Analytical study of a model of fluid flow through a channel with flexible walls

The present paper is devoted to mathematical analysis of the model that describes fluid flow moving in a channel with flexible walls, which are subject to traveling waves. Experimental data show that the energy of the flowing fluid can be consumed by the structure (the walls) inducing “traveling wave flutter.” In the problems involving two-media interactions (fluid/structure), flutter-like perturbations can occur either in the fluid flowing in the channel with harmonically moving walls, or in the solid structure interacting with the flow. In the present research, it is shown that there are no abrupt (or flutter-like) changes in the flow velocity profiles. Using the mass conservation law and incompressibility condition, we obtain the initial boundary value problem for the stream function. The boundary conditions reflect that (i) there is no movement in the vertical direction along the axis of symmetry and (ii) there is no relative movement between the near-boundary flow and the structure (“no-slip” condition). The closed form solution is derived for the stream function, which is represented in the form of an infinite functional series.