Dust Free Zones in Natural Convection Boundary Layers Over Horizontal Plate

We present a study on the dust free region above hot horizontal surfaces of uniform temperature in natural convection and propose relations to predict its height in the limit of small particle inertia. Trajectory analysis of the particles inside the flow field revealed the existence of two special trajectories, called separatrices, originating from a saddle point and forming the boundary of the dust free region. These separatrices follow a curve of constant $\eta$, denoted as $\eta_{df}$, where $\eta$ is the boundary layer similarity variable. An equation developed for the separatrices showed that $\eta_{df}$ is a function of Prandtl number, $Pr$, and the thermophoretic number, $Th$. Scaling laws are developed for $\eta_{df}$ using the boundary layer equations of Rotem and Classen corresponding to the asymptotic cases of $Pr$, i.e, $Pr>>1$ case and $Pr<<1$ case. Interestingly these scaling laws obtained for the asymptotic cases of $Pr$ are found to be valid even for the intermediate $Pr$ regime except for intermediate $Pr>1$ in the large $\eta$ limit. Brownian effects on the particles are neglected for the entire analysis. Gravitational effects are also ignored for the major portion of the work except for the initial discussions on the effect of inclusion of gravitational term.