= Thermal dike-widening similarity
{title2=$b\propto\sqrt t,\quad z\propto\sqrt t$}
With equal solid and liquid densities, <magma> entering a <dike> at constant flux $Q_0$ has $bw=Q_0$: melt added by the walls exactly supplies the increasing cross-sectional storage. If wall heat flux is $\alpha_T\rho c_pw(T-T_m)$, both walls give $b_t=2\alpha_Tc_pw(T-T_m)/L$. Combining this with sensible-heat conservation admits $b=B(t)f(\eta)$, $\eta=2\alpha_Tz/B$, $B^2=4\alpha_TQ_0t/S$. The shape equations are $f(f-\eta f')=\vartheta$ and $(S-\eta f)\vartheta'=-(S+\vartheta)\vartheta/f$.
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