Interaction of waves in one-dimensional dusty gas flow

The present study uses the theory of weakly nonlinear geometrical acoustics to derive the highfrequency small amplitude asymptotic solution of the one-dimensional quasilinear hyperbolic system of partial differential equations characterizing compressible, unsteady flow with generalized geometry in ideal gas flow with dust particles. The method of multiple time scales is applied to derive the transport equations for the amplitude of resonantly interacting high-frequency waves in a dusty gas. These transport equations are used for the qualitative analysis of nonlinear wave interaction process and selfinteraction of nonlinear waves which exist in the system under study. Further, the evolutionary behavior of weak shock waves propagating in ideal gas flow with dust particles is examined here. The progressive wave nature of nonresonantwaves terminating into the shockwave and its location is also studied. Further, we analyze the effect of the small solid particles on the propagation of shock wave.


Introduction
The study of elementary wave interactions consist of either interaction between two waves colliding, or one wave overtaking another, or one wave meeting a discontinuity. Such a phenomenon frequently happens while studying the wave propagation in the field of space science, space re-entry, astrophysical phenomenon and aerodynamics etc. Also, the analysis of how shock waves interact with each other, as well as with the exhaust plume of an aircraft, has been an area of great interest among scientists and engineers.
The propagation of waves in a medium is governed by quasilinear hyperbolic system of PDEs. One often encounters certain kinds of discontinuities known as acceleration waves, shock waves and weak waves. The study of these waves has been of great significance in engineering science and nonlinear science due to its application in various fields such as nuclear physics, plasma physics, geophysics, astrophysical sciences and interstellar gas masses.
In the present paper, we use the method of asymptotic analysis to study the evolutionary behavior of shock wave which is widely used by many researchers, e.g. Choquet-Bruhat [1], Hunter et al. [2], Majda et al. [3] over decades. Also, the qualitative analysis of interaction of nonlinear waves can be obtained by the interaction coefficients which occur in transport equation and these coefficients are measure the coupling strength between different types of wave modes. The method of asymptotic analysis has been widely used to study the propagation of weak shock waves governed by the nonlinear hyperbolic system of partial differential equations. The study of resonantly interaction of shock waves by using "Asymptotic analysis method" for one-dimensional ideal gas flow in presence of the solid dust particles have not been analyzed by any author till now. To analyze the evolutionary behavior of shock wave in ideal gas with dust particles is more complex in comparison to ideal gas flow. A different kind of physical phenomenon which occurs in various processes such as space re-entry, chemical explosion, nuclear explosions, supersonic flow and collision of two or more galaxies are described by mathematical model of quasilinear hyperbolic system of partial differential equations.
In last few decades, many attempts have been made to analyze the asymptotic properties of shock waves in various gasdynamic regimes where the governing equation is a system of quasilinear hyperbolic partial differential equations. The "Weakly nonlinear geometrical acoustics theory" provides a methodical technique for dealing with the interaction of nonlinear high-frequency small amplitude waves. The wave propagation phenomenon with an added effect of nonlinearity has been analyzed in past but the closed form exact analytic solution of the equations governing the motion of waves has never been obtained. In most of the literature, only approximate analytical or numerical solutions are discussed. In this context, it is worth to mention the contributions made by many authors like Hunter and Ali [4], Gunderson [5], He and Moodie [6], Whitham [7], Moodie et al. [8], Arora and Sharma [9], Arora [10], Fusco [11].
