= Zero-contact-angle wall meniscus
{title2=$z_0=\sqrt2\ell_c$}
A static liquid meniscus rising on a perfectly wetting vertical wall has dimensional height $z_0=\sqrt2\ell_c$, where $\ell_c=\sqrt{\sigma/(\rho g)}$ is the <capillary length>. In capillary-length coordinates, write the interface $x=h(z)$. The <Young–Laplace equation> gives $h_{zz}/(1+h_z^2)^{3/2}=z$ and the first integral $h_z/\sqrt{1+h_z^2}=z^2/2-1$. The integration constant follows from the horizontal far bath, where $z\to0$ and $h_z\to-\infty$. Zero contact angle means $h=h_z=0$ at the wall, so $z_0=\sqrt2$. Expanding and integrating there gives $h\sim(z-z_0)^2/\sqrt2$, the parabolic shape used to match an entrained film.
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