Stability of Relative Trajectories of Contaminated Spherical Drops in Combined Gravitational and Thermcapillary Motion

The motion of two interacting, spherical drops in the presence of

gravity and a constant, parallel temperature gradient is studied, with negligible

thermal convection and nearly uniform surfactant coverage.

At small Stokes and Reynolds numbers, the governing equations are

linear, and the trajectories are symmetric. 

In this linear case, it is possible for two drops to fall as a pair with

constant horizontal separation. 

Moreover, a saddle point can occur in the trajectory phase plane.

By finding the eigenvalues of the

system, the stability of the arrangements is investigated.

At finite Stokes numbers but small Reynolds numbers, the governing

equations become non-linear, and the trajectories lose symmetry. 

Asymmetric limit cycles can be observed.  In addition, trajectories with

a constant, finite horizontal gap become stable, with the drops oscillating into

the final steady state arrangement.  Complicated, retrograde motion is also possible.

The effect of viscosity and thermal conductivity ratios on stability of

the linearized system at realistic physical values is presented, by

considering perturbations to the orbits.