Nonlinear isochronous oscillations of a fluid in a paraboloid: theory and experiment

Abstract

Within the framework of the shallow-water model, the nonlinear axisymmetric oscillations of a fluid in a paraboloid of revolution and in an unbounded parabolic channel are investigated. It is established that in the paraboloid of revolution the oscillation period does not depend on the amplitude, that is, the oscillations are isochronous. Experimental investigations of free fluid oscillations in a paraboloid confirm this theoretical result.

Nonlinear radial oscillations of a fluid in a parabolic basin with regard for the external action

“…It is reasonable to say that the natural number n is the order of solution (5)- (7). For all n, the radial velocity is a linear function of r. Substituting (4)- (7) in Eqs. (1)-(3), we obtain the following system of 5n -1 first-order ordinary differential equations and algebraic relations: …”

“…The frequency of nonlinear oscillations of a fluid ω in a basin is independent of the amplitude of oscillations and the order n of solution, which means that they are isochronous [7]. On the contrary, the indicated frequency is determined by the Coriolis parameter, the geometric characteristics of the basin, and the coefficient x 1 of the linear term (in r) in the polynomial representation of the radial bulk force X.…”

“…On the contrary, the indicated frequency is determined by the Coriolis parameter, the geometric characteristics of the basin, and the coefficient x 1 of the linear term (in r) in the polynomial representation of the radial bulk force X. In the absence of external actions (X = 0), the frequency of oscillations of the fluid is superinertial and equal to the frequency found in [1,[4][5][6][7]14]. If the external radial force is directed from the boundary to the center of the basin (x 1 < 0), then the frequency of oscillations of the fluid shifts toward the region of higher frequencies, whereas the bulk force directed from the center of the basin to the coastline (x 1 > 0) decreases the frequency of oscillations of the fluid in the basin as compared with the case where the external bulk force is absent.…”

“…The application of the model of long waves in which the distribution of pressure in a moving fluid is assumed to be hydrostatic makes it possible to enlarge the list of nonlinear problems for which it is possible to find exact analytic solutions. Within the framework of this approach, the problems of axially symmetric oscillations of fluids in basins of parabolic shape [4][5][6][7] and fluctuations of eddy formations in the ocean [8][9][10][11][12][13][14] were solved. The investigations of the fluctuations of eddy formations were initiated by the problems of oceanology.…”

Two-dimensional isochronous nonstandard Hamiltonian systems

“…In particular, the Liénard class of oscillators appear in the study of a wide range of fields such as seismology [14], biological regulatory systems [15], in the study of a self graviting stellar gas cloud [16], optoelectronics, fluid mechanics [17]. Many of these Liénard class of equations admit limit cycle/periodic oscillations which are used to model many physical phenomenon.…”