From the physical and mathematical view point, the discussion of shock waves in an ideal gas consisting of solid particles is a topic of great interest because of its numerous applications such as underground explosions, interstellar masses, lunar ash flow and explosive volcanic eruptions etc. Dusty gas is composed of small solid particles and gas in which solid particles do not attain more than 5% of its entire volume. In mixture of gas and solid particles, the study of shock wave has more significance due to its wide applications in several areas such as supersonic-vehicle in sand storms, supersonic flights in polluted air, nuclear reaction, aerospace engineering science etc. Vishwakarma et al. [12][13][14] have discussed the propagation of shock wave in dusty gas with varying density. Chaturvedi et al. [15][16][17][18] have discussed the evolution of weak shock wave in two-dimensional, steady supersonic flow in dusty gas. Sharma et al. [19] have used the scheme of multiple time scales to study the wave interaction in a nonequilibrium gas flow. Pooja et al. [20] and Nath et al. [21,22] have used an asymptotic technique to analyze the evolution of weak shock waves in nonideal magnetogasdynamics and nonideal radiating gas flow. Singh et al. [23] have theoretically investigated the propagation of shock wave in radiative magnetogasdynamics. Propagation of shock wave in a mixture of gas and dust particles has been widely investigated by several authors such as Nath et al. [24], Nath [14,25] and Nandkeolyar et al. [26]. Singh et al. [27][28][29][30] have studied the evolutionary behavior of shock wave in various gasdynamic regimes. Bhattacharyya et al. [31,32] have discussed about the simulation of Cattaneo-Christov heat flux on the flow of single and multiwalled carbon nanotubes between two stretchable coaxial rotating disks. Seth et al. [33] have studied the partial slip mechanism on free convection flow of viscoelastic fluid past a nonlinearly stretching surface. Jena et al. [34] and Radha et al. [35] have applied the methods of relatively undistorted waves and weakly nonlinear geometrical optics to study the situations when the disturbance amplitude is finite, arbitrarily small and not so small in nonideal gas flow and relaxing gas.
The main motive of the present paper is to apply the method of resonantly interacting multiple time scales to study the small amplitude high-frequency waves for one-dimensional, unsteady planar flow, cylindrically symmetric flow and spherically symmetric flow in a dusty gas. The transport equations for the amplitude of resonantly interacting high-frequency waves in a dusty gas are derived. Also the existence of weak shock waves in a dusty gas is discussed here. Further, the evolutionary behavior of weak shock waves propagating in ideal gas flow with dust particles is examined here.
This paper is organized as follow: in Section 2, we describe the basic equations for the dusty gas flow. Also, we reformulate the governing equations into quasilinear system and derive the characteristic for the system. In Section 3, we use the multiple time scale method to obtain high-frequency small amplitude asymptotic solution to the system written in Section 2. The transport equations for the propagation of shock are derived in Section 4. In Section 5, we investigate the conditions which explain the evolutionary behavior of shock wave for the planar and nonplanar cases. In the last Section 6, we discuss the results and conclusion of the present work.

Problem formulation and characteristics
The basic equations governing the one-dimensional compressible, inviscid, unsteady planar and nonplanar flows in a dusty gas mixture following the equation of state of Mie Grüneisen type are written as [16-18, 21, 22] ∂ρ ∂t where v is the velocity of the particle along the spatial coordinates x. The symbols ρ, p and t represent the density, pressure and time, respectively. T denotes the temperature and R is the gas constant. Here, m = 0 exhibits the planar flow, m = 1 exhibits the cylindrically symmetric flow and m = 2 exhibits the spherically symmetric flow. The entity Z is the volume fraction and k p is the mass fraction of solid particles in the mixture which are defined as where m sp is the total mass of the solid particles, V sp is volumetric extension of the solid particles, V g is the total volume of the mixture and m g is the total mass of the mixture. The quantity where γ here c sp is the specific heat of the solid particles, c p and c v are the specific heats of the gas at constant pressure and at constant volume respectively. The relation between the entities Z and k p is defined as Z ϕρ, where ϕ kp ρ sp with ρ sp as the specific density of solid particles and e is the internal energy per unit mass of the mixture which is given by Now, we write Equation (2) in the following matrix form where U (ρ, ν, p) tr , Q (mρν/x, 0, ρc 2 mν/x) tr and P is the coefficient matrix of order 3 × 3 having the components P ij , P 11 P 22 P 33 ν, P 13 P 21 P 31 0, Here the superscript "tr" represents transposition. The eigenvalues of the matrix P are given as λ 1 ν + c, λ 2 ν, λ 3 ν − c. Therefore, the system (6), which has distinct eigenvalues, is strictly hyperbolic and has three families of characteristics corresponding to three distinct eigenvalues, out of these three characteristics two represent waves moving in ±x directions with speed v ± c. The remaining one characteristic exhibits the particle path propagating with velocity v. Now, we suppose that the shock waves are propagating into an initial back ground state U 0 (ρ 0 , 0, p 0 ) tr . At constant speed v 0 = 0, the characteristics speeds are provided by λ 1 c 0 , λ 2 0, λ 3 −c 0 , where the subscript 0 indicates the evaluation at U U 0 which is identical with an equilibrium state.

