FLOW OF POWER-LAW FLUIDS IN THIN STRAIGHT TUBES OF NON-UNIFORM CROSS-SECTION

We analyze the asymptotic behavior of solutions of a boundary value problem describing the flow of a non-Newtonian so called power-law fluid in a thin tube, with variable cross-section varying with a small parameter ε, the ratio between the radius of the cross-section and the length of the tube. The flow is assumed to be driven by an external pressure which is applied as a normal stress along of the tube’s ends. On the remaining part of the boundary we impose a no-slip and no-penetration conditions. We study the limiting behavior of the pressure and velocity field a small parameter ε in two-direction tends to zero, deriving the one-dimensional nonlinear limit problem for the pressure with a coefficient called “flow factor”. Depending on the of the geometry as well as the rheology of the fluid and the limit velocity is a generalized form of the Poiseuille-law, i.e. is a nonlinear function of the limit pressure derivative.