= Spherical freezing inside a porous cavity
{title2=$\tau=x^2/2+x^3/(3K)$}
A spherical solid of radius $a$ growing inside a fixed cavity of radius $R$ expels volume at rate $4\pi(1-\rho_s/\rho_l)a^2\dot a$. The <Darcy flux> outside the cavity gives $p-p_m=\mu(1-\rho_s/\rho_l)a^2\dot a/(\Pi R)$. The <common-pressure melting-point slope> and quasi-steady spherical <thermal conduction> give $\rho_sLa\dot a=k(T_i-T_\infty)$. Define $x=a/R$, $\tau=tk(T_m-T_\infty)/(R^2\rho_sL)$ and $K=L^2\Pi/[kT_m\mu(1/\rho_s-1/\rho_l)^2]$. Elimination yields $(x+x^2/K)x_\tau=1$ and the displayed zero-seed integral. The pressure rise is $p^*x/(x+K)$, with $p^*=L(T_m-T_\infty)/[T_m(1/\rho_s-1/\rho_l)]$. Easy drainage gives square-root growth; low permeability gives an initial square-root regime followed by cube-root growth and pressure saturation. The model stops at $x=1$ and uses a uniform exterior conductivity.
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