Weakly nonlinear resonant waves
In this segment, the multiple time scale method will be applied to obtain high-frequency small amplitude asymptotic solution to the system of Equation (6) when the attenuation time scale (τ at ) is large in comparison to the characteristic time scale (τ ch ), it means ξ = τ ch /τ at ≪ 1. Let l (i) and r (i) (i = 1,2,3), respectively, exhibit the left and right eigenvectors of the matrix P 0 corresponding to the eigenvalues λ 1 c 0 , λ 2 0, λ 3 −c 0 . The eigenvectors l (i) and r (i) (i 1, 2, 3) satisfy the normalization conditions where δ ij is Krӧnecker delta. Now under the above assumptions the left and right eigenvectors are given by Now, we explore the asymptotic solution of Equation (6) as ξ → 0in the following form where U 1 is smooth bounded function of its arguments and U 2 is bounded function in (x, t) coordinate in a definite bounded portion containing atmost sublinear growth ϑ → as ϑ → → ±∞ Sharma and Srinivasan [19]. Here ϑ being the phase function of ith wave communicated with the characteristics speed λ i . Now using the Taylor's series expansion of matrices P and Q, about the constant state U 0 in powers of ξ and using Equation (9) in Equation (6) and then substituting the partial derivatives ∂/∂Y(Y will either be x or t) by∂/∂Y + ξ −1 ∑ i 3 i 1 (∂θ i /∂Y)∂/∂ϑ i and equating to zero the coefficients of ξ 0 and ξ 1 in resulting expression, we get where ∇ is the gradient operator with regard to the dependent variable U, I is the identity matrix of order 3 × 3. Now all the phase functions θ i (1 ≤ i ≤ 3) propitiate the Eikonal equation: where ⟨det⟩ represents the determinant. Now we consider a simplest phase function given as We infer from Equation (10) that for all phase functions θ i ,the derivative terms ∂U 1 ∂ϑ i are parallel to the right eigenvectors r (i) of the matrix P 0 , thus we have where the scalar function σ i (l i ⋅ U 1 ), is recognized as wave amplitude that depends on the ith fast variable ϑ i . The wave which forms here, whether it is an oscillatory wave or a pulse, is based on dependency of σ i on ϑ i . Let us suppose that σ i (x, t, ϑ i ) has zero mean value with regard to the fast variable ϑ i , it means We utilize the Equation (14) in Equation (11) which gives the following relationship for U 2 Further, utilizing Equation (16) in Equation (11) and pre-multiplying the resulting expression by l (i) yields the system of decoupled inhomogeneous first-order PDEs which are given as Now the term ∂ ∂t + λ i ∂ ∂x , which exhibits the ray derivative, is denoted by dσ i ds , hence the Equation (17) may be recast as The ith characteristics in Equation (18) is provided by Therefore, we determine the asymptotic average of Equation (18) along the characteristics and then supplicate to the sub-linearity of U 2 in ϑ, which ensures that there is no secular term in Equation (11). In view of Equation (18) we have, along the characteristics, ϑ i are stable and the asymptotic mean value of m • vanishes which implies that the wave amplitude σ i (1 ≤ i ≤ 3) propitiating the succeeding system of coupled integrodifferential equations: where σ k ∧ ∂σ k ∂ϑ k and the coefficient β i and Ω i jk are written as Now we have to determine the coefficients β i which vanish for plane waves (m = 0) therefore in this case the governing system does not contain any source term. Further, in the absence of dust particles, Equation (19) changes to the same equation as discussed in the study by Majda and Rosales [3]. Furthermore, the interaction coefficients Ω i jk are asymmetric in j and k which quantify the coupling strength between jth and kth wave modes where j ≠ k that can produce an ith wave (i ≠ j ≠ k). The interaction coefficients Ω i jk for i j k assign to the nonlinear self-interaction. Here for genuinely nonlinear waves, the interaction coefficients are nonzero and zero for linearly degenerate waves. Also, it is noticed that if each of the coupling coefficients Ω i jk (i ≠ j ≠ k) are zero or the integral in Equation (19) becomes zero, then we conclude that the wave do not resonate and Equation (19) detracts to the uncoupled system of Burger's equations. Now from the Equation (20), in the governing system, the coefficients β i and Ω i jk provide the qualitative picture of the nonlinear interaction procedure and it can be demonstrated by the formulas as given in (20).
