= Quasisteady drag-out matching law
{title2=$H_\infty=0.948v_0^{2/3}$}
In a thin wall-withdrawal model with large dimensionless surface-tension parameter $\widehat C$, matching a parabolic static meniscus fixes inner longitudinal scale $\epsilon=\widehat C^{-1/3}$ and thickness scale $\delta=\epsilon^2$. The leading inner equation is $v_0H'+(H^3H^{(3)})'/3=0$, matching $H\sim\bar z^2/\sqrt2$ below and a bounded film above. Scaling $H=v_0^{2/3}g$, $\bar z=v_0^{1/3}\zeta$ preserves the quadratic matching coefficient and gives the universal equation $3g'+(g^3g^{(3)})'=0$. Its selected bounded solution tends to $0.948$, giving the displayed dimensionless law. It applies to positive slowly varying wall speed in the leading matched regime; the numerical constant comes from that universal inner problem, not a dimensional identity.
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