Nonlinear hydroelastic response of the wall of a narrow channel filled with pulsating viscous liquid due to longitudinal vibrations of its opposite wall

The problems of hydroelasticity that arise during the mathematical modeling of the nonlinear response of the wall of a narrow channel filled with pulsating viscous liquid were formulated and solved. The plane channel has parallel rigid walls, where the bottom wall with nonlinear elastic supports at the ends undergoes longitudinal vibrations due to its interaction with the opposite vibrating wall through the liquid layer. The liquid dynamics in the channel were analyzed as a pulsating Couette flow with the consideration of the liquid inertia. The movement of the bottom wall of the channel was described using the mass-on-spring model characterized by symmetric stiffness with cubic nonlinearity. With the dissipative properties of the viscous liquid taken into account, the influence of the initial conditions became negligible, making it possible to focus on the formulation of a boundary value problem of mathematical physics for steady-state forced vibrations of the channel wall. Following the asymptotic analysis by the perturbation method, the problem was reduced to a nonlinear ordinary differential equation that generalizes the Duffing equation. The equation was solved by the Krylov–Bogolyubov method, and the nonlinear hydroelastic response of the wall to the primary resonance was determined in the form of its amplitude- and phase-frequency characteristics. The nonlinear hydroelastic response characteristics were expressed as implicit functions and require further numerical investigation. An example of such an investigation was provided, demonstrating that taking into account the liquid inertia and varying thickness of the liquid layer in the channel significantly affects the amplitude of vibrations, resonant frequencies, as well as the range of unstable vibrations with sudden amplitude changes.