pm(MMir)21/(2Pm) (15)

and

P2 [a+ (MMt) 2j/(P1 Pm) (16)

Thus the solutions to Eq. (13) are given in exponential form by

Umn(t) =CieP- + C2eP~ (17)

Hmn(t) = aCje -Pit + PC 2e P2t (18)

where a = (P1I - a)/(Mmir) and P3 (P2 -a)/(Mnir). Utilizing theaforementioned initial conditions yields

CI + C2 = Umn(9

0C + /3C2 = s (20)

having solutions of

'I.*~*CI = (USM~n-P(Hs~s)/(a -P1) (21)

C2 = (-aUSM~+Hs~s)/(a-3) (22)

Swhere Uss and Hss are the Fourier coefficients of the steady solution1 when itM M

is expressed as a double Fourier series of the forms of Eqs. (9) and (10). To* reduce the amount of tedious algebra resulting upon the invocation of the

orthogonality of tk-e sinusoidal functions we have tried to express the steady

state results1 as simply as possible. These solutions are given by

'5-. 00

USS(y'z) = I Un(y)cos(nirz/(2k)) (23)n= 1,3,5...

and

£2 00Hss(y,z) = I Hn(y)cos(nirz/(2k)) (24)

Sn= 1,3,5...

6

where Un(y) = [g2(e, - e2)/d 1 + gj(e 3 - e4)/d21F ,

Hn(y) = [-gI(eI +e 2)/dI + gI(e 3 +e 4)/d 2]F

F = (4sin(nir/2))/((gj + g2)nn) ,

g, = (M+/M 2 + n2 2/k 2 )/2

2= (-M+\/M2 + n2n2/k 2 )/2

eI = e-g(-Y)

e2 e-gj(l+Y)

e3 e-g2(1 -y)

e4 e-92(1+Y)

d = (I - e) -2g, and

* d2 = -e)- 292.

The standard use of orthogonality yields expressions for U'n and Hsn as givenby:

I

n= -2 f Un(y) sin(mTry) dy (25)0

Op -

Hsn = -2 f Hn(y) cos(mny) dy (26)0

The indicated integration yields

(2cos(mn)mn( -g2 + gig 2-g2 _m2T2))

mn (gig2 + g2m2n2 + g2M2m2 + m4Tr4)

and

Hss (2cos(mT)g g 2(g1 - g2))mn (g292 + g2mu2 + g2m2n2 + mrn4 4 )

