Convective scalar transport from spherical drops in complex shearing flows

Abstract

We calculate the scalar transport rate, as characterized by the Nusselt number\,($Nu$), from a neutrally buoyant spherical drop in an ambient linear flow, in the absence of inertia and in the strong convection limit. This corresponds to the regime $Re \ll 1, Pe \gg 1$, where $Re$ and $Pe$ are the Reynolds and P\'eclet numbers, and denote the ratios of the diffusive and convective time scales associated with momentum and scalar transport. The focus is on the exterior problem, with the drop-phase transport resistance assumed negligible, and the scalar field being a constant on the drop surface. While $Nu \propto Pe^{\frac{1}{2}}$ for $Pe \gg 1$, owing to the transport occurring across a thin $O(a Pe^{-\frac{1}{2}})$ boundary layer\,($a$ being the drop radius), the proportionality factor in this relation depends sensitively on ambient flow geometry via the surface-streamline topology. Unlike a rigid sphere, a variety of surface-streamline topologies can drive transport across the boundary layer at widely differing rates. In contrast to earlier studies which almost exclusively focus on axisymmetric ambient flows, we calculate $Nu$ for a pair of non-axisymmetric linear flow families: (i) 3D extensional flows with aligned vorticity and (ii) Axisymmetric extensional flows with inclined vorticity, using a non-orthogonal surface-streamline aligned coordinate system. Taken together, the families span the entire gamut of surface-streamline topologies in the space of incompressible linear flows. Independent numerical simulations of the interior problem reveal the emergence of an $O(aPe^{-\frac{1}{2}})$ boundary layer beneath the drop surface, driven by chaotic streamlines, pointing to the possibility of $Nu \propto Pe^{\frac{1}{2}}$ for the conjugate problem, for sufficiently large $Pe$.