1 Introduction

Many physical problems are modelled by hyperbolic system of PDEs in the form of either conservation laws or balance laws.Nowadays,engineering and science researchers routinely confront problems in finding the solution techniques for such mathematical models formulated in terms of nonlinear differential equations.Multiphase flows model is one of such models,arising in the context of multi-component flows or in the gas transport in pipe networks in which phase change takes place due to geometrical or physical forces.Two-phase flow occurs in many scientific and technical disciplines from natural and man-made process to the modelling of normal operations or accidental conditions in nuclear,petroleum or process engineering installations. These requirements provide a productive problem solving environment for the researchers in this field.Since there is no existing general theory for solving such nonlinear PDEs exactly, their solution relies on some analytical methods.Lie group analysis,based on the symmetry and invariance principles^[1,2] ,is the only systematic method for solving nonlinear differential equations analytically.Lie group analysis is a classical method discovered by Norwegian mathematician Sophus Lie for finding invariant and similarity solutions.The main advantage of such methods is that they can be successfully applied to nonlinear differential equations.Lie group analysis has played an important role in the construction of particular solutions of differential equations in terms of their symmetries.Today,the Lie group theoretical approach to differential equations has been extended to new situations and has become applicable to the majority of equations that frequently occur in applied sciences.This technique has been applied by many researchers to solve different flow phenomena over different geometries.Abd-el-Malek and Amin^[3] applied the Lie group method twice in studying nonlinear inviscid flows with a free surface under gravity,whereas the same method is used to determine symmetry reductions of the nonlinear PDEs^[4] ,i.e.,Hirota-Satsuma coupled Korteweg-de Vries(KdV)and the reduced ODEs are solved analytically.Bira and Sekhar^[5] studied the one-dimensional isentropic magnetogasdynamics and derived some exact solutions via Lie group analysis.The authors in Ref.^[6] derived the symmetry group for the one-dimensional ideal isentropic magnetogasdynamics and found some new exact group invariant solutions,while symmetry reductions and exact solutions for governing equations has been studied extensively in Ref.^[7].Sharma and Radha ^[8] obtained exact solutions of Euler equations of ideal gasdynamics using the Lie group analysis. Lie group transformations for self-similar shocks in a gas with dust particles have been discussed by Jena ^[9] .Donato and Oliveri ^[10] studied the reduction to autonomous form by group analysis and exact solutions of axisymmetric MHD equations.Lie group analysis and basic similarity reductions are performed for MHD aligned creeping flow and heat transfer in a second-grade fluid by neglecting the inertial terms by Afify ^[11] .Sharma and Radha ^[12] determined the class of self-similar solutions to a problem concerning plane and radially symmetric flows of a relaxing gas involving shocks of arbitrary strength using Lie group invariance.Interaction of a characteristic shock with a weak discontinuity in a non-ideal gas was discussed in Ref.^[13].Similarity solutions for three-dimensional Euler equations using symmetry analysis have been discussed by Sekhar and Sharma ^[14] .Sekhar and Sharma ^[15] studied the evolution of weak discontinuities in shallow water equations.Behavior of weak discontinuities for hyperbolic quasilinear systems has been discussed by Sharma^[16].

The structure of this article is organized as follows.In Section 2,we present the Lie group analysis for the governing PDEs and obtain the group of transformations which allows the system to remain invariant.The group of transformations is used to reduce the system of PDEs to system of ODEs.Then,the solutions of the ODEs which consequently produce some exact solutions for the governing system of PDEs are derived.Lastly,we discuss the evolutionary behavior of weak discontinuity.

2 Symmetry analysis

We consider an isothermal no-slip drift-flux model for multiphase flow in the following form^[17]:

where ρ₁ is the density of phase1,ρ₂is the density of phase 2,uis the common velocity of the two phases,and a is a constant which depends on the both phases.The independent variables t and x represent the time and spacecoordinates,respectively.

In order to reduce the governing PDEs into ODEs and to obtain exact solutions,we seek a one parameter Lie group of transformations^[1,2]

where φ₁,φ₂,μ₁,μ₂,and μ₃ are the infinitesimal group of transformations,which are to be determined in such a way that the PDEs(1)remain invariant; the entity is a small group parameter.The existence of such a group allows the number of independent variables in the problem to be reduced by one,and thereby allowing the system(1)to be replaced by a system of ODEs.

