 Gravity-capillary drop with a meridional curvature approximation

If an axisymmetric thin-film equation is deliberately closed using only the meridional curvature $\kappa =-h^{\prime \prime }$, its regular equilibrium obeys $h^{\prime \prime \prime }=h^{\prime }/l^2$, with capillary length $l=\sqrt{\gamma /(\rho g)}$. The boundary conditions $h^{\prime }(0)=0$, $h(R)=0$, $h^{\prime }(R)=-\tan {\theta }_c$ give the displayed profile and $\alpha =\tan {\theta }_c/\sinh (R/l)$. Its volume is $2\pi \alpha l^3[s^2\cosh s/2-s\sinh s+\cosh s-1]$, $s=R/l$. This model omits the azimuthal contribution of axisymmetric free-surface curvature. Retaining that term instead gives $h=l\tan {\theta }_c[I_0(R/l)-I_0(r/l)]/I_1(R/l)$ in terms of modified Bessel functions.