Inertial effects on the transfer of heat or mass from neutrally buoyant spheres in a steady linear velocity field

Abstract

Microscale inertia is found to break the degenerate closed-streamline configuration that occurs in a shearing flow past a neutrally buoyant torque-free spherical particle in the inertialess limit. The broken symmetry at small but finite Re allows heat or mass to be convected away in an efficient manner in sharp contrast to the inertialess diffusion-limited scenario. Inertial forces scale with the particle Reynolds number, defined as Re=γ̇a2∕ν, where a is the radius of the particle, γ̇ is the characteristic magnitude of the velocity gradient, and ν is the kinematic viscosity of the suspending fluid. The dimensionless heat or mass transfer rate is then given by Nu=C(RePe)1∕3+O(1) when Re≪1 and RePe≫1, the constant C being a function of the flow in the vicinity of the particle. Here, Nu is the Nusselt number defined as Q∕(4πkaΔF), where Q is the dimensional heat/mass flux, k the appropriate transport coefficient, and ΔF the driving force viz. the temperature or concentration difference between the particle and the ambient fluid; for pure diffusion, Nu=1. The Peclét number (Pe) is a dimensionless measure of the relative dominance of the convective and diffusive transfer mechanisms. It is shown that C equals 0.325(1+λ)2∕3 for a two-dimensional linear flow, where λ measures the relative magnitudes of extension and vorticity. For simple shear (λ=0), knowledge of the inertial velocity field to O(Re3∕2) enables one to determine the next term in the asymptotic expansion for Nu; one finds Nu=(RePe)1∕3(0.325−0.0414Re1∕2)+O(1) in the limit 1≫Re≫Pe−2∕5. It is argued that the convective enhancement at finite Re via symmetry-breaking streamline bifurcations will occur in generic shearing flows with nonlinear velocity profiles; the degenerate Stokes streamline pattern around a neutrally buoyant torque-free particle in a quadratic flow serves to reinforce this assertion. The above mechanism represents a possible means for heat or mass transfer enhancement from the dispersed phase in multiphase systems. Implications for particles in turbulent flows are also discussed.