Example 11.4-2. Tangential Flow in an Annulus with Viscous Heat Generation
>  restart;
There is only one velocity component (the tangential one) and it depends only on r.
The temperature also depends only on r. Thus in eq. B.9-2, all terms on the LHS
are zero. Only the first term in the first bracket on the right is non-zero.
Only one term in eq. B.7-2 is non-zero, leaving eq. 11.4-4 as the energy equation.
>  vtheta:=r->Omega*R*((r/(kappa*R)-kappa*R/r))/((1/kappa)-kappa); eq. 3.6-29
>  tde:=mu*(r*diff(vtheta(r)/r,r))^2: second term in eq. 11.4-4
>  tde:=simplify(%); compare with second term in eq. 11.4-5
>  de:=k*diff(r*diff(T(r),r),r)/r + tde;
[Maple Math]
>  mu:=N*k*(T1-Tk)*(1-kappa^2)^2/(Omega*R*kappa^2)^2; replacing the viscosity with the dimensionless variable N
>  tr:={r=xi*R,T=Tk+Theta*(T1-Tk)}; The transformation set to be used in dchange
>  with(PDEtools,dchange); dchange is part of the PDEtools package
>  newde:=dchange(tr,R^2*de/(k*(T1-Tk)),[xi,Theta(xi)]);
>  newde:=simplify(newde); Compare to eq. 11.4-9
>  s:=dsolve({newde,Theta(kappa)=0,Theta(1)=1},Theta(xi)); solving the DE and inserting the B. C. 11.4-11&12
>  assign(s);
>  Theta:=unapply(Theta(xi),xi,N,kappa);
[Maple Math]
>  Thetbook:=(1-log(xi)/log(kappa))+N*((1-1/xi^2)-(1-1/kappa^2)*log(xi)/log(kappa)); eq. 11.4-13
>  simplify(Thetbook-Theta(xi,N,kappa)); Confirms eq. 11.4-13
[Maple Math]
>  with(plots):
>  p1:=plot(Theta(xi,0,.5),xi=.5...1.0):
>  p2:=plot(Theta(xi,0,.9),xi=0.9...1.0):
>  p3:=plot(Theta(xi,1,.5),xi=.5...1.0,color=green):
>  p4:=plot(Theta(xi,20,.9),xi=.9...1.0,color=blue):
>  display({p1,p2,p3,p4});
[Maple Plot]
>