Variational approach to constructing hyperbolic models of two-velocity media

Abstract

A generalized Hamilton variational principle of the mechanics of two-velocity media is proposed, and equations of motion for homogeneous and heterogeneous two-velocity continua are formulated. It is proved that the convexity of internal energy ensures the hyperbolicity of the one-dimensional equations of motion of such media linearized for the state of rest. In this case, the internal energy is a function of both the phase densities and the modulus of the difference in velocity between the phases. For heterogeneous media with incompressible components, it is shown that, in the case of low volumetric concentrations, the dependence of the internal energy on the modulus of relative velocity ensures the hyperbolicity of the equations of motion for any relative velocity of motion of the phases.Introduction. At least three approaches to constructing mathematical models of two-velocity media are known at present. The averaging method is used most widely, especially to constructing models of motion for heterogeneous two-velocity media. A distinguishing feature of heterogeneous media is that each phase occupies only part of the volume of the mixture, unlike in homogeneous mixtures, in which each phase is uniformly distributed over the entire volume of the mixture. Applying an appropriate averaging operator to the equations of conservation of mass, momentum, etc. that are valid within each phase, one obtains averaged equations of motion. The main problem that arises in this approach consists in closing the resulting system: the system contains more unknowns than equations. Different experimental and theoretical assumptions on the flow structure, the mechanism of interaction between the phases, etc. [1-3] (see also a review [4]), are used for closing. As was noted by many authors [5, 6], if the pressure in the phases coincide, the corresponding equations of motion in a nondissipative approximation turn out to be nonhyperbolic even when the relative difference between the phase velocities is slight. This means that the Cauchy problem for the corresponding nonlinear equations of motion is incorrect.In [7-9], hyperbolic (nonequilibrium in pressure) models of two-layer liquid flows were obtained by the averaging method. For closure of the equations of motion, a series of hypotheses relating the pressure and velocity at the interface between the liquids to their average values in the layer were invoked [7], or the process of mixing of the liquid at the interface was taken into account by introducing a third liquid layer [9]. An interesting two-velocity model of a bubble liquid which takes into account oscillations of bubbles is proposed in [10]. In the approximation of an incompressible liquid and a small bubble concentration, the model is hyperbolic for a low relative velocity of the bubbles and gives steady wave modes. Interesting hyperbolic models are also proposed in [11][12][13][14]. Their hyperbolicity was reached using closing relations, which are usually Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian A...

A new form of governing equations of fluids arising from Hamilton's principle

“…Due to the invariance of the governing equations with respect to the Galilean transformation we assume, without loss of the generality, that u 1e = u 2e = 0. Suppressing the index "e" to avoid double indices, we get Taking into account (7.12) -(7.13) we obtain by straightforward calculations of the determinant (7.11) The following result is proved in [21]:…”

“…Conditions (7.14) mean that internal energy U = W − w ∂W ∂w , where w = |w|, is convex [15][16]21]. Hence, if the internal energy is a convex function and the relative velocity w is small enough, the equilibrium state is stable in time direction in the time-space.…”

“…Now we are able to calculate the dispersion relation(7.11) which is the determinant of a square matrix of dimension 10. +ρ2 ) + ρ 1 ρ 2 β 4 1 − a(ρ 2 1 w 11 + 2ρ 1 ρ 2 w 12 + ρ 2 2 w 22 ) + ρ 1 ρ 2 (ρ 2 w 22 + ρ 1 w 11 ) +ρ 2 )+ρ 1 ρ 2 λ 4 − a(ρ 2 1 w 11 +2ρ 1 ρ 2 w 12 +ρ 2 2 w 22 ) + ρ 1 ρ 2 (ρ 2 w 22 +ρ 1 w 11 ) λ 2If potential W (ρ 1 , ρ 2 , µ) satisfies conditions(7.14), all the roots λ of the polynome (7.16) are real.|w|, is convex[15][16]21]. Hence, if the internal energy is a convex function and the relative velocity w is small enough, the equilibrium state is stable in time direction in the time-space.…”

Optimization of consistent two-equation models for thin film flows

“…In the same way, the energy ( 40) is the sum of Ũ 2 /2 which is not Galilean invariant and which assumes the role of the kinetic energy of the model and of ( ũ 2 + h2 Fr −2 cos θ)/2 which is Galilean invariant and which can therefore be treated as the potential energy (see Gavrilyuk and Perepechko [24] for a discussion on what are the kinetic energy and the potential energy of a system in the context of a variational method). It follows that, in the energy of the model Ũ 2 /2 + E 2 + α 3 h2 Fr −2 cos θ/2, E 2 is Galilean invariant.…”

Global Unique Solvability of the Initial-Boundary Value Problem for the Equations of One-Dimensional Polytropic Flows of Viscous Compressible Multifluids

“…A survey of possible formulations of the model and known results can be found in [5], [6]. Related multi-velocity models are discussed in [7], [8], [9], [10], [11]. As the first results on the solvability of multifluid models in the multidimensional case (but in approximate formulations), we can indicate [12], [13], [14].…”