Interaction of a Singular Surface with a Characteristic Shock in a Relaxing Gas with Dust Particles

1 Introduction

Waves are a natural outcome of technical progression, so are the discontinuities. As the environment is inundated with waves, the study of the interaction of a wave with discontinuities is likely to have a positive impact on the propagation of waves. The topic carries a high empirical utility that makes it an interesting area to explore further. A wave can be viewed as a surface which moves in accordance with some flow variables describing the material medium. Some interrelated discontinuities are also carried along by the surface, which arise out of the flow variables or their derivatives.

Singular surface theory of the first order uses the approach that the functions itself are continuous and undergo jump discontinuity in at least one of its first order derivatives. In a recent paper, this theory was used to study wave propagation and its termination into discontinuities in three different contexts of flow of a non-ideal dusty gas, and further among other results, the effects of the ratio of species densities and the ratio of specific heat on shock formation were shown [1]. The nonlinear theory of wave propagation was discussed by Varley and Cumberbatch [2] and the propagation of weak discontinuities in different materials using singular surface theory has been studied by many authors from an applicative point of view [3], [4], [5]. The studies of concern from the pre-computer era to date, have been the interaction of these weak discontinuities with shock wave using the general theory of wave interaction. The work done by Jeffrey [6], [7] on the problem of interaction has been further extended by many authors. Radha et al. [8] studied the general theory for problems of wave interactions leading to the results obtained by Boillat and Ruggeri [9]. Boillat and Ruggeri [10] studied the characteristic shock and constructed examples to analyze the same for completely and strictly exceptional systems. For a clear understanding one may refer the fundamental works carried out in [9], [10], [11], [12], [13]. Mentrelli et al. [14], taking into consideration a perfect gas, studied the interaction between a shock and an acceleration wave with varying shock strength. Many more results have been concluded using the theory in different material media [15], [16], [17], [18], [19], [20], [21], [22]. In dusty gases, self similar solutions and converging shocks followed by wave propagation were discussed in [23], [24], [25], [26]. Recently, Shah and Singh studied the collision between a blast wave and a steepened wave in dusty real reacting gases [27], and the interaction of a singular surface with a strong shock in reacting polytropic gases [28].

The relaxation mechanism in gas is basically the rate of process of attaining equilibrium through non-equilibrium which occurs due to some external forces and the time taken to get the equilibrium state back is known as the relaxation time [29], [30], [31]. In general, the energy in a gas molecule is distributed among translational modes, vibrational modes, and rotational modes. The time taken by a particular mode to attain equilibrium is called the relaxation time of that mode. The relaxation times for translational and rotational modes are very short, whereas the same for vibrational modes is much longer [32]. Hence, the propagation of waves in a vibrationally relaxing gas depends on the relaxation time.

In this paper, we discuss the interaction of a singular surface with a characteristic shock in a relaxing gas with dust particles. It is imperative to highlight some of the assumptions being made to formulate the system of the fluid flow [33], [34], [35], [36]. It is considered that the solid particles are spherical in nature with uniform size and the specific heat of the gas and the dust particles are constant. We have further assumed that the volume occupied by the solid particles in the mixture is negligible. Using the singular surface approach, the transport equation for the jump in first order derivative of the velocity, which is of a Bernoulli-type equation is obtained. The solution of the transport equation is discussed considering a particular case. The evolution of a characteristic shock in the medium is discussed considering the same particular case as in the case of the singular surface. Next, the interaction between the singular surface with the characteristic shock resulting in reflected and transmitted waves and the jump in the shock wave acceleration is studied. The results obtained are analyzed for various dust parameters along with the relaxation effects involved in the flow by performing numerical calculations and depicting the same.

