On small motions of a two joined bodies system with cavities partially filled with a heavy viscous fluid

Let $G_1$ and $G_2$ be two joined bodies with masses $m_1$ and $m_2$. Each of them has a cavity partially filled with homogeneous incompressible viscous fluids situated in domains $\Omega_1$ и $\Omega_2$ with free boundaries $\Gamma_1(t), \Gamma_2(t)$ and rigid parts $S_1, S_2$. Let $\rho_1, \rho_2$ be densities of fluids. We suppose that the system oscillates (with friction) near the points $O_1, O_2$ which are spherical hinges.

We use the vectors of small angular displacement
\begin{equation*} \vec{\delta}_{k}(t)=\sum\limits_{j=1}\limits^{3}\delta_{k}^{j}(t)\vec{e}_{k}^{ j}, \quad k=1,2, \end{equation*}
to determine motions of the removable coordinate systems $O_{k}x_{k}^{1}x_{k}^{2}x_{k}^{3}$ (connected with bodies) with respect to stable coordinate system $O_{1}x^{1}x^{2}x^{3}$. Then angular velocities $\vec{\omega}_k(t)$ of bodies $G_{k}$ is equal to $d\vec{\delta}_k/dt$.

Let $\vec u_k(x,t)$ and $p_k(x,t)$ be fields of fluids velocities and dynamical pressures in $\Omega_k$ (in removable coordinate systems), $\zeta_k (x,t)$ are functions of normal deviation of $\Gamma_k(t)$ from equilibrium plane surfaces $\Gamma_k(0)=\Gamma_k$. Then we consider initial boundary value problem (25), (26), (28)–(30) with conditions (34)–(39).

We obtain the law of full energy balance (44). Using the method of orthogonal projections initial problem can be reduced to the Cauchy problem for the deferential equation
\begin{equation*} \mathcal{C} \frac{d^2 X}{dt^2} + \mathcal{A} \frac{d X}{dt} + \mathcal{B} X = \mathcal{F} \end{equation*}
in Hilbert space $\mathcal{H}:=H_1 \oplus H_2:= (\vec{J}_{0, S_1}(\Omega_1) \oplus \vec{J}_{0, S_2}(\Omega_2)) \oplus (\mathbb{C}^3 \oplus \mathbb{C}^3)$. Here operator $\mathcal{C}$ is bounded and positive definite, operator $\mathcal{A}$ is positive definite, $\mathcal{B}$ is bounded below self-adjoint operator. General properties of such problem is known. It has a unique strong solution for $t\in [0;T]$ if the natural conditions for initial data and function $\mathcal{F}$ are satisfied. As a corollary we obtain theorem on solvability of initial Cauchy problem.

Corresponding spectral problem reduces to the operator pencil of S. G. Krein. The spectrum consists of $\lambda=0$, two branches of positive eigenvalues with limit point $+0$ 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 $H$ (probably with finite defect). We obtain sufficient conditions for absence of negative eigenvalues (stability of hydromechanics system) and for equality of negative eigenvalues of the problem and the operator of potential energy $B$.