Interaction of an acceleration wave with a characteristic shock in a non-ideal relaxing gas

International Journal of Non-Linear Mechanics 82 (2016) 17–23

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Interaction of an acceleration wave with a characteristic shock in a non-ideal relaxing gas Monica Saxena, J. Jena n Department of Mathematics, Netaji Subhas Institute of Technology, Sector -3, Dwarka, New Delhi 110078, India

art ic l e i nf o

a b s t r a c t

Article history: Received 27 December 2015 Received in revised form 24 February 2016 Accepted 24 February 2016 Available online 3 March 2016

The amplitude of an acceleration wave propagating along the characteristic associated with the largest eigenvalue in a non-ideal relaxing gas is evaluated. The evolution of a characteristic shock and its interaction with the acceleration wave is studied. The amplitudes of the reflected and transmitted waves and the jump in the shock wave acceleration after interaction are computed. The effects of relaxation and non-ideality on the amplitude of acceleration wave are discussed. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Acceleration wave Characteristic shock Non-linear wave interaction

1. Introduction The term acceleration wave means an isolated geometric surface that moves relative to the material and across which the acceleration (but not the velocity) is discontinuous [1]. In Continuum Mechanics, the acceleration waves are also known as weak discontinuity waves (i.e. C ð1Þ waves) and are important kind of solutions of non-linear hyperbolic systems. These waves are characterized by a discontinuity in a normal derivative of the field but not in the field itself [2]. The evolution of a weak discontinuity or acceleration wave for a hyperbolic quasi-linear system of equations satisfying Bernoulli's law is described extensively in the literatures [3–5]. For a characteristic shock, the shock surface coincides with a characteristic surface and its velocity with an eigenvalue of the system, both ahead and behind the shock. The corresponding eigenvalue may be single or have multiplicity j 41. In the case where the eigenvalue is single, the shock is characteristic if and only if the exceptionality condition ∇λ  R ¼ 0 is satisfied, where λ is the eigenvalue (and also the velocity of the characteristic shock), R is the corresponding right eigenvector of the system and ∇ is the gradient with respect to the field vector [6,7]. The shock corresponding to the multiple eigenvalue that has multiplicity j 41 is always exceptional [8–10]. The works of Jeffrey [11] and Boillat and Ruggeri [12] are the origin of the general theory of wave interactions. The shock n

Corresponding author. E-mail addresses: [email protected] (M. Saxena), [email protected] (J. Jena). http://dx.doi.org/10.1016/j.ijnonlinmec.2016.02.007 0020-7462/& 2016 Elsevier Ltd. All rights reserved.

undergoes an acceleration jump as a consequence of an interaction with a weak wave [12–14]. Radha et al. [15] verified that the general theory of wave interaction problem which originated from the work of Jeffrey [11] leads to the results obtained by Brun [14] and Boillat and Ruggeri [12]. The theory has been successfully applied to study the interaction of discontinuity wave with a characteristic shock or a strong shock in the mediums like shallow water, relaxing gas, dusty gas, transient pinched plasma and nonideal gas [16–23]. At high temperatures, the internal energy of the gas molecules consists of translational, rotational and vibrational components. When a gas is compressed by a receding piston or by the passage of a shock front, the whole energy goes initially to increase the translational energy, and it is followed by a relaxation from translational mode to rotational mode and also from translational mode to vibrational mode until the equilibrium between these modes is re-established [24]. The process is called relaxation and the departure from equilibrium is due to vibrational relaxation; the rotational and translational modes are assumed to be in local thermodynamical equilibrium throughout. Arora et al. [25] used a similarity method to study imploding strong shocks in a non-ideal relaxing gas with van der Waals equation of state. One dimensional steepening of waves in non-ideal relaxing gas is studied in [26] and it is observed that the transport equations for the discontinuities in the first order derivatives of the flow variables lead to Bernoulli type of equations. In this paper, we considered a system of partial differential equations describing the one dimensional unsteady plane and radially symmetric flow of an inviscid vibrationally relaxing gas with van der Waals equation of state. The evolution of a characteristic shock is studied and the amplitude of the acceleration

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wave propagating along the characteristic associated with the largest eigenvalue is evaluated. The interaction of the acceleration wave with the characteristic shock is considered and the jump in the shock acceleration and the amplitudes of reflected and transmitted waves after interaction are evaluated by using the results of general theory of wave interaction [15].

