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Invariant solutions of the special model of fluids with internal inertia
Abstract
This paper is devoted to the detailed study one of the models, obtained in [Hematulin A, Meleshko SV, Gavrilyuk SG. Group classification of one-dimensional equations of fluids with internal inertia. Math Meth Appl Sci 2007;30:2101–20] by group analysis. The main feature of this model is that the equations describing the motion of fluids with the potential function W=-αρ-3ρ˙2+βρ3 admit projective transformations. Optimal systems of subalgebras are constructed. Representations of all invariant solutions are obtained.
An application of group analysis provides a regular procedure for mathematical modeling by classifying differential equations with respect to arbitrary elements. This paper presents the group classification of one-dimensional equations of fluids where the internal energy is a function of the density, the gradient of the density and the entropy. The group classification separates all models into 83 different classes according to the admitted Lie group. Some invariant solutions are studied.
A systematic application of the group analysis method for modeling fluids with internal inertia is presented. The equations studied include models such as the nonlinear one-velocity model of a bubbly fluid (with incompressible liquid phase) at small volume concentration of gas bubbles (Iordanski Zhurnal Prikladnoj Mekhaniki i Tekhnitheskoj Fiziki 3, 102–111, 1960; Kogarko Dokl. AS USSR 137, 1331–1333, 1961; Wijngaarden J. Fluid Mech. 33, 465–474, 1968), and the dispersive shallow water model (Green and Naghdi J. Fluid Mech. 78, 237–246, 1976; Salmon 1988). These models are obtained for special types of the potential function
(Gavrilyuk and Teshukov Continuum Mech. Thermodyn. 13, 365–382, 2001). The main feature of the present paper is the study of the potential functions with W
S
≠ 0. The group classification separates these models into 73 different classes.
Group analysis provides a regular procedure for mathematical modeling by classifying differential equations with respect to
arbitrary elements. This article presents the group classification of one-dimensional equations of fluids, where the internal
energy is a function of the density and the gradient of the density. The equivalence Lie group and the admitted Lie group
are provided. The group classification separates all models into 21 different classes according to the admitted Lie group.
Invariant solutions of one particular model are obtained.
Group classification of the three-dimensional equations describing flows of fluids with internal inertia, where the potential function W = W(ρ,ρ·), is presented. The given equations include such models as the non-linear one-velocity model of a bubbly fluid with incompressible liquid phase at small volume concentration of gas bubbles, and the dispersive shallow water model. These models are obtained for special types of the function W(ρ,ρ·). Group classification separates out the function W(ρ,ρ·) at 15 different cases. Another part of the manuscript is devoted to one class of partially invariant solutions. This solution is constructed on the base of all rotations. In the gas dynamics such class of solutions is called the Ovsyannikov vortex. Group classification of the system of equations for invariant functions is obtained. Complete analysis of invariant solutions for the special type of a potential function is given.
La méthode d'équivalence de Cartan permet de décider de l'équivalence locale de deux objets de nature géométrique sous l'action d'un pseudo-groupe de difféomorphismes locaux. En utilisant cette méthode, nous donnons des conditions explicites pour qu'une équation différentielle ordinaire du 3eme ordre soit linéarisable par une transformation de contact. Pour citer cet article : S. Neut, M. Petitot, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 515–518.
All invariant and partially invariant solutions of the Green-Naghdi equations are obtained that describe the second approximation of shallow water theory. It is proved that all nontrivial invariant solutions belong to one of the following types: Galilean-invariant, stationary, and self-similar solutions. The Galilean-invariant solutions are described by the solutions of the second Painleve equation, the stationary solutions by elliptic functions, and the self-similar solutions by the solutions of the system of ordinary differential equations of the fourth order. It is shown that all partially invariant solutions reduce to invariant solutions.
We calculate in detail the conditions which allow the most general third-order ordinary differential equation to be linearised in X
(T)=0 under the transformation X(T)=F(x,t), dT=G(x,t)dt.
This chapter focuses on the theory and applications of differential invariants of Lie transformation groups, which can be infinite. It discusses the concepts of a differential invariant and special transformation of a differential equation, the so-called group stratification. The central theorem is about the finiteness of bases for differential invariants of arbitrary Lie groups. In the proposed variant Tresse's theorem is completely established in infinitesimal language by the means of invariant differential operators. An equation E is called automorphic relative to the given group G if all its solutions are situated on the orbit of one of them. In other words, from one given solution U, it is possible to find all solutions by applying to U the transformations of the group G. Classical problems of the construction and classification of automorphic systems provide excellent examples of the applications of the general theory, and these play an important role in the problems of invariant transformations for differential equations.
