Rotary vibrations of a porous spherical shell with an impermeable core in a viscous liquid

Background. The investigation of the viscous liquid flows in contact with the oscillating submerged porous bodies of various configurations is of a considerable interest for hydrodynamics in the connection with a great theoretical importance and various practical applications. The aim of the present paper is to determine the fields of filtration velocities and free liquid velocities respectively inside and outside the porous spherical shell with the impermeable spherical core which performs rotational-oscillatory motion in a viscous liquid.
Materials and methods. To solve the problem of the viscous liquid flow induced by rotational-oscillatory motion of the submerged porous spherical shell with the impermeable spherical core we used methods of mathematical physics, vector analysis and ordinary differential equations solving. To describe the viscous liquid flows in the porous medium the nonstationary Brinkman equation is used. The motion of the viscous liquid outside the porous medium with the Navier-Stokes equation is described.
Results. Exact analytical solutions for the nonstationary Brinkman equation in the region inside the porous medium and for the Navier-Stokes equation outside the porous medium respectively are obtained. The fields of the filtration velocity and the free liquid velocity inside and outside the porous medium respectively are determined.
Conclusions. It is shown that the liquid velocity fields in the cases of the spherical porous shell with an impermeable core and a porous sphere without the core differs considerable. It is shown also that the results obtained earlier for particular case of the viscous fluid flows induced by a rotational-oscillatory motion of a submerged porous sphere can be obtained from the solution presented in this paper.