singularities

Singularities in charged droplets

In this project we study the evolution of charged droplets of a conducting viscous liquid. The flow is driven by electrostatic repulsion (which pulls the droplet apart) and capillarity (that stabilizes the droplet). Charged droplets are known to be linearly unstable when the electric charge Q is above the Rayleigh critical value.

Here we investigate the nonlinear evolution that develops after the linear regime. Using a boundary elements numerical method, we found that a perturbed sphere with critical charge evolves into a fusiform shape with conical tips at time T, and that the velocity at the tips blows up as (T-t)^alpha, with alpha close to -1/2. In the neighborhood of the singularity, the shape of the surface is self-similar, and the asymptotic angle of the tips is smaller than the opening angle in Taylor cones.

The figure above depicts a charged conducting droplet surrounded by a dielectric (for example water in air). The charges cannot escape the droplet if ionization is negligible. The self-generated electric field tends to deform the drop, while the surface tension has a stabilizing effect.

Experimental work by T. Achtzehn, R. Müller, D. Duft and T. Leisner, suggest that when the droplet becomes unstable, it develops a fusiform shape with apices, and later jets of liquid emanate from these tips, as shown in the following figure:

In this work we want to see if we can justify the formation of tips by using a simple fluid dynamical model. We would like to know what happens after the droplet destibilizes, if there are non-spherical equilibria, and to describe how the droplet splits into smalles droplets.

Mathematical formulation

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.

Linear stability of the surface

We can get some insight on the stability of the droplet by looking at the linearized solutions of a perturbed sphere of radius R

where the perturbing term is a spherical harmonic of indices l and m, which describe the shape of that particular mode. By substituting this expression into the equations of motion and dropping all quadratic terms we obtain g in terms of the charge Q, surface tension and the viscosities of the fluid in the exterior and interior of the drop

When this quantity is positive, the drop is unstable, and when is negativ, it is stable. For the case l=2, it gives the critical charge described above.

Numerical solution of the initial value problem

These equations of motion can be solved using the Boundary Elements Method. First we write the relation between the electric potential and the charge density at the surface of the drop

Then we discretize the shape of the droplet with a series of conical rings (for axi-symmetric solutions) or with triangles (for general 3D shapes), and the charge density sigma is solved from the resulting linear system.
Then we repeat the same procedure for the velocity of the fluid

where we have used the Green functions and the force per unit area at the surface. At this point we have the solved the velocity u. Then we move the surface using this velocity where f is the capillary and electric force.
As the surface may develop regions of high curvature, we use an adaptive grid, the size of the triangles adapt to the local curvature of the surface: they are smaller in regions of high curvature. We use a combination of techniques: elastic relaxation, addition-deletion of triangles and modified Delaunay triangulation.

Selfsimilar dynamics

In the following figures, we perturbed a sphere with a low amplitude (0.1) mode Y20, and then we let it evolve, the tips formation is evident (in the above figure, the viscosities inside and outside the drop are equal, in the lower one, the viscosity outside the droplet is near zero)

Now we rescale the above figures with the radius of curvature of the tip, and all the asymptotic profiles lie in the same curve. This is evidence of selfsimilarity. The selfsimilarity exponent is close to 0.5.

Non axisymmetrical simulations

Now, instead of discretizing the droplet with conical rings, we use triangles, so we can describe arbitrary three dimensional shapes (see the Master's Thesis of Orestis Vantzos for details).

The following image shows that axi-symmetric droplets develop tips even if we do not restrict the symmetry of the numerical grid to axial symmetry. It suggests that the selfsimilar solution is stable respect to non-axi-symmetric perturbations.

The next figure shows that even if the shape of the droplet is FAR from axisymmetric, still it may develop tips which are locally axisymmetric.
gota3d2

This is anothe example of the same fenomenon, with a droplet with tetrahedral symmetry. The initial shape, not shown in this images, is a sphere perturbed with the mode Y32.
gota3d3

Finally, not all droplets develop tips! If the initial charge is more than twice the critical charge, the droplet just begins to split without showing tips in the intermediate steps. This example is also interesting because it shows that the axi-symmetry is mantained, however the numerical grid is not axi-symmetric.

And the corresponding triangulation

And finally, the result that really matters: the comparison with the experiments of Beauchamp (simulation by Orestis Vantzos). The red drop is the numerical simulation, the grey shapes are photographs of real drops. In this setup, the drop is subject to an external electric field.