Example 10.5-4 Free Convection Heat Transfer From a Vertical Plate:

Part a: Deriving the dimensionless equations.

>  restart;
>  with(linalg):
>  v:=(y,z)->vector([vy(y,z),vz(y,z)]);
>  s:=vector([y,z]);
>  econt:=diverge(v(y,z),s)=0;
>  emot:=rho*multiply(v(y,z),grad(vz(y,z),s))=
mu*diverge(grad(vz(y,z),s),s)+rho*g*beta*(T(y,z)-T1);
>  een:=rho*Cp*multiply(v(y,z),grad(T(y,z)-T1,s))=
k*diverge(grad(T(y,z)-T1,s),s);
>  with(PDEtools,dchange);
>  tr:={T=T1+Theta*(T0-T1),z=zeta*H,
y=eta*(mu*alpha*H/B)^(1/4),vz=phiz*(B*alpha*H/mu)^(1/2),
vy=phiy*(alpha^3*B/(mu*H))^(1/4)};
>  tr[2..5]; In the next step we will need to use only four of 
the transformations in tr. In fact Maple will object if we try 
to use them all. We can select those four, but we need to see 
where Maple put them in tr. Maple does not store the elements 
in a set in the same order you give them!
>  newcont:=dchange(tr[2...5],econt,[eta,zeta,phiy(eta,zeta),
phiz(eta,zeta)]);
>  simplify(newcont,assume=positive); This gives 10.5-44
[Maple Math]
>  newmot:=dchange(tr,emot,[eta,zeta,phiy(eta,zeta),
phiz(eta,zeta),Theta(eta,zeta)]);
>  simplify(%,assume=positive);
>  k:=alpha*rho*Cp;beta:=B/(rho*g*(T0-T1));mu:=Pr*k/Cp;
>  simplify(newmot/B,assume=positive); This is the same 
as 10.5-45 except for the second derivative of phiz wrt zeta. 
This came from the corresponding term in 10.5-34 that was neglected 
in the text.
[Maple Math]
[Maple Math]
>  newen:=dchange(tr,een,[eta,zeta,phiy(eta,zeta),
phiz(eta,zeta),Theta(eta,zeta)]);
>  simplify(%,assume=positive);
>  simplify(sqrt(Pr*H/(rho*B))*%/(Cp*(T0-T1)));
>  simplify(%,assume=positive); This is identical to eq. 10.5-46 
except it includes a term that was neglected in the text.
[Maple Math]
[Maple Math]