The substitutions of the results of Eqs. (27) and (28) into Eqs. (21) and (22),

and the quantities defined after Eq. (24) and considerable amounts of algebraic

manipulation yields C1 and C 2, which are given by

7

C1 (16cos(mirsin((nir)/2)mPm(a 2g9 2p - a2g 1g2OP + a2g~Op + a2M2Opir 2

ag19 2MOP + agRa-ag + ag~g2MOp - ag~g2Ra + 2agIg 2

+ ag2Ra - 2ag2 + am2ir2Ra - 2am~ir 2 - g2 g2MRa - 2g2M2M2rr2

+ 9 g2MRa + 2gIgM9 i2 - 2g2M2M2ir 2 - 2M2M4Tr4))/(n(a g9~O

2 a2 2Op2 20I2 24prgm + a2g9 mOyr + am4i4+ 2ag2 g OpRa

2 4a 2gpM + 2ag2O2ptr2Ra -4ag2i2 pM + 2ag 2Opir2Ra

- 4ag~m 2n2Pm + 2aM4Opir4Ra -4am4Tr4pM - 4g2 g M2M2T[2pM

2 2f~R - M2M4 7r4pM + 42Z2Ra - gM2M4Tw4pM-~~~~~~ 412i6 m ~rR~)(7

2 22

A and

C2 =(8cos(miT)sin((nn)/2)m(a g9O P 2a 9p~ - a12P

+p 2ag~g2OPM + a~ p- 2a gOpPm+ mop2

- 2a2M2OpT[2pM + 2ag~g2MOPm + 2ag4OpRa - 2agP~

-2agMOpPm - 2ag~g2OpRa + 2ag~g2PmRa + 2ag~OpRa

2 2 R2- 2ag2PmRa + 2am2Opir2Ra - 2am 2n2pmRa + 2g g2MPmRa + gRa

29 2ggMPmRa - g1g2Ra + g~R + M~tr2Ri)/( 292g1O2

+ 2g 0pO~r2 2 a2 2OpTr2 + a2M40p,4 + 2agg0pRa

- 4ag2 g pM + 2agfm 2Opir 2Ra -4ag~m2 pM + 2ag2O2pir2Ra

*- 4ag M2njirn + 2am4Optr4Ra -4am4Tr4Pm - 4g2 g M2m2pM

2 1 2

+ g M21r2Ra - 4M2M6tr6pM + m4nr'Ra)) (28)

For documentation we give the expressions for aC, and PC2 which are used forthe calculation of the induced magnetic intensity. These expressions are

8

I

ac1 ~ 12OP + (~o~~sn(u/)a~g 20 2a g2OpPm + a gpg 2O

- 2a 3gM2 ppM - ag2O 2agOm+amOnr2a1m2MO +P 2a2 22OpRa

a~~g 912MO PP + 2agg2 pm

- 2a2g9 Op -2a2g~pmRa + 4a2g9 pM + a2gig 2MO2

- 2a2gj1g MOppM- 2a2g 1g2QpRa + 2a 2g 1g20P + 2a 2g91g2PmRa

- 4a2g91g2PM + 2a2g~OpRa - 2a2g~Op - 2a 2g~pmRa + 4a2g9 pM

+ 2a2m2Optr2Ra - 2a2M2Opir2 - 2a2M2ir2pmRa + 4a 2M2ir2pM

- 2aglg 2MOpRa + 2aglg 2MPmRa - 2agMmOp~

+ agM 2M21T2pM + ag2R 2 _ 2Ra + 2aglg2MOpRa

- 2aglg~MPmRa + 2agIg 2MmOl 2-4g 2MmnP - aglg2Ra

I ~~+ 2ag 1g2Ra - 2ag 2mOut + 4ag2m~rP 2 a 2R2

- 2aM2M4Opn 4 + 4aM2M4 nr4pM + amxn 2Ra - 2am2ir2Ra

2M 2 2~nRg~g 2M~a -2M2M

21r2Ra + glg aR + 2g 192M~~r

-2M2M2ir2Ra - 2M2M4ir4Ra))/(Mnn(a2g2g 2 2 a2g 2O2

a22 2 2 22pa-4g p

+ a 2 02 pi 2 + a2m4Opn4 + 2aggO~ -1ag2g192

" 2ag~m2OpT 2Ra - 4ag~m27r2pM + 2ag~m 2Opir2Ra - 4ag~m21r2pM1a~P 1 2 2

2am4Opn4 Ra - 4rnft 4g g2M 2M2n2pM+ g2g 2Ra-~~~ 19Mmi4 m+gm12~ gMmi 4 m+gm2R

22

- 4M2M6T[6pm + m4n4Ra)) (29)

and

P3C, (8cos(miT)sin((niT)/ 2)(-a 9 0 + 2a3g 2Oplm +

-2a3 g91g2OPM a~g2O0' + 2a39g2OpPm - a3rn2Q2 ff

* 2a 3M2Opir2pM- 2a2g 2 M~~ 22OpRa, + 2a2g2~

* 2a2g2PmRa - 4a12 2pm + 2a2glg2MOppM + 2a2gIg 2 OpRa

9

-2a2g91g20P- 2a 2g91 g2PmRa + 4a2g 1g2PM - 2a2g~OpRa + 2a2g~op

+ 2a2g2pmRa -4a2g2pm -a2m2OpiR2Ra + 2a2M2Opit 2

A.+ 2a2m2iT2PmRa -4a2m27T2pm - 2ag 2 MPmRa + 4ag21g 2MPM

+ 2agMmOpr- 4ag2M2m2T[2pm - ag2,Ra + 2ag2Ra

+ 2aglg2MPmRa -4ag~g2MPm - 2agIg 2 M2m2Opr 2

4a~gM~~i2P +2 2M2m2Op7T2+ 4agI2 M2M2~pm +agjg 2Ra -2aglg 2Ra +2g

- 4agMmiyP - ag2Ra + 2agRa+2MmOi'

