Effects of blockage and fluid inertia on drag and heat transfer of a solid sphere translating in FENE-P viscoelastic fluids in a tube

Abstract

An extensive numerical investigation of the flow and heat transfer phenomena of a solid sphere translating in a cylindrical tube filled with FENE-P viscoelastic fluids is reported herein. The governing equations, namely, mass, momentum, energy, and viscoelastic constitutive equations, have been solved over the following ranges of conditions: Reynolds number, 1 ≤ Re ≤ 100, Weissenberg number, 0 ≤ Wi ≤ 10, polymer extensibility parameter, 10 ≤ L2 ≤ 500 and blockage ratio, 0 ≤ BR ≤ 0.7 for a fixed value of the polymer viscosity ratio β = 0.5 and Prandtl number Pr = 10. Limited simulations with the FENE-CR viscoelastic fluid model have also been carried out to make a comparison between the two viscoelastic models. At low Reynolds numbers, the velocity overshoot and/or negative wake downstream the sphere has been observed under appropriate conditions. This tendency of their appearing decreases with the increasing Reynolds number, and decreasing blockage ratio, polymer extensibility parameter and Weissenberg number. The size of the recirculation region (wake length) increases with the Weissenberg number at low values of the polymer extensibility parameter, whereas a reverse trend is seen at high values of L2 . The drag coefficient decreases with the Reynolds and Weissenberg numbers, whereas it increases with the blockage ratio. On the other hand, the average Nusselt number always increases with the Reynolds number irrespective of the values of Wi, L2 and BR. However, the corresponding effect of the blockage ratio and polymer extensibility parameter is seen to be more complex, modulated by the values of Re and Wi. For instance, at high Reynolds numbers, the average Nusselt number always increases with the blockage ratio; however, at low values of it, there is a critical value of the blockage ratio present up to which the average Nusselt number increases, and beyond that, it decreases. Furthermore, the average Nusselt number always initially increases up to a certain value of the Weissenberg number, and after that, it remains almost constant or decreases depending upon the values of L2 , BR and Re. Finally, simple correlations for the average Nusselt number and drag ratio are presented, which not only capture the functional dependence of the governing parameters, but also can be used for the interpolation of the present results for the intermediate values of the governing parameters in a new application