Statements of Linearized Boundary Value Problems of Continuum Mechanics with a Spectral Parameter in the Boundary Conditions

We derive statements of linearized boundary value problems in small perturbations arising in continuum mechanics for incompressible viscous media and inviscid media. The known main three-dimensional flow is assumed to be steady-state; along with this flow, a perturbed flow of the same medium induced by the same bulk and surface forces is considered in a domain with unknown moving boundary. The arising linearized statements are reduced to a system of four equations for the perturbations of pressure and velocity components in the unperturbed domain and to a system of homogeneous boundary conditions carried over to the unperturbed boundaries. It turns out that in such statements, the spectral parameter α—the complex vibration frequency—occurs linearly in three equations of motion and one boundary condition. In special cases of the perturbation pattern, reduction is possible to one equation for the stream function amplitude linearly containing the parameter α and four boundary conditions, two of which contain the parameter α (occurring linearly in one condition and quadratically in the other). Examples include a layer of a heavy Newtonian fluid flowing down a sloping plane and vibrations in a two-layer system of heavy perfect fluids.