 Quasisteady drag-out matching law

In a thin wall-withdrawal model with large dimensionless surface-tension parameter $\hat{C}$, matching a parabolic static meniscus fixes inner longitudinal scale $ϵ={\hat{C}}^{-1/3}$ and thickness scale $\delta ={ϵ}^2$. The leading inner equation is $v_0{H}^{\prime }+(H^3{H}^{(3)}{)}^{\prime }/3=0$, matching $H\sim {\bar{z}}^2/\sqrt{2}$ 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 $3g^{\prime }+(g^3{g}^{(3)}{)}^{\prime }=0$. 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.