Steady flows of second-grade fluids in a channel

In this paper, we study mathematical models describing steady flows of second-grade fluids in a plane channel. The flows are driven by constant pressure gradient. We consider various boundary conditions on the channel walls, namely, the no-slip condition, the free-slip condition, threshold slip conditions, and mixed boundary conditions. For each of the boundary value problems, we construct exact solutions, which characterize the velocity and pressure fields in the channel. Using these solutions, we show that the pressure significantly depends on the normal stress coefficient $\alpha$, especially in those subdomains, where the change of flow velocity is large (in the transverse direction of the channel). At the same time, the velocity field is independent of $\alpha$, and therefore coincides with the velocity field that occurs in the case of a Newtonian fluid (when $\alpha= 0$). Moreover, we establish that the key point in a description of stick-slip flows is value of $\xi h$, where $\xi$ is module of the gradient pressure, $h$ is the half-channel height. If $\xi h$ exceeds some threshold value, then the slip regime holds at solid surfaces, otherwise the fluid adheres to the channel walls. If it is assumed that the free-slip condition (Navier's condition) is provided on one part of the boundary, while on the other one a stick-slip condition holds, then for the slip regime the corresponding threshold value is reduced to a certain extent, but not by more than half. Refs 15.