- 4aM2M4ir4pM - am2iT2Ra + 2am2ir2Ra + 4g 192M~~r

+ 2g2M2m~n 2Ra - 4glg2M3M2T2pM - 2gIg 2 M2m2Tr2Ra + 2g2M 2n2Ra+ 1 22 gm2,t2

" a'm~Op2Tr + 2ag2g2OpRa - 4ag2g2pm + 2ag~m2Opnr2 Ra

- 4agfm27g2pM + 2ag~m2Opir2Ra - 4ag~m2n2Pm + 2am4Opir4Ra

+ gm4r2RPm 4g M2mr'pm + gm2ir2R - 4 2 m~r

+ m~ir4Ra)) (30)

The coefficients of the time functions in the exponentials are given by

p1 (ii/64M2k4M2pM - 16k4M4Tr2P2M+ 32k4M47r2pM - 16k4M4 T[2

- 8~m~~itPM+ 16k2M2n2ir2pm - 8k 2M2n2 r2 - n4,p

+ 2n4712pm - n47r2) + 4k2m2 itpm + 4k2M~n + n2nr~m

+ n 21t))/(8kPm) (31)

01

.4lj

and

P2 =(27u(4M2k2m2 + 4k2m2 + n2))/(i\/(64M2k4m2Pm- 16k4m4T[22in

+ 32k4M4i'r2Pm - 16k4m~rr2 - 8kim2n~ir2p~n + 16k2mn2n2rr2pm

- 8kn 2n2 r2 - n4Tr2pmn+ 2n 47r2Pm - n47T2) + 4k2m27jpM

+ 4k2m2Tn + n2 nrnm + n2ir) (32)

M =Hartmann number, Op 1 + Pmn, Omn = 1 - Pmn and Ra

M~a2 - 4PM(MTM)2

Thus the transient parts of the velocity and magnetic intensity mathematicalexpressions are determined by Eqs. (9) and (10) and the auxiliary Eqs. (17), (18),and (27) through (32). In addition the above given definitions of g1 , 92, 0OP,Omn, and Ra must be used.

LIMITING CASE SOLUTIONSWhen the Hartmann number approaches zero the solution given above

approaches the nonmagnetohydrodynamic limit of zero induced magnetic* intensity and a purely hydrodynamic transient fluid dynamic velocity

distribution. The complete nonmagnetohydrodynamic solution (including boththe steady and the transient parts) is given by

00

U(y,z,t) = 4 1 [sin(nir/2)cos(ntrz/(2k))/[nn(l - e - li/k)1 1

[enni(y 1)/(2k) - e-n(y+ 1)/(2k)]

m cos(nrn)sin(n/2)cos(nirz/(2k))e - +(/2) 2

+2 1m = 1,,. 2n [m2 +(n/(2k)) 2 (33)

In most laboratory experiments Pmi is of the order of 2 x 10 -6 and theexpressions for P, and P2 Simplify to

p, = tr2[m2 + n2 /(4k2)]2/Pm (34)

and

P2= 1 + M2m2/[M2 +n2/(4k 2)J (35)

where it is clear that the P2 terms contained in the second time function of Eqs.* (9) and (10) do not aid in the convergence of the series, which is made up of

11110'', 111 1

double summations. On the other hand the P1 terms, in addition to being verylarge, increase in size with the square of the indices m and n. From theinspection of these variables it is expected that the first term of the exponentialsolutions not only converges faster with index but also vanishes faster than thesecond term.

Exponential diminishing time solutions were assumed for the temporalvariations as a matter of convenience. However, inspection of the radicalcontained in the solution of P1 indicates the possibility of complex roots. It isclear that oscillatory solutions exist when

M > (1 - Pm)[m2 + n2/(4k 2)]/(2mnPm) (34)

requiring extremely large Hartmann numbers in the terrestrial laboratory,assuming high temperature liquid metal superconductors are not available. If themagnetic Prandtl number can be increased, then some of the early terms in thesummations may be damped sinusoids.

