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 β=110^{∘} (hydrophobic) and β=160^{∘} (superhydrophobic) which correspond to the experimental study of Dash et al. [Phys. Rev. E 90, 062407 (2014)PLEEE81539-375510.1103/PhysRevE.90.062407]. The droplet with β=110^{∘} is stable while the one with β=160^{∘} is unstable to the azimuthal wave number m=1 mode. This suggests that the experimental observation for a droplet with β=110^{∘} corresponds to the steady toroidal base flow, while for β=160^{∘}, the m=1 instability promotes the rigid body rotation motion. A marginal stability analysis for different β shows that a 3-μL water droplet is unstable to the m=1 mode when the contact angle β is larger than 130^{∘}. A marginal stability analysis for different volumes is also conducted for the two contact angles β=110^{∘} and 160^{∘}. The droplet with β=110^{∘} becomes unstable when the volume is larger than 3.5μL while the one with β=160^{∘} is always unstable to m=1 mode for the considered volume range (2-5μL). In contrast to classical buoyancy driven (Rayleigh-Bénard) 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.