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摘要
We study the heat or mass transfer from a neutrally buoyant spherical drop embedded in an ambient Newtonian medium, undergoing a general shearing flow, in the strong convection limit. The latter limit corresponds to the drop Peclet number being large (Pe >> 1). We consider two families of ambient linear flows: (i) planar linear flows with open streamlines (parametrized by alpha with 0 <= alpha <= 1, the extremal members being simple shear flow (alpha = 0) and planar extension (alpha = 1)) and (ii) three-dimensional extensional flows (parameterized by epsilon, with 0 <= epsilon <= 1, the extremal members being planar (epsilon = 0) and axisymmetrie extension (epsilon = 1)). For the first family, an analysis of the exterior flow field in the inertialess limit (the drop Reynolds number, Re, being vanishingly small) shows that there exist two distinct streamline topologies separated by a critical drop-to-medium viscosity ratio (lambda) given by lambda(c) = 2 alpha/(1 - alpha). For lambda < lambda(c) all streamlines are open, while the near-field streamlines are closed for lambda > lambda(c). For the second family, the exterior streamlines remain open regardless of lambda. The two streamline topologies lead to qualitatively different mechanisms of transport for large Pe. The transport in the open streamline regime is enhanced in the usual manner via the formation of a boundary layer. In sharp contrast, the closed-streamline regime displays diffusion-limited transport, so there is only a finite enhancement even as Pe -> infinity. For Re = 0, the drop surface streamlines in a planar linear flow may be regarded as generalized Jeffery orbits with a flow and viscosity dependent aspect ratio Jeffery orbits denote the aspect-ratio-dependent inertialess trajectories of a rigid axisymmetrie particle in a simple shear flow; see Jeffery (Proc. R. Soc. Lond. A, vol. 102 (715), 1922, pp. 161-179). A Jeffery-orbit-based non-orthogonal coordinate system thus serves as a natural candidate to tackle the transport problem from a drop, in a planar linear flow, in the limit Pe >> 1. Use of this system allows one to derive a closed-form expression for the dimensionless rate of transport (the Nusselt number Nu) from a drop in the open-streamline regime (lambda < lambda(c)). Symmetry arguments point to a Jcffery-orbit-based coordinate system for any linear flow, and a variant of this coordinate system is therefore used to derive the Nusselt number for the family of three-dimensional extensional flows. For both classes of flows considered, the boundary-layer-enhanced transport implies that the Nusselt number takes the form Nu = F(P,lambda)Pe(1/2), with the parameter P being alpha or epsilon, and F(P,lambda) given as a one and two-dimensional integral, respectively, which is readily evaluated numerically.
- 出版日期2018-7-6
