Section 10.9: Free Convection
Looking at an element dy by H by W in an energy balance.
Looking at an element dy by dz by W in a momentum balance.
>  restart;
>  A:=H*W;Ay:=W*dz;Az:=W*dy;V:=dz*dy*W;  H is the height and W the width of the plate. A is the area of each plate. Ay is the area of the element used in the momentum balance perpendicular to the y direction and Az is the area of the element perpendicular to the z direction.
Working on an energy balance first. The temperature varies only in the y direction.
>  Q:=y->A*qy(y); heat transfer rate in the y direction
>  de:=limit((Q(y)-Q(y+dy))/(A*dy),dy=0); eq. 10.9-1, left eq.
>  qy:=y->-k*D(T)(y); Fourier's Law.
>  de; eq. 10.9-1 right equation
>  s:=dsolve({de=0,T(-B)=T2,T(B)=T1},T(y)); solving the de and using the bc: 10.9-2,3
>  assign(s);T:=unapply(T(y),y);
>  eq1:=(T1+T2)/2=Tm;eq2:=T2-T1=dT;s:=solve({eq1,eq2},{T1,T2}); replacing T1 and T with the mean temp. Tm and the difference in temp: dT.
>  assign(s);T(y); eq 10.9-4
Now using a momentum balance on an element dy by dz by W
>  tauyz:=y->-mu*D(vz)(y);
Newton's Law: eq. 1.1-2 giving the z component of the momentum acting on the face perpendicular to the y direction.
>  dem:=((tauyz(y)-tauyz(y+dy))*Ay+(p(z)-p(z+dz))*Az)-rho(y)*g*V;
The momentum balance.
>  de:=limit(limit(dem/V,dy=0),dz=0); eq. 10.9-6
>  rho:=y->rhobar*(1-beta*(T(y)-Tbar)); Truncated form of 10.9-6
>  de; eq. 10.9-9, but only if we take: Tm=Tbar.
>  Tm:=Tbar;de;
>  s:=dsolve({de,vz(-B)=0,vz(B)=0},vz(y));
>  assign(s);vz:=unapply(vz(y),y);
>  vzbook:=y->rhobar*g*beta*dT*B^2/(12*mu)*((y/B)^3-y/B)+B^2/(2*mu)*(diff(p(z),z)+rhobar*g)*((y/B)^2-1);
>  simplify(vz(y)-vzbook(y));
>  eq:=int(rho(y)*vz(y),y=-B...B);
>  de:=simplify(eq/rhobar*mu/B^3*3/2); This comes up with an extra term over what the book says. If we neglect it:
>  eq:=D(p)(z)+rhobar*g=0;
>  s:=dsolve(eq,p(z));
>  assign(s);p:=unapply(p(z),z);

>  vz(y); eq. 10.9-15
>  vzbook:=y->rhobar*g*beta*dT*B^2/(12*mu)*((y/B)^3-y/B);
>  simplify(vz(y)-vzbook(y));
>  int(vz(y)*rho(y),y=-B...B); This is still hanging around.
>  vztil:=ytil->B*vz(ytil*B)*rhobar/mu;
>  beta:=Gr*mu^2/rhobar^2/g/dT/B^3;simplify(vztil(ytil)); eq.10.9-17 with Gr in 10.9-18 replacing beta

>