RESULTS AND DISCUSSIONWe present a few calculations of velocity and induced magnetic intensity

transients, roughly covering the area of the channel. The calculations are donein double precision and require the summation of 50,000 terms for each point inspace9 (m and n ranging up to 1000 and 50 respectively). By the use ofminimum-maximum method10 only 400 terms are necessary for engineeringaccuracy. The optimization of the method for double series summations mightfurther reduce the needed number of terms. To focus our attention upon thetemporal variation of the velocity and the induced magnetic intensity, bothquantities are normalized by dividing them by their final or steady state value.The steady state values are the same as those presented in an earlier paper.1 Bythis technique the transients at different points in the channels can be comparedmost easily.

TRANSIENTS

In Figs. 2, 4, 6 and 8 are shown normalized dimensionless velocitytransients for an aspect ratio of unity a Hartmann number of 5 and a magneticPrandtl number of 2.54x 10-6. In each case the velocity was normalized withrespect to its steady state value. In each figure there are curves of normalizeddimensionless velocity as a function of dimensionless time at y = 0.25, 0.5 and0.75. A different value of z is used in each of the figures. In the case where z= 0 (see Fig. 2), the transients are almost identical at all values of y presented.

Inspection of Figs. 2, 4, 6 and 8 indicates the differences as the transientsincrease slightly when z increases from the channel center to the insulating wallsand the risetime of the transient also increases slightly. The risetime for thevelocity in the cases presented here is roughly 0.1 units of dimensionless time.Also, for the velocity transients the exponential solution having the small timeconstant has small amplitude compared to the large time constant exponentialsolution. That is, C1 is less than C2.

12

0O

t0

These curves are presented in Figs. 3, 5, 7 and 9 for the same parametersas used in the corresponding velocity figures. In each case the induced magneticfield was normalized with respect to its steady state value. In all cases thenormalized dimensionless induced magnetic intensity rises rapidly from zero toan apparent initial value (see Fig. 3) and subsequently undergoes less rapidchanges. The expansion of the dimensionless time scale shows that thenormalized dimensionless induced magnetic intensity actually begins at a valueof zero (as it should) and increases very rapidly at first. This is the small timeconstant exponential part of the solution and as it fades the long time constantexponential part of the solution takes over. The magnetic intensity initiallybuilds up faster at stations nearer the moving plate. We feel that the fast part ofthese transients is due to the passing of Alvin waves while the slow part is dueto viscous and magnetic diffusion. The curves are similar for the different valuesof z and show a slight tendency to have a slower approach to the steady state asz increases.

CONCLUSION

A formal solution to the transient magnetohydrodynamic Couette problemhas been obtained and we have presented but few of the situations that could beexamined. We have disclosed the presence of fast and slow transients and the

* possibility of oscillatory behavior in both the fluid velocity and the inducedmagnetic field. In future work we plan to investigate these phenomena and start-up losses.

ACKNOWLEDGMENTS

The authors wish to express their appreciation to Dr. John S. Walker ofthe University of Illinois at Urbana-Champaign for reviewing this report and

-. supplying many helpful comments. A condensed version of the report was-. presented at the 4th Symposium on Electromagnetic Launch Technology at the

University of Texas at Austin on April 12-14, 1988.

-13

yy

x MOVING

z B PERFECTCONDUCTOR

(U=U 0)

y =1

1; ELECTRICAL INSULATORS

y =0

z =-k =oz=k1*.

Ii. STATIONARY PERFECT CONDUCTOR (u=0)

Fig. 1. Rectangular channel cross section.

0.250.75

>.8U)

w

z

w

0* 0 0

.0 .5 1.0 1.5 2.0

DIMENSIONLESS TIME, t

Fig. 2. Velocity transients at z - 0 and y = 0.25, 0.5 and 0.75 for k =1 anda Hartmann number of 5.

14

1.0-

Cn

0Cnz

0.

