Hyperbolic-secant porous plume profile
= Hyperbolic-secant porous plume profile
{title2=$w=W\operatorname{sech}^2(x/b)$}
The symmetric steady <porous thermal plume> has $w=W\operatorname{sech}^2(x/b)$, $T-T_0=w/\beta$ and $\psi=-Wb\tanh(x/b)$. Its exact normalizations are $Wb^2=6\kappa z$ and $(4/3)W^2b=Q$, so $b=(48\kappa^2z^2/Q)^{1/3}$ and $W=(3Q^2/(32\kappa z))^{1/3}$. The <streamfunction> supplies lateral entrainment required by continuity.