Dust-free region over horizontal hot surfaces

We study the formation of dust-free regions above hot horizontal surfaces of uniform temperature and propose relations for its height in the limit of small particle inertia and gravitational effects. By including particle inertia, thermophoretic, gravitational and viscous effects, we conduct Lagrangian simulations of particle dynamics in a natural convection boundary layer over a horizontal surface. Trajectory analysis of the particles inside the boundary layer on the surface reveals the existence of two separatrices originating from a saddle point, which form the boundary of the dust-free region. These separatrices for low gravitational effects follow the boundary layer thickness, but are of much lower height and also depend on the dimensionless thermophoretic number ( $Th$ ) and Prandtl number ( $Pr$ ). We obtain a relation for the dimensionless height of the dust-free region ( $\eta _{df}$ ) as a function of $Pr$ and $Th$ , for low dimensionless gravitational number ( $Gn$ ); the numerical solution of this equation gives us the dust-free region height for any $Th$ and $Pr$ . We then obtain scaling laws for $\eta _{df}$ using the boundary layer equations corresponding to the $Pr \gg 1$ and $Pr \ll 1$ cases; these scaling laws are shown to be valid respectively for $Pr>1$ and $Pr<1$ , except in the large $\eta$ limit for $Pr>1$ , where $\eta$ is the boundary layer similarity variable. We then obtain an empirical relation in this large $\eta$ limit using the numerical solutions of the boundary layer equations for the intermediate $Pr$ case to obtain scaling laws for dust-free region height for the whole range of $Pr \ll 1$ to $Pr \gg 1$ .