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.