Viscoelastic fluids with pressure-dependent viscosity; exact analytical solutions and their singularities in Poiseuille flows

Abstract

We study the pressure-driven unidirectional flow of a viscoelastic fluid with pressure-dependent viscosity in a straight channel and a circular tube. The upper convected Maxwell (UCM) constitutive equation is utilized to model the response of the viscoelastic fluid under deformation. The starting point of the analysis is based on the second-order, symmetric conformation tensor. As such, the final model is slightly different from previous models in terms of the extra-stress tensor as the primary variable. The solution of the governing equations is found analytically for all the dependent variables except for a constant which is introduced by the separation of variables method. A physical singular point of the solution is also revealed explicitly. The singularity is taken into account in order to construct very accurate high-order asymptotic solutions for the unknown constant. Further reprocessing of the asymptotic solutions by means of non-linear techniques which accelerate the convergence of series, result in more accurate analytical expressions even close to the singularity of the solutions.