Section 9.4: Heat Conduction with a Viscous Heat Source

pp. 276-279

>  restart;
The velocity is assumed to vary linearly between the cylinders
>  Vz :=x-> (x/b)*V;  equation 9.4.2
>  Sv := mu*(diff(Vz(x),x))^2;  equation 9.4.1: the rate of 
viscous dissipation per volume
>  A:= W*L;  define area of conduction
>  Q:=x->-k*A*D(T)(x); Heat flow at any x
>  eq:=limit((Q(x)-Q(x+dx))/dx,dx=0)+A*Sv=0;  energy balance: 
In - out + rate of production =0.
>  s:= dsolve({eq,T(0)=T0, T(b)=Tb},T(x));  solve the equation 
with BCs #1 and #2: equation 9.4-9 and 9.4-10
>  assign(s); T:=unapply(T(x),x);   
>  Theta:=xi->simplify((T(xi*b)-T0)/(Tb-T0)); The LHS of 
eq. 9.4-11 and xi=x/b
>  Theta(xi); 
>  mu:=Br*k*(Tb-T0)/V^2; Defining the Brinkman No. and replacing 
the viscosity with it.
>  Thet:=simplify(Theta(x/b));
Thetbook:=((x/b)+(1/2)*Br*(x/b)*(1-(x/b))); 
difference:=simplify(Thet-Thetbook);  Here is what we found 
for the left hand side of eq. 9.4-11 compared to the book's answer.