On instability of axially symmetric equilibrium figures of rotating viscous incompressible liquid. (English) Zbl 1075.35042

Solonnikov, V. A.

Zap. Nauchn. Semin. POMI 318, 277-297 (2004); translated in J. Math. Sci., New York 136, No. 2, 3812-3825 (2006).

Let \(F\) be the equilibrium figure of an incompressible liquid subjected to the capillary and self-gravitation forces and rotating as a rigid body with the angular velocity \(\omega\) about the \(x_3\)-axis. The evolution free boundary problem for the perturbations of velocity, pressure and of the figure \(F\) is considered in the paper. This problem written in the coordinate system rotating with the angular velocity \(\omega\) about the \(x_3\)-axis has the form \[ \begin{aligned} &\frac{\partial v }{\partial t }+(v\cdot\nabla)v+2\omega(e_3\times v)-\nu\Delta v+\nabla p=0,\quad \text{div\,}v=0, \quad x\in \Omega_t,\;t>0\\ &T(v,p)n=(\sigma H+\frac{\omega^2}2| x'| ^2+p_0+k\nabla U(x,t))\,n, \quad V_n =v\cdot n,\quad x\in\Gamma_t\equiv \partial\Omega_t\\ &v(x,0)=v_0(x),\quad x\in \Omega_0 \end{aligned} \] Here \(\Omega_t\subset \mathbb R^3\) is a bounded domain occupied by a liquid, \(v(x,t)\) is the velocity, \(p\) is the pressure. \(\Omega_t,\;v\) and \(p\) are unknown objects, \(\nu, \sigma\) are given positive constants. \(T(v,p)\) is the stress tensor, \(H\) is twice the mean curvature of \(\Gamma_t\), \(V_n\) is the velocity of evolution of \(\Gamma_t\) in the direction of the outward normal \(n\). \[ U(x,t)=\int\limits_{\Omega_t}\frac{d\,y}{| x-y| } \] is the Newtonian potential. The initial domain \(\Omega_0\) is given and close to the \(F\). The initial data \(v_0\) is small.
It is proved that the solution \((v,p)\) is unstable when the second variation of the energy functional can take negative values.

Reviewer: Il’ya Sh. Mogilevskij (Tver’)

MSC:

35Q30	Navier-Stokes equations
76E07	Rotation in hydrodynamic stability
35R35	Free boundary problems for PDEs
76U05	General theory of rotating fluids

Keywords:

Navier-Stokes equations; free boundary; rotating liquid; instability