In the paper, the nonlinear oscillations of a two-layer fluid that completely fills a limited tank are theoretically studied. To determine any smooth function on the deflected interface, the Taylor series expansions are considered using the values of the function and its normal derivatives on the undisturbed interface of the fluids. Using two fundamental asymmetric harmonics, which are generated in two mutually perpendicular planes, the differential equations of nonlinear oscillations of the two-layer fluid interface are investigated. As a result, the frequency-response characteristics are presented and the instability regions of the forced oscillations of the two-layer fluid in the cylindrical tank are plotted, as well as the parametric resonance regions for different densities of the upper and lower fluids. The Bubnov-Galerkin method is used to plot instability regions for the approximate solution to nonlinear differential equations. At the final stage of the work, the nonlinear effects resulting from the interaction of fluids with a rigid tank that executes harmonic oscillations at the interface of the fluids are theoretically studied.
KeywordsThe purpose of this work was to study spectral and Cauchy problem for the mechanical system consisting of three rods, two of them being identical and connected with the third one by linear elastic elements. We stated the corresponding spectral problem and studied its spectrum. Findings of the research show that eigenfunctions of the considered spectral problem are classified according to the irreducible representations of the finite group of transformations despite the fact that the initial equations system admits continuous (Lie) transformation groups. We considered the weak solution of Cauchy problem and revealed its simplification in case of special "symmetrical" form of initial conditions and right-hand side of the corresponding operator equation system Introduction. In the natural vibrations problem symmetry of mechanical system plays an important role. The representation theory of symmetry groups is an approach which allows recognizing and exploiting the influence of the system symmetry on the corresponding spectral problem. The main result of the representation theory with respect to spectral problems can be formulated in the following theorem [1]. Theorem. Let linear operator A commutes (permutable) with the representation operators of symmetry group G and has a discrete spectrum of eigenvalues with finite multiplicity, then its eigenfunctions are basis functions of group G irreducible representations.As a result, with help of projection operators on subspaces of irreducible representations, spectral problem can be solved in these subspaces [2] which usually have lower dimension than initial space [3].This approach has been successfully applied to mechanical systems with a finite number of degrees of freedom [4] (molecular vibrations [5], mass-spring models of mechanical systems [6,7], finite element and finite difference models [8][9][10] etc.). In this case the symmetry group is a finite group, usually represented by spatial symmetry of the system, and the irreducible representations and their respective projectors can be easily found with the tables of characters of the finite groups irreducible representations [11].
Due to the continued research into chemistry, biology, pharmaceutics and rocket space technology, interest in the study of the dynamics of layered fluids has increased significantly. The paper focuses on oscillations of a three-layer viscous fluid, gives the formulation of the viscous fluid free oscillations problem. Within the research, we determined natural frequencies and damping coefficients of oscillations of the three-layer viscous fluid in a cylindrical vessel by means of the boundary layer method and a mechanical analog. Oscillations of the three-layer viscous fluid were considered as joint oscillations of two partial hydrodynamic systems, one of which corresponds to oscillations of the upper and middle viscous fluids, and the other one - to oscillations of the middle and lower fluids. Then, we determined the coefficients of viscous resistance in partial hydrodynamic systems of a two-layer viscous fluid. Using the mechanical analog of oscillations of the three-layer liquid, we derived the characteristic equation for determining natural frequencies of the hydrodynamic system under consideration. Next, we calculated the dependency of natural frequencies and liquid-liquid interface damping coefficients on the height of the middle layer and the density of the upper fluid. Finally, we analyzed and compared theoretical calculations with the results obtained by other researchers and experimental investigation. The paper gives the results of experimental studies of oscillations of the three-layer fluid in a stationary cylindrical tank.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.