Example 10.5-1. Tangential Flow in an Annulus with Viscous Heat Generation

>  restart;
>  vt:=r->Omega*R*((r/(kappa*R)-kappa*R/r))/((1/kappa)-kappa); 
eq. 10.5-4
>  tde:=mu*(r*diff(vt(r)/r,r))^2: second term in eq. 10.5-5
>  tde:=simplify(%); compare with second term in eq. 10.5-6
>  de:=k*diff(r*diff(T(r),r),r)/r + tde;
>  tr:={r=xi*R,T=Tk+Theta*(T1-Tk)};
>  with(PDEtools,dchange);
>  newde:=dchange(tr,R^2*de/(k*(T1-Tk)),[xi,Theta(xi)]);
>  mu:=N*k*(1-kappa^2)^2*(T1-Tk)/(Omega^2*R^2*kappa^4);
>  newde;
>  newde:=simplify(%); Compare to eq. 10.5-10
>  s:=dsolve({newde,Theta(kappa)=0,Theta(1)=1},Theta(xi)); 
solving the DE and inserting the B. C. 10.5-12&13

>  assign(s);
>  Theta:=unapply(Theta(xi),xi):
>  Thetmayb:=((N+1)-N/xi^2)-((N+1)-N/kappa^2)*ln(xi)/ln(kappa); eq. 10.5-14
>  simplify(Thetmayb-Theta(xi)); Confirms eq. 10.5-14