3i y=0.25

, .822

w 20

0L 1.00IMESIOLS TIME, t20

Fig. 3. Vageocinity transients at z 0.2 , y 0.25, 0.5 and 7 for k 1 and

a Hartmann number of 5.

1.15

00

1.0 !

LU .8-j

= .2

0z.0I I I

.0 .5 1.0 1.5 2.0

* DIMENSIONLESS TIME, t

I%.2

Fig. 5. Magnetic intensity transients at z = 0.25, y = 0.25, 0.5 and 0.75,-. for k = 1 and a Hartmann number of 5.

1.0 0- .75

w

> .8 y=0.25UI)uJI: -J2.60

2 .4

0 .0

2 0 .5 1.0 1.5 2.0DIMENSIONLESS TIME, t

-. Fig. 6. Velocity transients at z = 0.5, y = 0.25, 0.5 and 0.75 for k = 1 and* a Hartmann number of 5.

16

1.0-

4,'-, y02

0F

NZ

ir .20

VP' z

.0.,,.0 .5 1.0 1.5 2.0

* DIMENSIONLESS TIME, t

Fig. 7. Magnetic intensity transients at z = 0.25, 0.5 and 0.75 for k = 1 anda Hartmann number of 5.

1.0" 0.75

0y=0.25

.4-.

. 0N .

z _ ... 0155.

LU .

cnLU

z 6

0LU

"44

*~ 0.0 Iz .0 .5 1.0 1.5 2.0

DIMENSIONLESS TIME, t

Fig. 8. Velocity transients at z m 0.75, y = 0.25, 0.5 and 0.75 for k = 1 and*" a Hartmann number of 5.

17

1.0

... ... 1 -...

z

00

.0.7

0.5 101. .

DIENIOLES.2M,5Fi .2 antcitniytaset tz=07,y=02,05ad07 o

* ~ k=I n aHrmannubr0f5

I."0

18

0

.oJ ... . .I ...... .I : ... .. . U .. . .. q . - IrJ . .... . . fl . . .. P U, l . . Ifl .. 'U n fI ... .. .. -! . .. U"; - ... ..

REFERENCES

I. S.H. Brown, P.J. Reilly and N.A. Sondergaard, "Magnetohydrodynamicliquid-metal flows and power losses in a rectangular channel with a movingconducting wall," J. Appl. Phys., Vol. 62, No. 2, pp. 386-396, July 1987.

2. Lord Rayleigh, "On the motion of solid bodies through viscous liquid,"Phil. Mag., Series VI, Vol. 21, pp. 697-711, 1911.

3. V.J. Rossow, "On flow of electrically conducting fluids over a flat plate inthe presence of a transverse magnetic field," Natl. Advisory Comm.Aeronaut. Tech., Note No. 3971, 1957.

" 4. C.C. Chang and J.T. Yen, "Rayleigh's problem in magneto-hydrodynamics," Phys. Fluids, Vol. 2, pp. 393-403, 1959..I

5. G.S.S. Ludford, "Rayleigh's problem in hydromagnetics: the impulsemotion of a pole-piece," Arch. Rat. Mech. Anal., Vol. 3, pp. 14-27, 1959.

6. M.M. Stanisic, B.H. Fetz, H.P. Mickelson, Jr. and F.M. Czumak, "Onthe flow of a hydromagnetic fluid between two oscillating flat plates," J.Aerospace Sciences, Vol. 29, pp. 116-117, 1962.

7. F.H. Shair, "The transient interaction of a transverse magnetic field withfluid in Couette Flow, NASA Tech. Inf., Series N63-13559, Code 3, 1963.

8. W.F. Hughes and F.J. Young, The Electromagnetodynamics of Fluids,New York, John Wiley, 1966, ch. 9, p. 399.

9. W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling,Numerical Recipes, The Art of Scientific Computing, Cambridge,Cambridge University Press 1987, ch. 5.5, p. 145.

10. P. Albrecht and G. Honig, "Die numerische Inversion der Laplace-Transformierten", Angew. Informatik, Vol. 8, pp. 336-345, 1977.

~19

U,

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