= Conserved momentum flux of a porous thermal plume
{title2=$M=\beta F_h/C_h,\quad Q=(36\kappa Mz)^{1/3},\quad B=g\alpha F_h/C_h$}
For a slender <porous thermal plume>, <Darcy law> gives $w=\beta(T-T_0)$, with $\beta=Kg\alpha/\nu$. If $C_h=\rho_0c_p$ and $F_h=C_h\int w(T-T_0)dx$ is the conserved excess convective <heat flux> per span, then the kinematic <momentum flux> is $M=\int w^2dx=\beta F_h/C_h$, and the <buoyancy flux> is $B=\int g\alpha(T-T_0)w dx=g\alpha F_h/C_h$. The <hyperbolic-secant porous plume profile> has $Q=(36\kappa Mz)^{1/3}$. Multiplication of $Q$ and $M$ by $\rho_0$ gives physical <mass flux> and the density-weighted Darcy second moment. Intrinsic pore-velocity momentum flux instead has an additional inverse-porosity factor. Momentum conservation here follows from the heat/velocity proportionality, not from an inertia balance: porous drag balances buoyancy locally.
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