Linear stability of thermocapillary convection in non-volatile sessile droplets on a heated substrate

The linear stability of the incompressible steady axisymmetric thermocapillary flow in spherical sessile droplets is calculated numerically. The governing equations are discretized on Taylor-Hood finite elements using FEniCS. A combination of Newton's law of cooling and radiative heat transfer is imposed on the free surface. We compute the dependence of the critical Marangoni number on the contact angle for a range of Prandtl and convective and radiative Biot numbers. As the contact angle is increased from small values the basic flow is destabilized and the critical Marangoni number reaches a minimum. The minimum is followed by a very strong stabilization which is associated with a frequent change of the critical mode and a partial re-stabilization when the neutral curves turn backward. We find a range of intermediate contact angles where the basic flow is stable up to high Marangoni numbers. When the contact angle is increased even further, the basic flow is destabilized again.