Steepened wave in two-phase Chaplygin flows comprising a source term

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Abstract

This manuscript brings some qualitative features of steepened wave in isentropic Chaplygin two-phase flows with a non-constant source term via Lie group transformation. The transport equation for steepened wave is determined. The behaviour of amplitude of steepened wave is investigated using the numerical solution of the system. The effects of inclination of the flow on the amplitude of singular surface are also shown.

Introduction

Most of the heat exchangers (approximately 60%) in industrial area include two-phase flow models [1]. The two phase flow model has wide range of applications in the nuclear explosions, power and process industries as well as in refrigerating, air conditioning and reservoir simulators etc. The presence of phasic momentum, interfacial pressure terms, surface tension and other factors make the model non-linear and more complicated. When we take such physical factors into account that predominate affect the steepened wave, the problem becomes very complex. It has commonly been assumed that dark energy and dark matter occupied the biggest part of universe. Different kinds of models have been introduced to illustrate the behaviour of dark energy. Modified and standard Chaplygin gas is one of the appreciable model to demonstrate the dark energy. The generalised Chaplygin gas which is an exotic fluid, portray the acceleration of universe. Firstly, Chaplygin [2] introduced the equation of state for an exotic background fluid and also known as pure Chaplygin gas. The modified Chaplygin equation of state for negative pressure at low energy density and high pressure at high energy density is used widely in [3], [4], [5]. Later on, Yang and Wang [6] obtained the analytical solution for the Riemann problem of the isentropic Euler equations with modified Chaplygin equation of state, which was proposed by Benaoum [7]. The recent studies show that the investigation of multiphase flows are getting much attention and is an young field of intense research. A fully conservative and hyperbolic system of two phase flow is studied in [8], [9]. Evje and Flåtten [10] investigated the wave structure of two fluids and drift-flux models by using a perturbation technique. Evje and Karlsen [11] studied the global existence of weak solution for viscous two phase liquid-gas model. Purnima and Raja Sekhar [12] determined the evolution of weak discontinuity for isentropic drift flux model by using similarity solution. For an isothermal no-slip compressible gas-liquid drift flux equation of two-phase flows, Minhajul and Raja Sekhar [13], [14] studied the interaction of elementary waves of the Riemann problem with a weak discontinuity.

The solution of hyperbolic systems are nonlinear waves, which may be considered as a moving surface. Thomas [15] defined the compatibility conditions of first and higher order which connects the first and higher order derivatives of flow variables across the singular surface. Using these conditions, Shah and Singh studied the behaviour of steepened wave in dusty real reacting gases and investigated its interaction with a blast wave [16] and a characteristic shock [17]. Furthermore, with the aid of a particular solution, the problem of interaction of a characteristic shock with a singular surface in a relaxing dusty gas have been investigated by Mittal and Jena [18].

In this manuscript, we have introduced a novel generalised Chaplygin equation of state for a model of two phase flow. The outline of the present manuscript is as follows. In Section 2, we presents the model describing two phase flow of Chaplygin gas with a non-constant source term. Section 3, Lie symmetry analysis is used to obtain the equivalent ordinary differential equations to the system under consideration. The behaviour of steepened wave is depicted by using numerical solution. The effects angle of the flow direction with respect to the horizontal on the steepened wave are also investigated in Section 4. The manuscript end up by the conclusions in Section 5.

Section snippets

Model formulation

The analysis of two-phase flow which covers all possible flow patterns would be too complicated hence certain assumptions have to be made in order to simplify the system so that the flow can be analysed. We consider the following assumptions to make the model little simpler

  • (i)

    velocities of both phases are same i.e. no slip relation;

  • (ii)

    no mass transfer between the both phase take place;

  • (iii)

    no interphasic momentum exchange term present.

Thus the equations for the conservation of mass and momentum for

Symmetry analysis

Here we devote our attention on finding the Lie symmetries of the systems under consideration (1). Further details on theoretical aspects of the Lie group analysis can be found in the well known monographs [17], [26], [27], [28], [29]. A one parameter (ε) Lie group of infinitesimal transformations is contemplated as followst*=t+εΥ1(t,x,ρ1,ρ2,u),x*=x+εΥ2(t,x,ρ1,ρ2,u),ρ1*=ρ1+ες1(t,x,ρ1,ρ2,u),ρ2*=ρ2+ες2(t,x,ρ1,ρ2,u),u*=u+ες3(t,x,ρ1,ρ2,u),where the form of infinitesimals Υ1,Υ2,ς1,ς2,ς3 are to be

One-dimensional steepened wave

In order to investigate the behaviour of steepened wave, we assume that steepened wave Ψ:x(t), generated from x=xs, is propagating with velocity dx/dt=Φ in the both phases of flow, across which all the variables are continuous but possesses jump in their first and second order derivatives. We use the following compatibility condition [15], [30] to obtain the transport equation for steepened wave[Ux]=V,[Ut]=−VΦ,[Uxx]=V¯,[Uxt]=Φ(dVdx−V¯),where |[F]|=Fs−F represents the jump in the entity F across

Conclusions

In this study, we have obtained the equivalent ordinary differential equations to the model of two phase flow of Chaplygin gas with a non-constant source term via Lie group analysis, which is worthwhile to analyze the assembly phenomenon for the dark energy and dark matter in the evolution of the universe. Such type of solutions for two phase flow of Chaplygin gas with a variable source term are only one of the efficient way to study the propagation of waves in the dark energy and dark matter

Acknowledgment

Research support from CSIR India (Ref no. 09/045(1444)/2016-EMR-I), is gratefully acknowledged by the first author (SS).

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