PHOTON USE AUTOPLOT file phi 5 cl msg MHD PLANE COUETTE FLOW msg Hartmann number = 4 msg Velocity (W1) profile msg Blue line --- PHOENICS solution msg crosses --- analytical solution da 1 w1;da 1 w1a col3 1;blb4 2 msg pressto continue pause msg press to end pause end END_USE DISPLAY 123456789-123456789-123456789-123456789- TEXT(1D Laminar MHD Couette Flow TITLE DISPLAY This problem concerns the steady fully-developed laminar flow of an incompressible electrically-conducting fluid in the positive z-direction of a plane channel. The top wall of the channel is moving and the bottom wall is stationary. A uniform magnetic field By is imposed normal to the walls and a current jx is induced in the fluid in the x-direction, together with a magnetic field Bz in the z-direction. The problem neglects end effects, secondary flows, Hall effect and ion-slip phenomena. ENDDIS The dimensionless momentum equation to be solved is: d/dy(dw/dy) + Ha*Ha(K-w) = 0 where y =y/yin z =z/yin w =w/wtop Ha=sig*(By*yin)**2/(rho1*enul) K =Ex/(wtop*By) Here, yin is the channel half width (m), By is the imposed magnetic flux density in the +ve y direction (volt.s/m**2), sig is the electric conductivity (ohm/m) and Ex is electric field intensity in the +ve x-direction (volt/m). The Hartmann number Ha represents the ratio of the electromagnetic forces to the viscous forces. The voltage ratio K is the ratio of the voltage to the open-circuit voltage. When K=1 the net current flow is zero, which is known as the open-circuit condition. When K=0 the channel is short-circuited. The analytical solution to this problem has been presented in 'Engineering Magnetohydrodynamics', Chapter 10, G.W.Sutton and A.Sherman, McGraw Hill, (1965). REAL(YIN,WTOP,HA,KVOLT);YIN=1.0;WTOP=1.0 HA=4.0;KVOLT=0.0 GROUP 2. Transience; time-step specification ** set parab=t to activate spot & residual monitoring print out as a function of lithyd PARAB=T GROUP 4. Y-direction grid specification NY=50;GRDPWR(Y,NY,YIN,1.) GROUP 7. Variables stored, solved & named SOLVE(W1);STORE(W1A) GROUP 8. Terms (in differential equations) & devices TERMS(W1,N,N,P,P,P,P) GROUP 9. Properties of the medium (or media) RHO1=1.;ENUT=0.;ENUL=1. GROUP 11. Initialization of variable or porosity fields ** compute analytical solutions REAL(WA,GR,HAY1,HAY2);INTEGER(JJM1) DO JJ=1,NY +PATCH(IN:JJ:,INIVAL,1,NX,JJ,JJ,1,NZ,1,1) +GR=0.5*YFRAC(JJ) IF(JJ.NE.1) THEN +JJM1=JJ-1 +GR=YFRAC(JJM1)+0.5*(YFRAC(JJ)-YFRAC(JJM1)) ENDIF +GR=GR*YVLAST;HAY1=GR*HA;HAY2=(1.-GR)*HA +WA=KVOLT+(1.-KVOLT)*(SINH(HAY1)-KVOLT*SINH(HAY2))/SINH(HA) +INIT(IN:JJ:,W1A,ZERO,WA) ENDDO GROUP 13. Boundary conditions and special sources PATCH(WALLTOP,NWALL,1,NX,NY,NY,1,NZ,1,1) COVAL(WALLTOP,W1,1.0,WTOP) PATCH(WALLBOT,SWALL,1,NX,1,1,1,NZ,1,1) COVAL(WALLBOT,W1,1.0,0.0) PATCH(MHDFOR,VOLUME,1,NX,1,NY,1,NZ,1,1) COVAL(MHDFOR,W1,HA*HA,KVOLT) GROUP 15. Termination of sweeps LSWEEP=1;LITHYD=10 GROUP 22. Spot-value print-out IYMON=NY;TSTSWP=-1 GROUP 23. Field print-out and plot control NPLT=1;NYPRIN=1;NZPRIN=1 GROUP 24. Dumps for restarts