Therefore, these coefficients are written as Now after some simplification, the resonant Equation (19) can be written as Here k is kernel which is written as In Equation (22), the integral average term shows the contribution to the wave amplitude σ 1 as a result of the nonlinear interactions of the wave field σ 2 with the wave field σ 3 . Equivalently from Equation (24), the integral average term shows the contribution to the wave amplitude σ 3 as a result of the nonlinear interaction of the wave field σ 2 with the wave field σ 1 . The nonlinear term proportional to σ 1 σ ′ 1 and σ 3 σ ′ 3 in (22) and (24) account for self-interaction which generates higher harmonics leading to the distortions of the wave profile and consequent shock formation. The result is that the two acoustic wave fields σ 1 and σ 3 exhibits a strong effect from the nonlinearity present in the system under study. More detailed analysis may be presented on the similar lines as given in the study by Sharma and Srinivasan [19].
Let us consider that the initial value of σ i at time t = 0 given by σ 0 i (x, ϑ i ). Further from Equation (23), we obtain σ 2 (x, t, ϑ 1 ) σ 0 2 (x, ϑ 1 ). Therefore, the system of Equations (22)(23)(24) reconstruct to the pair of equations for the wave field σ 2 and wave field σ 3 which connect through the linear integral operator containing the kernel which is given by Now if the initial data σ 0 i (x, ϑ) are periodic function of period 2π of the phase variable ϑ, the pair of resonant asymptotic equations is written as The kernel k appearing in Equations (27) and (28) is given by the Equation (26).

Nonlinear geometrical acoustics solution
The approximate asymptotic solution of the form (9) of the system (2) or (6) satisfy the small amplitude oscillating initial data given by which is nonresonant when the functions U 0 1 (x, x/ξ ) are smooth with a compact support [3]. In fact the expression (9), with U 1 as provided by Equation (14), is uniformly valid to the leading order if the shock waves are present in the solution.
Further, the characteristics equations are written as where e i is either +1 or −1 for i = 1 and i = 3, respectively. Thus in view of Equation (29), the decoupled Equations (27) and (28) may be recast as Now Equation (31) produces on integration along the rays, s i x − e i c 0 t constant where the function σ 0 i is derived from the initial data (29) and the fast variable τ i parameterizes the set of characteristics curves which are given in Equation (30).
Therefore, one can derive from Equation (30), Here, we infer that the wave amplitude decays as x −m/2 which is similar to corresponding classical gasdynamic case Hunter and Keller [2]. In view of U 0 1 (x, x/t) and the initial data given by Equation (29), one can find the solution of the system of Equations (6) as

(34(c))
Now let us suppose that the fast variable τ i given by Equation (33), may be written at t 0, τ i x/ξ . Therefore, with the help of solution (34) at t = 0, i.e. the initial values of σ i (1 ≤ i ≤ 3) are written as Therefore, we have obtained the complete solution of the system of Equations (6) in view of (29) which is given by (34) and (35). Further if there is any multivalued overlap in the given solution then it has to be dealt by introducing shock waves into the solution. Now by using the Rankine-Hugoniot jump conditions, shock wave is introduced into the solution to prohibit the multivaluedness.

Shock waves
The shock location ϑ s i may be obtained from the following relation Hunter and Keller [2], which is the shock speed in the t − ϑ i plane, where σ (−) i and σ (+) i exhibit the value of the σ i just ahead and behind the shock wave serially. Furthermore, we have σ (−) i 0 in the undisturbed region for the shock front. Now by using the Equations (36) and (32) and leaving the superscripts on ϑ s i and σ (+) i in Equation (36), we obtain dϑ i dt Further, using the Equation (33) and Equation (37), we obtain the following relation between e i and t on the shock, Hence because of Equation (33), Equation (38) provides the following equation which negotiates the shock path parametrically.
