Heat transfer from a neutrally buoyant sphere in a second-order fluid

Abstract

In the absence of inertial effects, the heat or mass transfer from torque-free neutrally buoyant spheres in planar shearing flows of Newtonian fluids is diffusion limited at large Pe; here, the Peclet number ( Pe) is a dimensionless measure of the relative importance of convective and diffusive transfer mechanisms. In the inertialess Newtonian limit, the linearity and reversibility of the governing Stokes equations of motion leads to the existence of a region of closed streamlines around the freely-rotating particle that precludes convective enhancement in these flows. Non-Newtonian stresses act to break this symmetry, and the rate of heat/mass transfer from the particle is significantly increased for large Pe. It is possible to analytically determine the transport rate in the limit of weak non-Newtonian effects. For a torque-free particle in a planar linear flow of a second-order fluid, the dimensionless rate of heat or mass transfer, characterized by the Nusselt number, is found to be N u = 0.478 [ P e D e ( 1 + λ ) 2 ( 1 + ϵ ) ] 1 / 3 in the limit P e D e ≫ 1 and D e ≪ 1 , where ϵ is a dimensionless property of the fluid, and λ is a parameter that depends on the relative magnitudes of extension and vorticity in the ambient flow. In simple shear flow, corresponding to λ = 0 , the Nusselt number may be alternatively written as 6.01 ( P e D e ) 1 / 3 [ ( ( 1 / 2 ) + ( Ψ 2 / Ψ 1 ) ) / ( 1 + ( Ψ 2 / Ψ 1 ) ) ] 1 / 3 ; here, Ψ 1 and Ψ 2 are the first and second normal stress coefficients. The Deborah number ( De) is the ratio of the intrinsic relaxation time of the fluid to the macroscopic flow time scale in all above cases, and serves as a dimensionless measure of the relative magnitudes of the non-Newtonian (elastic) and Newtonian stresses; for simple shear flow, one may define D e = ( Ψ 1 + Ψ 2 ) γ ˙ / η in terms of the normal stress coefficients, η being the solvent viscosity.