New analytical solutions for weakly compressible Newtonian Poiseuille flows with pressure-dependent viscosity

Steady-state, isothermal, Poiseuille flows in straight channels and circular tubes of weakly compressible Newtonian fluids are considered. The major assumption is that both the mass density and the shear viscosity of the fluid vary linearly with pressure. The non-zero velocity components, the pressure, the mass density and viscosity of the fluid are represented over the flow domain as asymptotic expansions in which the dimensionless isothermal compressibility coefficient ɛ is taken as small parameter. A perturbation analysis is performed and asymptotic solutions for all variables are obtained up to first order in ɛ. The derived solutions, which hold for not necessarily small values of the dimensionless pressure-dependence coefficient, extend previous regular perturbation results and analytical works in the literature for weakly compressible fluids with constant viscosity (solved with a regular perturbation scheme), for incompressible flows with pressure-dependent viscosity (solved analytically), as well as for compressible fluids with pressure-dependent viscosity (solved with double regular perturbation schemes). In contrast to the previous analytical studies in the literature, a non-zero wall-normal velocity is predicted at first order in ε, even at zero Reynolds number. A severe reduction of the volumetric flow-rate at the entrance of the tube/channel and multiplicity of solutions in the flow curves (volumetric flow-rate versus pressure drop) are also predicted. Last, it is shown that weak compressibility of the fluid and the viscosity pressure-dependence have competing effects on the mean friction factor and the average pressure difference required to drive the flow.