Perturbation solution for the planar Poiseuille flow of a weakly-compressible second-grade fluid with Navier slip

We present a perturbation solution for the planar Poiseuille flow of a second-grade weakly-compressible fluid in isothermal regime. We assume that the relation between the density and the pressure is linear and that the bulk and shear viscosities are constant. Navier slip conditions are employed on the rigid walls of the channel. Expanding the main variables as a power series of the compressibility coefficient we determine explicit expressions of the main variables up to the second order. These are in the form of polynomials up to grade three in the longitudinal coordinate and grade ten in the transversal coordinate. We show that compressibility has a strong effect on velocity, pressure, shear stress and normal stress difference.