Buoyancy-driven convection of droplets on hot nonwetting surfaces

The global linear stability of a water drop on hot nonwetting surfaces is studied. The droplet is assumed to have a static shape and the surface tension gradient is neglected. First, the nonlinear steady Boussinesq equation is solved to obtain the axisymmetric toroidal base flow. Then, the linear stability analysis is conducted for different contact angles beta = 110. (hydrophobic) and beta = 160 degrees (superhydrophobic) which correspond to the experimental study of Dash et al. [Phys. Rev. E 90, 062407 (2014)]. The droplet with beta = 110 degrees is stable while the one with beta = 160 degrees is unstable to the azimuthal wave number m = 1 mode. This suggests that the experimental observation for a droplet with beta = 110 degrees corresponds to the steady toroidal base flow, while for beta = 160 degrees, the m = 1 instability promotes the rigid body rotation motion. A marginal stability analysis for different beta shows that a 3-mu L water droplet is unstable to the m = 1 mode when the contact angle beta is larger than 130 degrees. A marginal stability analysis for different volumes is also conducted for the two contact angles beta = 110 degrees and 160 degrees. The droplet with beta = 110 degrees becomes unstable when the volume is larger than 3.5 mu L while the one with beta = 160 degrees is always unstable to m = 1 mode for the considered volume range (2-5 mu L). In contrast to classical buoyancy driven (Rayleigh-Benard) problems whose instability is controlled independently by the geometrical aspect ratio and the Rayleigh number, in this problem, these parameters are all linked together with the volume and contact angles.