A high-order perturbation solution for the steady sedimentation of a sphere in a viscoelastic fluid

The steady, creeping and isothermal sedimentation of a rigid sphere in an incompressible viscoelastic fluid which follows the exponential Phan–Thien and Tanner constitutive equation is studied analytically. The solution of the governing equations is expanded as a regular perturbation series for small values of the Deborah number, and the resulting sequence of two-dimensional partial differential equations is solved analytically up to eighth order. Although the domain is unbounded, the solution is able to resolve features of the flow that cannot be revealed by a low-order theoretical analysis, such as very fine flow structures and a stress boundary layer close to the surface of the sphere. The calculation of the drag force exerted on the sphere by the fluid was done by developing two formulas. The first is based on the flow-field close to the sphere and the second is based on the far flow-field. Both formulas produce the same analytical expressions verifying the correctness and consistency of the series solution. At small Deborah numbers, a decrease of the drag force is predicted (i.e., an increase of the sedimentation velocity), followed by a significant drag enhancement at higher Deborah numbers. Investigation of the solution for the existence of a negative wake close to the rear stagnation point did not reveal such a phenomenon when physical constraints on the solutions were posed. On the contrary, a negative wake is predicted only as an artifact due to the loss of positive definiteness of the conformation tensor. It is also demonstrated that the effect of viscoelasticity is maximized in a region around the sphere with a radius which is about ten times the radius of the sphere. Last, it is shown that the fluid disturbances due to the viscoelasticity of the matrix fluid decay very slowly with the distance from the sphere, depending on the magnitude of the Deborah number. Streamlines, extra-stress and pressure contours as well as the extension of the polymer molecules are also presented and discussed.