Modeling Droplet Levitation Over a Heated Liquid Surface

Research into the interactions between droplets and liquid surfaces is of importance for a number of practical applications. These applications range from spray cooling techniques in engineering, to infectious disease transmission, to the distribution of aerosolized pharmaceuticals and various challenges in particle transport in multiphase flows. In this study, we focus on the dynamics of a slowly condensing droplet, suspended above an evaporating liquid layer. The key objective of the present study is to formulate comprehensive mathematical models that describe the phenomena of diffusion and heat transfer occurring within this system. Using this, we model the flow around the droplet and the force on the droplet. We employ the method of separation of variables in bipolar coordinates for both fluid flow and heat transfer models. We derive series expansions that describe the temperature distribution within the droplet itself and around it, as well as the vapor concentration in the air surrounding the droplet. This framework allows us to obtain the temperature profiles and condensation rates both at the surface of the droplet and along the surface of the liquid layer. Using a similar methodology, we find analytical expressions for the Stokes stream function and force on the droplet, and are able to make conclusions about the levitation height as a function of the droplet radius. The analytical method is then improved upon by considering the temperature distribution in the liquid layer as spatially variable. A coupled numerical and analytical approach is used to model the heat and mass transfer in the system. Above the liquid layer, we use separation of variables in bipolar coordinates. However, below the layer surface, the geometry is not suitable for the use of bipolar coordinates, so we employ a finite difference scheme in polar coordinates. The two solution methods are coupled at the boundary via the interface boundary conditions. The modification of the original analytical model leads to more accurate predictions for the force on the droplet and levitation height.