Flow of power-law fluids in thin straight tubes of non-uniform cross-section

We analyze the asymptotic behavior of solutions to a boundary value problem describing the flow of a non-Newtonian so-called power-law fluid in a thin tube with a variable cross-section depending on a small parameter ε , representing the ratio between the cross-sectional radius and the tube length. The flow is assumed to be driven by an external pressure applied as a normal stress at the tube ends. On the remaining part of the boundary, no-slip and no-penetration conditions are imposed. We study the limiting behavior of the pressure and velocity fields as the small parameter ε tends to zero in both transverse directions, deriving a one-dimensional nonlinear limit problem for the pressure with a coefficient referred to as the “flow factor”. The main physical finding is that the combined effects of the tube geometry and the non-Newtonian rheology lead to an effective nonlinear flow law. More precisely, we show that the limit velocity satisfies a generalized form of Poiseuille’s law, in which the flow rate depends nonlinearly on the limit pressure derivative, providing an explicit macroscopic description of how geometric variations and shear-dependent viscosity influence transport in thin channels.