Now we conclude that if σ 0 i (0) ≠ 0 then shock forms immediately right from the origin [11,15]. The approximation between τ i and t is provided as . Now it is indicating that the shock moves with velocity α 3 2 e i σ * i ( x s i ) −m/2 which is totally same as Equation (37). We also consider the case when only outgoing waves are generated from the source, that is compression, is followed by rarefaction implying as σ 0 i (τ i ) → 0 as τ i → τ * i which shows that in the surroundings of the wavelet τ i τ * i , the wavelets will never approach the shock and the shock is asymptotic to , and from Equation (32), the amplitude approximates as The effect of dust particles enters into the expression for amplitude through the parameter α 3 and c 0 . From the relation (40), we infer that the amplitude is inversely proportional to α 3 . Also, the amplitude σ i varies according to (α 3 ) −1/2 . Also an increase in α 3 causes to decrease the wave amplitudes. An increase in the dust particle density causes to decrease the wave amplitude which causes to enhance the wave decay process. Further, it is observed here that the decreasing (increasing) values of the mass fraction of dust particles causes to enhance (slow down) the amplitude of the shock waves, as a result the shock formation distance increases (decreases), i.e. the shock formation is delayed (early). Also, the amplitude has been computed to see the effect of dust particles which is presented in Table 1. It is clear from Table 1 that the presence of dust particles has same influence on the wave amplitude as reported qualitatively. Further, in the absence of dust particles, our results are in close agreement to the results as reported in [7].
It is clear from the Equation (40) that in the absence of dust particles, . We obtained that shock waves decay like t −1/2 for planar flow. In same manner shock waves decay like t −3/4 and t −1 (log t) −1/2 for cylindrically symmetric flow and spherically symmetric flow, respectively. Hence, in case of nonplanar flows, the shock formation distance increases in comparison to the corresponding planar flows. Also, the results obtained here are similar to the results as reported in [31].

Results and conclusions
The present study uses the multiple scales method to derive the small amplitude high-frequency asymptotic solution for the system of nonlinear partial differential equations characterizing one-dimensional compressible unsteady, planar and nonplanar flows in a dusty gas. The theory of weakly nonlinear geometrical acoustics is utilized to examine the resonant interaction of waves and to analyze the evolution of shock wave in a dusty gas flow. The transport equations for the wave amplitude along the rays for the dusty gas flow, comprising a system of inviscid Berger's equations with known kernel, has been derived. The qualitative analysis of nonlinear wave interaction process and self-interaction of nonlinear waves which exist in the system under study can be made by using the coefficients occurring in the transport equations. In our discussion the Euler equations reduce to a pair of asymptotically resonant equations for the fields of acoustic wave. The nonlinear interaction of the wave fields and selfinteractions which generate higher harmonics leading to the distortions of the wave profile and consequent shock formation has been discussed. A nonresonant multiwave mode matter has been discussed by Hunter and Keller [2]. Here, it is obtained that the wave fields do not get across with each other which is connected with the particle path way, however they interact with an acoustic wave field to yield resonant contribution regarding the other acoustic wave fields. The acoustic wave fields may or may not get across to each other but in either case their entire contribution toward the entropy field must be zero. We require a suitable value of ϕ, to assess the influence of dust particles, which enters into the solution through the parameter c 0 . Now any change in value of ϕ, affects the velocity, pressure and density of the high-frequency small amplitude type waves. It is clear from Equation (34) that the increasing values of ϕ, causes the density, velocity and pressure of the high-frequency small amplitude waves to increase. Also, we have discussed here about the presence of shock and their position in the dusty gas. It is noticed from Equation (33) that in a contracting piston motion having cylindrical symmetry, if the initial wave amplitude exceeds a critical value then the shock forms before the focus, but in case of spherical symmetry, a shock is always formed before the formation of focus, does not matter how small be the initial wave amplitude. The existence of shock wave in a dusty gas is also discussed here. Therefore, we conclude that for nonplanar flows the formation of shock wave delays in comparison to the similar cases of the planar flows. Also, the decreasing (increasing) values of the mass fraction of dust particles causes to increase (decrease) the amplitude of the shock waves, as a result the shock formation distance increases (decreases), i.e. the shock formation is delayed (early). Also, the results obtained here are similar to the results as reported in the study by Pai et al. [31].