If we apply the procedure outlined in Ref.^[14] to the system(1),we obtain the following forms of the infinitesimals:

where α₁,α₂,α₃,α₄,and α₅ are arbitrary constants.

The similarity variables can be obtained from the characteristic equations given as follows:

In order to reduce the system of PDEs(1)to a system of ODEs,we solve the characteristic equations by considering different cases as follows:

In this case,the similarity and new dependent variables are

Substitution of the above new dependent variables in(1)gives the reduced system of ODEs Solving(4)for U= constant =k₁,weobtain where C₁ and C₂ are arbitrary integration constants and A₁ = Combining(5)and(3) yields the solution of(1)as

Case II:α₁=0,α₁=0

Here,the corresponding similarity and dependent variables are

Using the variables from(6)in(1),we obtain the system of ODEs

We solve the above system of ODEs,considering U= constant =k₂,,and we obtain where C₃ and C₄ are arbitrary integration constants and .Combining(7)and (6),we obtain the solution for(1)as follows:

Case III:α₁=α₂=α₄=0

This case yields the similarity and new dependent variables

Substituting these new variables in the governing system,we obtain the following system of ODEs:

The solution of the ODEs(9)is obtained by considering ,and the solution is given by

where C₅ and C₆ are constants of integration.The equations(10)and(8)in turn yield the corresponding solution for(1)as follows:

Case IV:α₁=0,α₂=0

For this case,if we solve the characteristic equation,we obtain the following similarity and new dependent variables as follows:

The variables in(11)reduce the governing system of PDEs to system of ODEs as follows:

The ODEs(12)can be solved completely for U=ξ and α₅=−α₂,and the solution of(1) is obtained as

where C₇ and C₈ are integration constants.

3 Evolution of weak discontinuities

The governing system of equations can be written in the matrix form

where M=(ρ₁,ρ₂,u) ^T is a column vector with superscript T denoting transposition,while B is a matrix with elements B₁₁ =B₂₂ =B₃₃ =u,B₁₂ =B₂₁ =0,B₁₃ =ρ₁,B23 =ρ₂,and The coefficient matrix Bhas three distinct eigenvalues together with six linearly independent left and right eigenvectors as given by

with the corresponding left and right eigenvectors

The evolution of weak discontinuity for a hyperbolic quasilinear system of equations satisfying the Bernoulli’s law has been studied quite extensively in Refs.[^15-16].The transport equation for the weak discontinuities across the third characteristic of a hyperbolic system of equationsisgivenby

where Λ denotes the jump in M_x across the weak discontinuity wave and given by Λ =β_r ⁽³⁾, where β is the amplitude,and The jump Λ across the weak discontinuity wave propagates along the curve determined by originating from the point(x₀,t₀). Now,(16)with the solution(13)gives the Bernoulli type of equation for the amplitude βas follows: where

The solution of(17)can be written in quadrature form as ,where and

It is noticed that,in the interval [1,∞),both the integrals I()andJ()are finite and continuous.For β₀ >0,which corresponds to an expansion wave,it is clear that as → ∞, I()→0 where as J(∞)<∞,and the wave decays and dies out eventually,the corresponding situation is shown in Fig.1.From the Fig.2,it is seen that,for β₀ <0,there existsβcsuch that, for |β₀|<β_c, initially decreases fromβ0and reaches to minimum at finite time.However, for |β₀|&geβ_c, increases from β₀ and terminates into a shock; the corresponding situation is illustrated by the curve in Fig.3.Here, ,and are the dimensionless variables.

4 Conclusions

Using the Lie group analysis,we obtain some exact solutions of the system of nonlinear PDEs which governs a no-slip drift-flux model for multiphase flow.These analytical solutions play an important role in a better understanding of qualitative features of two-phase flow equations. In this context,analytical solutions of non-linear differential equations graphically demonstrate and allow unraveling the mechanisms of many complex non-linear phenomena such as spatial localization of transfer processes,multiplicity or absence of steady states under various conditions,existence of peaking regimes.Finally,we discuss the evolution of weak discontinuity where we study the existence of shock wave along the solution curve.