Viscoelastic Poiseuille flows with total normal stress dependent, nonlinear Navier slip at the wall

Abstract

The effect of slip at the wall in steady, isothermal, incompressible Poiseuille flows in channel/slits and circular tubes of viscoelastic fluids is investigated analytically. The nonlinear Navier law at the wall, for the dependence on the shear stress, along with an exponential dependence of the slip coefficient on the total normal stress is assumed. The viscoelasticity of the fluid is taken into account by employing the Oldroyd-B constitutive model. The flow problems are solved using a regular perturbation scheme in terms of the dimensionless exponential decay parameter of the slip coefficient, ɛ. The sequence of partial differential equations resulting from the perturbation procedure is solved analytically up to third order. As a consequence of the nonlinearity of the slip model, a two-dimensional, continuously developing, flow field arises. Spectral analysis on the solution shows that the velocity and pressure profiles are fully resolved even for high values of ɛ, which indicates that the perturbation series up to third order approximates the full solution very well. The effects of the dimensionless slip coefficient, isotropic pressure, and deviatoric part of the total normal stress in the slip model, as well as the other parameters and dimensionless numbers in the flow are presented and discussed. Average quantities, in the cross section of the channel/slit or tube, with emphasis given on the pressure drop and the skin friction factor, are also offered.