Spreading of charged droplets
What happens when a charged droplet spreads on a flat, smooth solid? Here we consider the spreading of a charged conducting droplet on a flat dielectric surface. Below we sketch the droplet with a charge Q, which is concentrated at the surface because the droplet is a conductor. We also show the electric field lines on the right half. These electric field tries to pull the droplet apart.
Then two forces drive the spreading: surface tension and electrostatic repulsion. By using the lubrication approximation we derive a fourth order nonlinear partial differential equation that describes the evolution of the height profile. We find that the equation has a two-parameter family of selfsimilar solutions. Some of the solutions are explicitly computed while the other solutions are studied numerically. We show that the solutions have moving contact lines and the radius of the drop is a power law of time with exponent one-tenth. We also construct explicit solutions corresponding to non-circular drops, whose interfaces are ellipses with constant focal length.

Mathematical formulation: general equations
As usual, we are going to assume that the inertia effects are negligible,

so we can describe the fluid flow with the Stokes equations, valid both inside and outside the droplet,
with normal stresses given by (using repeated index summation convention)
and zero tangential stress
These equations are written using the usual stress tensor for newtonian fluids
The electric field is assumed to be electrostatic on the exterior of the drop,
while in the interior, the electric field is zero, because the drop is made of a conductor fluid. That means that the electric potential is a constant at the surface. The surface evolves because the normal component of the surface velocity equals the normal  component of the fluid velocity. At the solid-liquid interface, the velocity of the fluid is assumed to be zero (the non-slip boundary condition).

Solution of the fluid flow equations: lubrication approximation
If the droplet is thin, we can use the lubrication approximation (the velocity field is supposed to be nearly horizontal, inertia effects negligible and viscous stressed assumed to be mainly due to the gradient of the horizontal velocity in the vertical direction). We assume that the shape of the surface of the drop is a graph of the form z=h(r,t), then the horizontal velocity field averaged in the z-direction reduces to
and the profile of velocity is parabolic in the z-direction

From the conservation of mass, we obtain a PDE for h

Within this approximation, the pressure is given by

The spreading occurs because the electric field and the charge density sigma concentrate near the border of the drop. Then a gradient of pressure is generated between the center of the drop and its border, which drives the flow.

Approximation for the electric field

We approximate the electric field by the field generated by a flat ellipse of negligible thickness with semiaxis a and b. (If the drop is circular, a=b). The corresponding charge density at a position (x,y) at the surface is

We did a comparison between this approximation and the ''exact''  charge density as computed with a 3D numerical code. The agreement is excellent provided the drop is thin  (height/radius<0.01). In the figure we show the charge density versus dimentionless radius rho=r/a for parabolic drops with a=b. The arrow indicates decreasing values of the ratio height/a= 0.1, 0.05, 0.01

Putting the lubrication equations together with the approximation of the charge density, we obtain a PDE for the drop shape

which must be solved subject to the following initial and boundary conditions:

Selfsimilar solutions
The above PDE admits solutions of the form

where H(rho) is the rescaled profile, and is a compactly supported function that satisfies the ODE and boundary conditions

The ODE has an explicit solution! it is given by

But this is not the only solution: Others can be found numerically, using a Runge-Kutta scheme for example. Below we depict some profiles:

All the solutions describe a spreading with a similarity exponent equal to 1/10. The explicit solution is the least singular of all of them, and it has a finite slope everywhere. The other solutions have logarithmic singularities for the derivative at the front, and they have infinite slope (violating the lubrication approximation), so we suspect that the only physical solution is the explicit one.
Non-circular solutions
It is possible to construct solutions where the droplet is an ellipse of semiaxis a(t) and b(t),

by using an ansatz of the form

From the conservation of the volume V, we obtain

and by simple substitution into the PDE one finds the equations for the radii:

These equations can be solved explicitly without difficulty, and one finds that the drops have constant focal length, and that they tend to circles as time approaches infinity. That appears to indicate that the elliptical drops are stable.