2. Basic equations We consider a system of partial differential equations describing the one dimensional unsteady plane (m ¼0), cylindrically symmetric (m ¼1) and spherically symmetric (m ¼2) flow of an inviscid vibrationally relaxing gas with van der Waals equation of state as [25] U t þAU x ¼ f ;

where U ¼ ðρ; u; p; σ Þ , f ¼ ð  mρu=x; 0;  γ pmu=ðð1  bρÞxÞ  ðγ  1Þ ρQ ; Q Þtr and 0 1 u ρ 0 0 B0 u 1=ρ 0 C B C A¼B C: @ 0 γ p=ð1  bρÞ u 0A 0

0

ðσ ðp; ρÞ  σ Þ

τ

;

where σ ¼ σ e þ cfpð1  bρÞ=ρ  ð1  bρe Þpe =ρe g is the equilibrium value of σ and the suffix e refers to an initial equilibrium reference state; the quantities τ and c are the relaxation time and the ratio of vibrational specific heat to the specific gas constant, respectively. If not stated otherwise, a variable as a subscript indicates partial differentiation with respect to that variable. The van der Waals equation of state is of the form p¼

ð4Þ

d ∂ ¼ ∂t∂ þ u∂x denotes the material derivative following the where dt shock. Now, using (1) and (3) in (4) we obtain the following transport equations for the quantities ζ and η    dζ mu 1  bð2ρ  ζ Þ ðγ  1Þðρ  ζ Þð1  bðρ  ζ ÞÞ ¼  ζ ux þ þ x 1  bρ dt γ pτ     cp0 ð1  bρ0 Þ  ζ σ0   σ þ η  cpb  ρη ;

ρ0

  dη 1 cpζ ð1  2bρÞ ¼  ηþ : dt τ ρðρ  ζ Þ

ð5Þ

2.1. Particular case Let us consider the case exhibiting the space–time dependence when the flow variables u; ρ; p; σ behind the characteristic shock are given by uðx; tÞ ¼ kðtÞx;

u

Here, x is the distance, t the time, ρ the density, u the particle velocity, p the pressure, σ the vibrational energy, γ the ratio of specific heats, b the van der Waals excluded volume and lies in the range 0:9  10  3 r b r1:1  10  3 (SI unit of b is m3 =kg) [27,28]. It may ne noted that b¼0 corresponds to the case of ideal relaxing gas [24]. The quantity Q is the rate of change of vibrational energy, and is a function of p, ρ and σ, given by Q¼

d½U dU n þ ½L ¼ L½f  þ½Lf n ; L dt dt

ð1Þ tr

0

Lð2;1Þ and Lð2;2Þ ; respectively, and then on forming the jumps across the characteristic shock, we get the evolutionary law for ζ and η

ρ ¼ ρðtÞ;

p ¼ pðtÞ;

σ ¼ σ ðtÞ:

ð6Þ

In this case the particle velocity exhibits linear dependence on x and such a state can be visualized in terms of an atmosphere filled with a gas which has spatially uniform pressure variation on account of the particle motion and the spatially uniform relaxation rate [29–31]. This type of velocity distribution is useful in modelling the free expansion of polytropic gasses [29]. Using (6) in equations ð1Þ1 and ð1Þ2 we get the following forms of flow parameters:

ρ ¼ ρ0 ð1 þ ðt  t 0 Þk0 Þ  ðm þ 1Þ ;

kðtÞ ¼ k0 =ð1 þ ðt  t 0 Þk0 Þ;

ð7Þ

whereas equations ð1Þ3 and ð1Þ4 lead to the following system of ordinary differential equations in p and σ     dp γ ðm þ 1Þk ðγ  1Þcð1  bρÞ ðγ  1Þρ cp ð1  bρ0 Þ σ0  σ  0 þ þ pþ ¼ 0; τ τ ρ0 dt 1  bρ   dσ cpð1  bρÞ 1 cp ð1  bρ0 Þ  þ σ  σ0 þ 0 ð8Þ ¼ 0; dt ρτ τ ρ0

where k0 and ρ0 corresponds to the initial reference state. Using the dimensionless variables

