I came across a somewhat interesting differential equation while studying the shape of a meniscus ring formed at the bottom of a cylinder. Here are some 2D cross sections through a cylinder symmetry plane, computed using numerical integration, assuming a "wetting" surface, i.e. a contact angle of zero:
The equilibrium condition is the requirement of constant mean interface curvature. This requirement can be derived from the assumptions of uniform ambient pressure outside the fluid and constant hydrostatic pressure inside the fluid, both valid for menisci small enough to justify the neglect of effects due to gravity. The Young-Laplace equation then asserts that the pressure difference across a curved interface is related by a multiplicative constant (called the surface tension) to the mean interface curvature. Different profiles can thus be characterized by their initial radii of curvature $0<r_o<1$ at the vertical walls in the vertical direction.
The dotted lines are circular profiles with radii equal to $r_o$. An interesting property of the meniscii is that the profiles are circular both in the limit $r_o \to 1$ and $r_o \to 0$. In between these limits one sees that the radius of curvature is non-constant, the necessity of which can be easily appreciated by noting that the curvature in the circumferential direction is 1 at the vertical walls and 0 at the bottom of the cylinder.
Parameterizing the left profile by the horizontal coordinate $x$ as a function of the angle $\alpha$ between the curve tangent (starting at the vertical wall and going down) and the horizontal axis we have $$ x = \int_{-90^\circ}^\alpha d \alpha \frac{\cos \alpha}{1 + \beta + \frac{\sin \alpha}{x}} - 1 $$
where $\beta \equiv 1 / r_o$. This integral equation is derived from the constraint of mean interface curvature, which by inspecting the curvature at the vertical wall is seen to be $\frac{1}{2}\left( 1 + \beta \right)$.
Do these profiles (parameterized by $\beta$) have a name? Can the profiles, whether they're expressed as a function $y(x)$ or $x(\alpha)$ or what have you, be expressed for arbitrary $\beta$ in terms of elementary functions? Just curious.
Capillary Miniscus depth Equation (1)
In its simplest form/cases and equilibrium consideration we get $ R_m$ for surface mean curvature
$$ \frac{1}{R_1} +\frac{1}{R_2} =\frac{2}{R_m}$$ belong to the family named after Delaunay, as Delaunay Unduloids.
There are three varieties. These roulettes have maximum volume for given surface area in contact with air.
They are classed as CMC surfaces. The ordinary differential equation is
$$ \cos \phi= \frac{a}{r} + \frac{r}{b} $$
when $r_{}$ is meniscus radius, $\phi$ is slope angle to be sure and $(a,b)$ are constants depending on geometry/ relative adhesive/cohesive force of liquid determining contact angle between glass tube/water.