Model problem on normal oscillations of partially dissipative hydrosystem

Let us consider the plain (two-dimensional) problem for two fluids situated in the rectangular container of a width $l$. We suppose that the lower viscous fluid takes the region $ \Omega_1:=\{\, (x_1;x_2)\, : \, 0<x_1<l, \,\, -h_1<x_2<0\, \} $ and the upper ideal fluid takes the region $ \Omega_2:=\{\, (x_1;x_2)\, : \, 0<x_1<l, \,\, 0<x_2<h_2\, \}. $ The boundary $\Gamma_1$ has the equation $x_2=0$, and the free surface $\Gamma_2$ of the ideal fluid has the equation $x_2=h_2$. Suppose that the homogeneous gravitational field with the acceleration $\vec g = -g \vec e_2$ acts on the fluid system opposite to the direction of the axis $Ox_2$ and capillary forces. Further, two cases will be considered: 1) fluids are considered to be heavy and capillary forces are not taken into account; 2) fluids are considered to be capillary, that is, being in a state close to weightlessness. In the second case the coefficients of surface tension $\sigma_i>0$ on the fluid boundaries $\Gamma_i$ are know physical constants, and the wetting angles (contact angles) between surfaces $\Gamma_i$ and the rigid wall $S$ of the vessel are right angles. In this paper, we consider a model spectral problem that preserves all the features of the original problem of normal oscillations of the hydrodynamic system described above. A qualitative and asymptotic investigation of the spectrum of the problem is carried out the base of a study of the transcendent characteristic equation for the complex fading decrement of normal oscillations.