Uniformly heated vertical plate in a porous medium ($f^{\prime \prime \prime }+\frac{2}{3}f{f}^{\prime \prime }-\frac{1}{3}(f^{\prime }{)}^2=0$)

For constant wall heat-flux density $F>0$, effective thermal conductivity $k_e$, buoyancy mobility $B={\rho }_{0}g\alpha K/\mu$ and thermal transport coefficient $\kappa =k_e/({\rho }_0{c}_f)$, 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 $T-T_a=(F\delta /k_e)f^{\prime }(\eta )$ gives the displayed equation with $f(0)=0$, $f^{\prime \prime }(0)=-1$ and $f^{\prime }(\infty )=0$. The exact integral is ${\int }_0^{\infty }(f^{\prime }{)}^{2}d\eta =1$. Consequently its kinematic Darcy momentum flux is $\int w^{2}dy=B\kappa Fz/k_e$ and its buoyancy flux is $g\alpha \kappa Fz/k_e$. These are per span on one heated face.

Assume an exponentially decreasing temperature profile while imposing wall heat flux, impermeability and the exact integrated heat budget. For $f=C(1-e^{-\eta /d})$, the wall condition gives $C=d^2$ and $\int (f^{\prime }{)}^{2}d\eta =1$ gives $d^3=2$. This yields the displayed integral approximation and discharge $\int wdy\simeq 2^{2/3}\kappa z/\delta$. It satisfies the boundary conditions and exact flux budget, but not the differential equation pointwise; its mass-flux coefficient is approximate.