2 Basic Equations

The system of partial differential equations representing the one-dimensional, unsteady flow of a dusty relaxing gas with planar (m = 0) and non-planar, i.e. cylindrical symmetry (m = 1) or spherical symmetry (m = 2) with the assumptions made for dust particles in [36], [37] is given by:

along with the equation of the state

where t is the time, u is the flow velocity along thex-axis, ρ and p are the density and pressure of the gas–particle mixture, σ is the vibrational energy, Q=1τ{σe+c(p(1−Z)ρ−pe(1−Ze)ρe)−σ}, Γ=γ(1+δβ)1+δβγ, a=(Γpρ(1−Z))1/2, R the specific gas constant, T the temperature, and Z=Vsp/V is the volume fraction. If otherwise not stated a variable as a subscript indicates a partial derivative with respect to that variable. However, the variable with ‘e’ as a subscript denotes the initial value of the variable in the equilibrium state.

Here, V_sp and V are the volumetric extension of the solid particles and total volume of the mixture, respectively; the parameters c and τ specifies the ratio of vibrational specific heat to the specific gas constant and the relaxation time, kp=φsp/φ is the mass fraction of the solid particles in the gas mixture, δ=kp/(1−kp), β=csp/cp, γ=cp/cv, θ=kp/ρsp where Z in terms of k_p can be written as Z=θρ, entailing φsp and φ as the total mass of the solid particles and the mixture, respectively, c_sp as the specific heat of solid particles, c_p the specific heat of the gas at constant pressure, and c_v the specific heat of the gas at constant volume. Also, we make use of an added variable G=ρsp/ρg, the ratio of the density of the solid particles to the species density of the gas, to compare the behavior of the flow variables as the time proceeds.

It may be noted that the equations of states for the gas and solid particles are given by pg=ρgRT and ρsp= constant, respectively, where p_g is the pressure of the gas. As the presence of dust particles have negligible contributions to the pressure of the mixture due to the larger particle size, it is assumed that p_g = p [34]. The densities of the gas, solid particles and the mixture are related to each other as per the following:

If the flow is vibrationless or translationally active, i.e. either in equilibrium (σ=σ¯) presupposing σ¯=σe+c(p(1−Z)ρ−pe(1−Ze)ρe) as the equilibrium value of σ, or frozen (σ=const) as τ → ∞, no relaxation is involved thereby implying Q = 0.

The system of (1) can also be written in the matrix form as:

where U=(ρ,u,p,σ)tr, f=(−mρux,0,−((Γ−1)(1−θρ)ρQ+mρa2ux),Q)tr and

in which tr represents transposition.

3 Transport Equation for the Jump Discontinuity

Let us consider flow variables ρ, u, p, and σ to be necessarily continuous while allowing discontinuities in their derivatives across the surface associated with a propagating wave and denoted by x=ε(t) the equation of the wave front Φ. For Φ to be a characteristic surface, the moving discontinuities of the first and second order must satisfy the geometrical and kinematical compatibility conditions. The compatibility conditions were derived by Hadamad [38] using the Lagrangian parameters rather than the space coordinates as Eulerian variables. He also considered an extension of the relations using Lagrangian parameters giving the discontinuities in the second and higher derivatives of quantities under the assumption that only the quantities themselves are continuous over Φ. The conditions were later presented by Thomas using Eulerian variables [39], [40], which are as follows:

where H takes the value of any of the flow variables, B is a function defined on the surface Φ, provided B ≠ 0 implying the existence of discontinuity and V=dεdt is the speed with which Φ propagates. If W is any variable then |[W]| = W − W₀ means the jump in W across the wave front Φ and W₀ and W represents the values of W just ahead and just behind Φ.

As for an advancing wave V is positive, the evaluating system (1) on the inner boundary of Φ gives

on using the following jump values

Also,

Differentiating system (1) with respect to x and evaluating behind the wave front Φ, we get:

