Viscoelastic shear flow past an infinitely long and freely rotating cylinder

We study analytically the effect of steady shear flow on an infinitely long and freely rotating circular cylinder using asymptotic methods. The ambient fluid is assumed viscoelastic and modeled with the Oldroyd-B constitutive equation under isothermal and creeping flow conditions. The solution for all the dependent variables is expanded as an asymptotic power series with the small parameter being the Weissenberg number, Wi, which is defined as the product of the single relaxation time of the fluid times the externally imposed shear rate. The resulting sequence of equations is solved analytically up to the eighth order in the Weissenberg number, leading to a series solution, the accuracy and validity of which is shown to be linked to the loss of positive definiteness of the conformation tensor. The solution derived here is the first analytical result in the literature for viscoelastic fluids past a freely rotating cylinder. It reveals the slowdown of the rotation of the cylinder with respect to the Newtonian case, as previously found for the same type of flow past a rigid sphere. It is also seen that the first correction to the velocity profile is of second order in Wi, while that for the pressure is of first order. This contrasts with the solution for the viscoelastic flow past a sphere for which the correction for both the velocity and pressure profiles are of first order in Wi. The analytical solution also shows that the region of closed streamlines around the cylinder is enlarged compared to the Newtonian case. Last, it is seen that the up-and-down and fore-and-aft symmetries of the streamlines and the vorticity contours, observed for Newtonian fluids, break in the viscoelastic case.