Gyroscopic waves in a rotating liquid layer

Abstract

The dispersion characteristics of gyroscopic waves in an incompressible liquid layer in a cavity of a rapidly rotating cylinder are studied. It is shown that in a viscous incompressible liquid layer, an inertial wave can be represented as the sum of six helical harmonics. The effects of the liquid viscosity and the ratio of the wave frequency to the angular velocity of rotation of the cylinder on the real and imaginary parts of the wavenumber are studied. Introduction.Wave processes in a liquid with the effect of the Earth's rotation taken into account were considered in papers dealing with waves in the ocean (see, for example, [1-4] and the bibliography therein). The strong effect of rotation on the liquid dynamics is exemplified by rotary systems. Rotors with a liquid-containing cavity, in which the centrifugal force can be several hundred thousand or even several millions of times the gravity. Disturbance of the balance between the pressure gradient and the Coriolis force can lead to the generation of wave motions in the rotating liquid, which are called inertial or gyroscopic waves. These waves play an important role in the problems of dynamics of rotors, turbines, separators, centrifuges, and rotating aircraft containing a liquid, and in some geophysical problems (flows in the Earth's liquid core [5]). Wave phenomena in a rotating liquid layer can have a significant effect on a number of technological processes (in particular, sedimentation processes), phase equilibria in multicomponent liquids, and on the aircraft flight trajectory. The dynamics of inertial waves in a liquid filling a rotating cylinder has been experimentally studied in a number of papers (see [6,7] and references therein). Stationary gyroscopic waves in a circular cylindrical layer of an ideal liquid bounded by solid walls were studied in [8,9]. Dispersion relations were obtained and dependences of dimensionless wavelengths on dimensionless frequencies were constructed. The behavior of a low-viscosity liquid in a rotating horizontal cylinder as a function of the rotation velocity and the degree of filling was studied in [10]. The resonant generation of waves in a liquid filling a rotor cavity, which is the main factor responsible for instability of steady-state rotation of rotary systems was investigated in [11][12][13][14][15][16][17][18][19].The present investigation of the properties of gyroscopic waves in a rotating liquid layer is motivated by interest in the problem of the stability of rotary systems with a fixed point at which the angular velocity of rotation of a rotor is maintained constant by a drive. A stability analysis method for similar rotary systems was proposed in [15,17]. In this method, one of the main stages involves calculation of the moments of the forces exerted by the liquid on the rotor walls during steady-state conical precession of the rotor. It is easy to show that, in the case of conical precession, the steady-state hydrodynamic problem is directly related to the problem of generation of inertial waves in a liq...

Magnetic field effect on waves in a centrifuged layer of a rotating conducting viscous fluid

“…(The exact expressions for the roots of Eq. (16) for S = 0 are given in [9].) However, these expressions are cumbersome.…”

“…Neglecting the effect of the viscosity and conductivity of the fluid, we write the dispersion equation in compact form [9] …”

“…Saltanov [8] has considered stationary gyroscopic waves in an ideal fluid layer confined by solid cylindrical walls. Dispersion relations for waves in a centrifuged layer of a viscous fluid were obtained and investigated by Soldatov [9].…”