The equilibrium and stability of sessile drops

The paper gives a mathematical analysis of the equilibrium and stability of two dimensional and axisymmetric sessile drops, when controlled by their volume or by their internal pressure. Sessile drops of either form when standing on a horizontal plane with a prescribed contact angle are shown to be stable when controlled by their volume or pressure, except for the neutrally stable sideways translation, without change of shape, which can occur formally. For two dimensional drops grown on a fixed base, and controlled by their volume, plane symmetric, plane asymmetric, and three dimensional forms of instability can occur, the last named being the first one to occur, at the position of the internal pressure maximum. For the same drop when controlled by its pressure plane symmetric and three dimensional instabilities arise together at the pressure maximum. For axisymmetric drops grown on a fixed base and controlled by their volume, axisymmetric and asymmetric instabilities can arise. Asymmetric instability with azimuthal wavenumber $m$ = 1 arises when the drop profile becomes horizontal at its base, and this instability always precedes the axisymmetric instability arising at the volume maximum. However, the same drop when controlled by its pressure becomes unstable first in the axisymmetric mode arising at the pressure maximum.