Use of (8) in (9) yields

Further, eliminating ρ¯, α(1)¯ and ξ¯ from (10) we obtain the following Bernoulli type of transport equation [6] for α(1)

where

3.1 A Particular Case

Using the following form of flow variables just ahead of the wave front Φ

(12)u0(x,t)=ℓ(t)x,ρ0=ρ0(t),p0=p0(t),σ0=σ0(t),

we integrate the governing system of (1) and hence obtain

(13)ρ0(t)=ρ0e(1+(t−t0e)ℓ0e)−(m+1),ℓ(t)=ℓ0e(1+(t−t0e)ℓ0e)−1,

and

(14)dp0dt+((m+1)Γℓ1−θρ0+(Γ−1)cτ)p0−(Γ−1)ρ0τ(1−θρ0)((σ0−σe)+cpe(1−θρe)ρe)=0,dσ0dt+(σ0−σe)τ−cτ(p0(1−θρ0)ρ0−pe(1−θρe)ρe)=0,

where ρ0e and ℓ0e are the reference values of density and velocity at t=t0e. Using the variables in the dimensionless form

x*=xt0ea0e,t*=tt0e,ℓ*=ℓt0e,ρ0*=ρ0ρ0e,p0*=p0ρ0ea0e2,σ0*=σ0a0e2,a0*=a0a0e,θ*=ρ0eθ,(α(1))*=t0eα(1),τ*=τt0e,ρe*=ρeρ0e,pe*=peρ0ea0e2,σe*=σea0e2,

and then suppressing the asterisk sign, the transport (11) reduces to

(15)2dα(1)dt+(Γ+11−θρ0)(α(1))2+Ωα(1)=0,

where

Ω=ℓ(3+Γ1−θρ0+Γθp0(1−θρ0)2a02+mΓ1−θρ0)+ma0x+mℓΓθp0(1−θρ0)2a02+(Γ−1)(1−θρ0)(Q0a02(1−θρ0)+(Γ−1)(1−θρ0)cτΓ).

The value of x appearing in the coefficient Ω in (15) is computed using the relation dxdt=u0+a0 from (6). Integrating (15), we obtain the following solution

(16)α(1)(t)=α0(1)exp(−12∫t0tΩ(w)dw){1+α0(1)∫t0t(Γ+12(1−θρ0))η(w)dw}−1,

with η(t)=exp(−12∫t0tΩ(w)dw).

Also, numerical integration of (15) using (13) and (14) is performed for 1⩽t⩽∞ and the results showing the behavior of jump in the velocity gradient are analyzed by plotting α(1) against 1/t. The analysis yields the following conclusions:

1. In the case of rarefaction wave, i.e α0(1)>0, (16) signifies that η(t)→0 along with I(t)=∫t0t(Γ+12(1−θρ0))η(w)dw<∞ resulting in α(1)(t)→0 as t approaches infinity for all the possible range of values of parameters involved in the flow such as k_p, β, G, c, and τ and hence concluding that with an increase in the time the wave flattens and eventually dies out. The behavior is shown graphically in Figures 1–4 for variations with the parameters k_p, β, G, and τ. It is observed that an increase in any of the parameters k_p, β, or G leads to an increase in α(1)/α0(1). As from the expression for Q in (1), an increase in the relaxation parameter c is equivalent to a decrease in the parameter τ, we have not added any figures showing variations with c. The presence of dust particles in the system drives the velocity gradient to increase thereby taking more time to converge to zero. However, an opposite trend is noticed with an increase in the parameter τ.

2. For a compression wave (α0(1)<0) the solution given by (16) breaks down when 1+α0(1)I(t) equals zero at some time t = t_c, implying the existence of steepened wave in a finite time only.

   

   Figure 1:

   Variation of α(1)/α0(1) with 1/t influenced by the parameter k_p where m = 1, c = 0.01, G = 1000, β = 0.5, τ = 10.

   

   Figure 2:

   Variation of α(1)/α0(1) with 1/t influenced by the parameter β where m = 1, c = 0.01, G = 1000, k_p = 0.6, τ = 10.

   

   Figure 3:

   Variation of α(1)/α0(1) with 1/t influenced by the parameter G where m = 1, c = 0.01, k_p = 0.6, β = 0.5, τ = 10.

   

   Figure 4:

   Variation of α(1)/α0(1) with 1/t influenced by the parameter τ where m = 1, G = 1000, β = 0.5, c = 0.01, k_p = 0.6.

   We study in detail the following cases arising from (16):

   1. The condition ‘|α0(1)|/αc(1)<1’, where αc(1)=1/I{∞} leads to α(1)→0 pointing towards the existence of flattened wave as time advances.

