The problem of normal oscillations of a viscous rotating stratified fluid

Assume that a viscous stratified fluid partially fills an arbitrary container and, in the unperturbed state, uniformly rotates with angular velocity
${\vec w_0 = \omega_0 \vec e_3}$, where ${\vec e_3}$ is the axial vector of rotation axis ${Ox_3}$ (assume that ${\omega_0>0}$).
In a state of relative equilibrium, the fluid occupies the region ${\Omega \in \Bbb R^3}$ bounded by solid wall $S$ and the equilibrium surface ${\Gamma}$.

Consider small motions of a fluid near equilibrium state. The problem is studied on the base
of an approach connected with application of so-called operator matrices theory. To this end, we
introduce Hilbert spaces and some their subspaces, also auxiliary boundary value problems. The
initial boundary value problem is reduced to the Cauchy problem for the differential first-order
equation in Hilbert space. After a detailed study of the properties of the problem, we proved
the theorem on the strong solvability of the Cauchy problem. This result based on factorization,
closure and accretivity property of operators of the resulting equation.

Finally, we consider the spectral problem on normal oscillations corresponding to the
evolution problem. This means that external forces equal to zero and dependence by time for
the unknown function has the form e^−λt. Here we obtain a spectral problem for a non-selfadjoint operator pencil. It has been established that the spectrum of the problem is discrete and
consists of a countable set of eigenvalues with finite algebraic multiplicity that are located in the
right half-plane and have the points ∞ and 0 as accumulation points. Proven to be if the fluid
rotates, then the spectrum of the fluid moves out of the real axis and all the eigenvalues, with
the exception of maybe a finite number, are located in the some region. Moreover, the property
of basicity of the corresponding systems of eigen- and associated elements can be replaced in the
case by the property of completeness (for a fluid with sufficiently high viscosity).

Keywords: stratification effect in viscous fluids, differential equation in Hilbert space, Cauchy
problem, normal oscillations.