Lectures on Geophysical Fluid Dynamics offers an introduction to several topics in geophysical fluid dynamics, including the theory of large-scale ocean circulation, geostrophic turbulence, and Hamiltonian fluid dynamics. Since each chapter is a self-contained introduction to its particular topic, the book will be useful to students and researchers in diverse scientific fields.
One-dimensional flows of fluids with internal inertia are studied in the manuscript. The given equations include such models as the non-linear one-velocity model of a bubbly fluid (with incompressible liquid phase) at small volume concentration of gas bubbles (Zh. Prikl. Tekh. Fiz. 1960; N3:102–111, Dokl. AN USSR 1961; 137:1331–1333, J. Fluid Mech. 1968; 33:465–474), and the dispersive shallow water model (J. Fluid Mech. 1976; 78:237–246, Lectures on Geophysical Fluid Dynamics. Oxford University Press: New York, 1998). These models are obtained for special types of the function . The group classification separates these models in 10 different classes. Optimal systems of subalgebras are constructed for all models. The knowledge of optimal systems of admitted subalgebras allows constructing essentially different invariant solutions. Copyright © 2007 John Wiley & Sons, Ltd.
An invariant submodel of the two-dimensional equations of gas dynamics, constructed on an operator which is a combination of the time-shift, rotation and projective operators, is investigated using the PODMODELI program [1]. A canonical form of the submodel is constructed and a preliminary analysis of it is carried out (the group property, the hyperbolicity region and the first integrals). The self-similar solution of the submodel is investigated in detail. It determines the solutions of the submodel in question with closed invariant streamlines. Using a hierarchy of submodels, first integrals are obtained in the “second-level” submodel. A qualitative description of the nature of the motion is given (the contact characteristics and the particle trajectories). It is shown that the solution possesses discrete symmetry - invariance under rotation around the origin of coordinates by an angle that is a multiple of 2π/N, with a certain natural N. It is pointed out that for certain values of the parameters, solutions of this type describe the gas motion with vacuum regions. The features of the flows obtained are illustrated by examples — the exact solutions of the gas — dynamic equations, which describe the expansion of a gas to a vacuum in an infinite time.
Within the scope of the three-dimensional theory of homogeneous incompressible inviscid fluids, this paper contains a derivation of a system of equations for propagation of waves in water of variable depth. The derivation is effected by means of the incompressibility condition, the energy equation, the invariance requirements under superposed rigid-body motions, together with a single approximation for the (three-dimensional) velocity field.
On the basis of previous work by the author, equations are derived describing one-dimensional unsteady flow in bubble-fluid mixtures. Attention is subsequently focused on pressure waves of small and moderate amplitude propagating through the mixture. Four characteristic lengths occur, namely, wavelength, amplitude, bubble diameter and inter-bubble distance. The significance of their relative magnitudes for the theory is discussed. It appears that for high gas content the dispersion is weak and then the conservation of mass and momentum lead to equations similar to the Boussinesq equations, describing long dispersive waves of finite amplitude on a fluid of finite depth. For waves propagating in one direction only, the corresponding equation is similar to the Korteweg–de Vries equation.
It is shown that for mixtures of low gas content the frequency dispersion is in most cases not small. Finally, solutions of the Korteweg–de Vries equation representing cnoidal and solitary waves in a bubble–liquid mixture are given explicitly.
The aim of this article is to study the main properties for a class of lagrangian models describing, in particular, bubbly liquids and dispersive shallow water flows. Important notions of generalized vorticity and generalized potential flows are introduced which permits one to prove the analogues of classical theorems of ideal Fluid Mechanics: Lagrange, Cauchy, Kelvin and Bernoulli theorems. A non-local Hamiltonian formulation of the generalized potential flows is obtained. A generalization of the classical singular solutions: 2-D vortex-source, axisymmetric swirl, and solutions with a uniform space distribution of the pressure have also been found.
We present here the solution of the problem on linearization of third-order ordinary differential equations by means of point and contact transformations. We provide, in explicit forms, the necessary and sufficient conditions for linearization, the equations for determining the linearizing point and contact transformations as well as the coefficients of the resulting linear equations.
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La géométrie de l'équation y 000 ¼ f ðx; y; y 0 ; y 00 Þ. Ser. I
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S. Neut and M. Petitot. La géométrie de l'équation y 000 ¼ f ðx; y; y 0 ; y 00 Þ. Ser. I, Équations différentielles/ordinary differential equations, vol.
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Ovsiannikov LV. Group analysis of differential equations. Nauka, Moscow, 1978. English translation, Ames, WF, editor. New York: Academic Press; 1982.
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