   2. For ‘|α0(1)|/αc(1)>1’, ∃ a finite time t_c such that I(tc)=1/|α0(1)| and |α(1)|→∞ as t→tc. This yields the existence of a shock wave in a finite time only at an instant t_c.

   3. For ‘|α0(1)|/αc(1)=1’, the wave behaves in the same way as in (1).

      These discussed outcomes for compression waves are illustrated in Figures 5–12. It is again observed that an increase in any of the parameters k_p, β or G leads an increase in α(1)/α0(1) for the interval −αc(1)⩽α0(1)<0, and a decrease in α(1)/α0(1) for the interval α0(1)⩽−αc(1)<0; however, an increase in the variable τ leads to an opposite trend.

3. Also the behavior of α(1)/α0(1) is observed to be decreasing and approaching to zero as time increases eventually to infinity.

Figure 5:

Variation of α(1)/α0(1) with t influenced by the parameter k_p in the case of α0(1)>0 where m = 1, τ = 10, c = 0.01, β = 0.5, G = 1000.

Figure 6:

Variation of α(1)/α0(1) with t influenced by the parameter k_p in the case of α0(1)<0 where m = 1, τ = 10, c = 0.01, β = 0.5, G = 1000.

Figure 7:

Variation of α(1)/α0(1) with t influenced by the parameter β, in the case of α0(1)>0 where m = 1, c = 0.01, k_p = 0.6, τ = 10, G = 1000.

Figure 8:

Variation of α(1)/α0(1) with t influenced by the parameter β in the case of α0(1)<0 where m = 1, c = 0.01, k_p = 0.6, τ = 10, G = 1000.

Figure 9:

Variation of α(1)/α0(1) with t influenced by the parameter G in the case of α0(1)>0 where m = 1, k_p = 0.6, c = 0.01, β = 0.5, τ = 10.

Figure 10:

Variation of α(1)/α0(1) with t influenced by the parameter G in the case of α0(1)<0 where m = 1, k_p = 0.6, c = 0.01, β = 0.5, τ = 10.

Figure 11:

Variation of α(1)/α0(1) with t influenced by the parameter τ, in the case of α0(1)>0 when m = 1, c = 0.01, k_p = 0.6, β = 0.5, G = 1000.

Figure 12:

Variation of α(1)/α0(1) with t influenced by the parameter τ, in the case of α0(1)<0 when m = 1, c = 0.01, k_p = 0.6, β = 0.5, G = 1000.

4 Characteristic Shock

Characteristic shock arises when the characteristic surface and the shock surface both occur simultaneously and the shock surface velocity, both behind and ahead of the shock, coincides with a latent root of the system [11], [41]. Characteristic shock exists when the corresponding eigenvalue is linearly degenerate (also referred to as exceptional). Mathematically it states:

where ∇ represents the gradient operator, R(i,j) the right eigenvectors with respect to the eigenvalue λ(i) having multiplicity m_i with j = 1, …, m_i. In particular, for the conservative hyperbolic system of equations, a wave with eigenvalue having multiplicity m > 1 is always exceptional (linearly degenerate) [9], [13]. The eigenvalues of the matrix A in (4) are:

As the multiplicity of λ(2)=u is 2, this implies the existence of characteristic shock with speed ϑ=u. Further, the eigenvectors of A are given by:

Considering X as the value just behind the characteristic shock and X_∗ ahead of the characteristic shock, let [X]=X−X∗ is the jump in X along the characteristic shock. Here, the Rankine-Hugoniot jump conditions are [u] = 0, [p] = 0, [ρ]=ζ, and [σ]=ω where ζ and ω are unknown functions of t to be determined.

On forming the jump across the characteristic shock in usual manner one obtains:

where d/dt=∂/∂t+u∂/∂x is the material derivative operator.

