Importance of Variable Density and Non-Boussinesq Effects on the Drag of Spherical Particles

What are the forces that act on a particle as it moves in a fluid? How do they change in the presence of significant heat transfer from the particle, a variable density fluid or gravity? Last year, using particle-resolved simulations we quantified these effects on a single spherical particle and on particles in periodic lattices when O($10^{-3}$)~\textless~Re~\textless~O(10). Let $\lambda$ be the normalized particle-fluid temperature difference. Large deviations (\textgreater 50\%) in the absolute drag are observed as $\lambda$ approaches unity. Oppenheimer, et al (2016) [1] have proposed a theoretical formula for the drag of a heated sphere at extremely low Re. We show that when Re~\textgreater~O(10), inertial effects completely dominate the drag while when Re~\textless~O($10^{-3}$), viscous effects completely dominate the drag and our simulations agree well with [1]. In the middle, there is honest competition between inertial and viscous effects and the drag modification strongly depends on the thermally induced near-particle density variation causing a non-zero volumetric dilation rate. In the limit of $\lambda$ approaching 0 (Stokes' limit), the drag modification can also be captured as a correction to Stokes' drag using a suitable scaling based on the dilation rate.