> restart;

Energy Balance using Fig 2.

S[v]=volume heat source produced viscous dissipation

> S[v]:=-tau[xz]*D(v[z])(x);

Since the fluid is Newtonian:

> tau[xz]:=-mu*D(v[z])(x);

> S[v];

For steady laminar flow of a fluid with constant viscosity, the velocity profile is linear:

> v[z]:=x->(x/b)*V;

V=R*omega as by definition of linear velocity due to angular rotation

> S[v];

> A:=W*L;

> Q:=x->q(x)*A;

Division by volume and taking the limit as dx->0 gives:

> eq1:=limit((Q(x)-Q(x+dx))/(A*dx),dx=0)=-S[v]*A*dx/(A*dx);

Since nothing is known about the heat flux for any value of x, we cannot integrate using boundary conditions

> s1:=dsolve({eq1},q(x));

Fourier's Law:

> q(x):=-k*D(T)(x);

> s1;

At x=0, T=To and at x=b, T=Tb

> s2:=dsolve({s1,T(0)=T[o],T(b)=T[b]},T(x));

> assign(s2); T:=unapply(T(x),x);

Checking equation 9.4-11

> Tbook:=x->(x/b+mu*V^2*x*(1-x/b)/(2*k*(T[b]-T[o])*b))*(T[b]-T[o])+T[o];

> simplify(T(x)-Tbook(x));

It works!