Asymptotic Study of the Nonlinear Velocity Problem for the Oscillatory Non-Newtonian Flow in a Straight Channel

The studies of non-Newtonian flows, such as blood flows in arteries and polymer flows in channels have very important applications. The non-Newtonian fluid viscosity is modelled by the Carreau model (nonlinear with respect to the viscosity dependence on the shear rate). In the present paper the oscillatory flow of Newtonian and non-Newtonian fluids in a straight channel is studied analytically and numerically. The flow in an infinite straight channel is considered, which leads to a parabolic non-linear equation for the longitudinal velocity. The Newtonian flow velocity is found analytically, while the non-Newtonian velocity is found numerically by the finite-difference Crank-Nicolson method. In parallel, the non-Newtonian (Carreau) velocity is developed in an asymptotic expansion with respect to a small parameter. The zero-th order term of this expansion is exactly the Newtonian velocity solution. The first order term of the velocity expansion is found analytically in terms of higher order harmonics in time. As an example, the polymer solution HEC 0.5\(\%\) is considered. It is shown that the obtained asymptotic solution and the numerical solution for the non-Newtonian (Carreau) velocity are close for different values of the small parameter.