load(ctensr)$
load(itensor)$
/*load("contrib/tensor/tentex.lisp")$*/
(bothcases:false,metric:G)$
M(L1,L2):=H([],[L2[1]])*H([],[L2[2]])/(4*%pi)
-1/(8*%pi)*G([],L2)*G([i1,i2])*H([],[i1])*H([],[i2])$
ST(L1,L2):=-g(L1,L2)*P([])+nu([])*(g([],[L2[1],N])
  *covdiff(V([],[L2[2]]),N)+g([],[L2[2],N])*covdiff(V([],[L2[1]]),N))$

show(M([],[i,j])+ST([],[i,j]))$

("the Navie-Stokes equation for the viscous axisymmetric azimuthal MHD flow")$
(metric:G,navie:comp([],[I])=
  lorentz(canform(rho([])*D_t([])*V([],[I])-
      covdiff(ST([],[I,J])+M([],[I,J]),J))),show(navie))$
("Now generate equations for ctensr")$


exp:generate(canform(navie))$

bothcases:true$
(Omega:[R,phi,z],depends(P,[R,z]))$
("Now define the ccoordinate representation for velocity and magnetic
field. The magentic field is expressed through potentials A,B")$

(V:[0,r*U,0],dim:3,depends(U,[R,z]))$

(depends([A,B],[R,z]),H:[-diff(A,z),B/R^2,1/R*diff(R*A,R)])$

(lg:matrix([1,0,0],[0,R^2,0],[0,0,1]),ug:invert(lg),constant(nu))$

christof(false)$

("evaluate the generated eqs")$
(MCS:mcs,LG:lg,UG:ug,DIM:3,ev(exp))$
("Now look the equation for the angular velocity")$
expand(ratsimp(ratexpand(comp[2])));

("the balance of the momentum on the hieght")$

expand(ratsimp(ratexpand(comp[3])));
("the radial balance of the momentum")$
expand(ratsimp(ratexpand(comp[1])));