= Uniformly heated vertical plate in a porous medium
{title2=$f'''+\tfrac23ff''-\tfrac13(f')^2=0$}
For constant wall heat-flux density $F>0$, effective <thermal conductivity> $k_e$, buoyancy mobility $B=\rho_0g\alpha K/\mu$ and thermal transport coefficient $\kappa=k_e/(\rho_0c_f)$, a slender <Darcy flow> boundary layer has width $\delta=(\kappa k_e z/(BF))^{1/3}$. Writing $\eta=y/\delta$, $\psi=\kappa zf(\eta)/\delta$ and $T-T_a=(F\delta/k_e)f'(\eta)$ gives the displayed equation with $f(0)=0$, $f''(0)=-1$ and $f'(\infty)=0$. The exact integral is $\int_0^\infty(f')^2d\eta=1$. Consequently its kinematic Darcy momentum flux is $\int w^2dy=B\kappa Fz/k_e$ and its buoyancy flux is $g\alpha\kappa Fz/k_e$. These are per span on one heated face.
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