On Small Motions of a Physical Pendulum with Cavity Filled with a System of Three Homogeneous Immiscible Viscous Fluids

Let $G$ be a physical pendulum of mass $m$. Suppose that it has a cavity filled with a system of three homogeneous immiscible viscous fluids situated in domains $\Omega _{1}, \Omega _{2}$ and $\Omega _{3}$ with free boundaries $\Gamma _{1}\left ( t \right ), \Gamma _{2}\left ( t \right )$ and rigid parts $S_{1},S_{2},S_{3}$. Let $\rho _{1}, \rho _{2}, \rho _{3}$ be densities of fluids, and $\mu _{1}, \mu _{2}, \mu _{3}$ be dynamical viscosities. We suppose that the system oscillates (with friction) near the fixed point $O$ of spherical hinge.
We use the vector of small angular displacement $\vec{\delta} \left ( t \right ) = \sum_{j=1}^{3}\delta ^{j}\left ( t \right ) \vec{e_{1}}^{j}$ to determine motion of the removable coordinate system $Ox_{1}^{1},x_{1}^{2},x_{1}^{3}$ (connected with pendulum) relative to stable coordinate system $Ox^{1}x^{2}x^{3}$. Then angular velocity $\vec{\omega}\left ( t \right )$ of body $G$ is equal to $d\vec{\delta }/dt$.
Let $\vec{u}_{k}\left ( x,t \right )$ and $p_{k}\left ( x,t \right )$ $\left ( k = 1,2,3 \right )$ be fields of fluids velocities and dynamical pressures in $\Omega_{k}$ (in removable coordinate systems), $\zeta _{j} \left ( x,t \right )$$\left ( j = 1,2 \right )$ are functions of normal deviation of $\Gamma _{j} \left ( t \right )$ from equilibrium plane surfaces $\Gamma _{j} \left ( 0 \right ) = \Gamma _{j}$. Then we consider linearized initial boundary value problem $\left ( 2 \right )-\left ( 4 \right )$.
We formulate the law of full energy balance. Using the method of orthogonal projections to the necessary Hilbert spaces and studying auxiliary boundary value problems initial problem can be reduced to the Cauchy problem for the deferential-operator equation $\left ( 19 \right )-\left ( 20 \right )$
$$C_{1}\frac{\mathrm{d} z_{1}}{\mathrm{d} t} + A_{1}z_{1}+gB_{12}z_{2}=f_{1}\left ( t \right ), z_{1}\left ( 0 \right ) = z_{1}^{0},$$
$$gC_{2}\frac{\mathrm{d} z_{2}}{\mathrm{d} t}+gB_{21}z_{1}=0,z_{2}\left ( 0\right )=z_{2}^{0},$$
in Hilbert space $H := H_{1}\oplus H_{2} := \left ( \vec{J}_{0,S,\Gamma } \left ( \Omega \right ) \oplus \mathbb{C}^{3}\right ) \oplus \left ( L_{2,\Gamma } \oplus \mathbb{C}^{2}\right )$. Here operator of kinetic energy $C_{1}$ is bounded and positive definite, operator of potential energy $C_{2}$ is bounded and selfadjoint, $A_{1}$ is unbounded positive definite, $B_{12}$ and $B_{21}$ are unbounded skew self-adjoint operators. General properties of such problem are studied earlier in paper $\left [ 4 \right ]$. It has a unique strong solution for $t\in \left [ 0;T \right ]$ if the natural conditions for initial data and function $f_{1}\left ( t \right )$ are satisfied. As a corollary we obtain theorem on solvability of initial boundary value problem.
Corresponding spectral problem reduces to the operator pencil of S. Krein. The spectrum consists of $\lambda = 0$, two branches of positive eigenvalues with limit points $+0$ and $+\infty$, and probably finite number of negative and complex eigenvalues. The systems of eigenelements corresponding to each of positive branches of eigenvalues form so called $p$-basis in Hilbert space $H_{1}$ (probably with finite defect). The number of negative eigenvalues in problem is equal to number of negative eigenvalues of the operator of potential energy $C_{2}$.
This work was partially supported by the Ministry of Education and Science of the Russian Federation (grant 14.Z50.31.0037), and by the V.I. Vernadsky Crimean Federal University development program for 2015 – 2024 within the framework of grant support for young scientists.
Key words: physical pendulum, viscous fluid, auxiliary boundary value problems, strong solution, self-adjoint operator, operator pencil of S. Krein.