Using (19), the transport equations for ζ and ω are obtained as follows

Using (12) in (20) and applying the following dimensionless variables in the resulting equation:

and then suppressing the hat sign, we obtain the following equations

The simultaneous differential (21) are solved using (13) and (14) together with the initial conditions at time t = 1 to study the evolutionary behavior of a characteristic shock. The results are depicted in Figures 13–20 for cylindrical (m = 1) flows. The profiles for other geometries, i.e. planar (m = 0) and spherical (m = 2) flows are found to be similar to the cylindrical case and hence are not plotted. The calculations are performed by taking γ = 1.4, σ₀ = 0.005, ζ0=0.01, ω0=0.01. The value of ζ and ω across the characteristic shock line tends to zero as time proceeds. As observed from Figures 13–16, with rise in the values of k_p, β, or τ leads to a rise in ζ, whereas a rise in G leads to a fall in ζ. From the Figures 17–20, it is observed that a rise in the values of k_p, β, or G, gives a fall to the parameter ω and a rise in τ gives a rise to ω.

Figure 13:

Variation of ζ vs. 1/t influenced by the parameter k_p where m = 1, c = 0.01, β = 0.5, τ = 10, G = 1000.

Figure 14:

Variation of ζ vs. 1/t influenced by the parameter β where m = 1, G = 1000, c = 0.01, k_p = 0.6, τ = 10.

Figure 15:

Variation of ζ with 1/t influenced by the parameter G where m = 1, β = 0.5, c = 0.01, k_p = 0.6, τ = 10.

Figure 16:

Variation of ζ with 1/t influenced by the parameter τ where m = 1, c = 0.01, G = 1000, k_p = 0.6, β = 0.5.

Figure 17:

Variation of ω vs. 1/t influenced by the parameter k_p where m = 1, G = 1000, c = 0.01, β = 0.5, τ = 10.

Figure 18:

Variation of ω vs. 1/t influenced by the parameter β where m = 1, k_p = 0.6, c = 0.01, τ = 10, G = 1000.

Figure 19:

Variation of ω vs. 1/t influenced by the parameter G where τ = 10, m = 1, c = 0.01, k_p = 0.6, β = 0.5.

Figure 20:

Variation of ω vs. 1/t influenced by the parameter τ where k_p = 0.6, β = 0.5, G = 1000, c = 0.01, m = 1.

5 Interaction Between Singular Surface and Characteristics Shock

With disturbances prevailing in every system, it is very obvious to study the impact of those disturbances or shock on the waves traveling in different material media when they interact with each other. In order to investigate the effect of the interaction between the singular surface traveling along the fastest characteristic with the characteristics shock, we adopted the method discussed in [16] for the interaction between shock waves and weak discontinuities. We consider the system of (1) in the following generalized conservation form behind and ahead of the shock:

where U and U_∗ represent the solution vectors behind and ahead of the characteristic shock, respectively, and G, F and K are given by the following column vectors

Let us consider the two adjacent intervals I_u and Iu* in accordance with the initial line, i.e. the characteristic shock:

where d₁ and d₂ are the constants chosen assuring the propagation of the wave surface in the defined domain of determinacy R. The solution vectors U and U_∗ are continuous in the region behind and ahead of the characteristic shock denoted by R_u and Ru*, respectively, with the initial conditions U(x,t0)=ϕ(x) and U*(x,t0)=ϕ*(x), but, on the other hand, discontinuous across the characteristic shock or the initial line that divides R. Let M(xm,tm) be the point where the fastest wave, i.e. the singular surface originating from (x0,t0) and moving along the characteristic dxdt=λ(1) denoted as ϕ(x,t)=0 interacts with the characteristic shock propagating with speed ϑ=u. After the interaction, the new wavefront from the point M moves along the characteristic dxdt=λ*(1) and is represented as ϕ*(x,t)=0. The jumps in U_x across the incident, reflected and transmitted waves at M, are given by:

with subscript s denoting the values calculated at the point M. The evolutionary equations determining the jump in the shock acceleration |[V˙]| and the amplitudes αk(i) and βk(i) of reflected and transmitted waves are given by the following system of algebraic equations [9]:

Lax’s [42] evolutionary conditions formulated as