ρRT ; ð1  bρÞ

where R is the specific gas constant and T is the translational temperature. The matrix A in Eq. (1) has eigenvalues

ζ~ ¼ ζ =ρ0 ;

λð1Þ ¼ u þ a;

and then suppressing the tilde sign, we can write Eq. (5) in the following form:   dζ 1  bð2ρ  ζ Þ ¼  ðm þ 1Þkζ þ ðγ  1Þðρ  ζ Þð1  bðρ  ζ ÞÞ 1  bρ dt

λð2;1Þ ¼ λð2;2Þ ¼ u ðλð2Þ ¼ u is a double rootÞ;

λð3Þ ¼ u  a; ð2Þ

with the corresponding left and right eigenvectors Lð1Þ ¼ ð0; ρa; 1; 0Þ;

Rð1Þ ¼ ð1=ð2a2 Þ; 1=ð2ρaÞ; 1=2; 0Þtr ;

Lð2;1Þ ¼ ð  a2 ; 0; 1; 0Þ; Lð2;2Þ ¼ ð0; 0; 0; 1Þ;



Rð2;1Þ ¼ ð  1=a2 ; 0; 0; 0Þtr ;

Rð2;2Þ ¼ ð0; 0; 0; 1Þtr ;

¼ ð0;  ρa; 1; 0Þ; R ¼ ð1=ð2a Þ;  1=ð2ρaÞ; 1=2; 0Þ ; ð3Þ  12 where a ¼ ρð1γpbρÞ is the frozen speed of sound. Since, the ð3Þ

L

ð3Þ

2

ð2Þ

t~ ¼ t=t 0 ;

tr

multiplicity of the eigenvalue λ ¼ u is 2, there exists a characteristic shock propagating with the speed V¼ u. Using the fact that across a characteristic shock, no mass flow takes place, the Rankine–Hugoniot conditions across this shock are given by ½u ¼ 0, ½p ¼ 0, ½ρ ¼ ζ , ½σ  ¼ η where ζ and η are functions of t. Here, ½X ¼ X  X n denotes the jump in X across the characteristic shock where X n and X are the values just ahead of the shock and behind the shock, respectively. Multiplying (1) by eigenvectors

ρ~ ¼ ρ=ρ0 ; p~ ¼ p=p0 ; η~ ¼ ηρ0 =p0 ; σ~ ¼ σρ0 =p0 ; τ~ ¼ τ=t 0 ; k~ ¼ kt 0 ; k~0 ¼ k0 t 0 ; b~ ¼ bρ0 ;

ðζ ðσ 0  cð1  bÞ  σ þ η  cpbÞ  ρηÞ ; ðγ pτÞ

ð9Þ

ð10Þ

  dη 1 cpζ ð1  2bρÞ ¼  ηþ : dt τ ρðρ  ζ Þ Also, Eqs. (8) in dimensionless form are given by   dp γ ðm þ 1Þk ðγ 1Þcð1  bρÞ ðγ  1Þρðσ 0 cð1  bÞ  σ Þ þ þ ¼ 0; pþ dt 1  bρ τ τ dσ cpð1  bρÞ 1  þ ðσ  σ 0 þ cð1  bÞÞ ¼ 0; dt ρτ τ

ð11Þ

where ρ ¼ ðk0 ðt  1Þ þ 1Þ  ðm þ 1Þ ; k ¼ k0 =ðk0 ðt  1Þ þ 1Þ, with p ¼ p0 , σ ¼ σ 0 , ζ ¼ ζ 0 and η ¼ η0 as the initial conditions as at t¼1.

M. Saxena, J. Jena / International Journal of Non-Linear Mechanics 82 (2016) 17–23

19

Fig. 2. Variation of ζ with τ; c ¼ 0:1; m ¼ 1; b ¼ 0:0009.

Fig. 1. Variation of ζ with c; τ ¼ 5; m ¼ 1; b ¼ 0:0009.

Eqs. (10) along with the Eqs. (11) are solved numerically to study the behavior of the jumps in the flow parameters ζ and η. The results are depicted in Figs. 1–8 for different values of flow geometry m, relaxation parameters c and τ, and non-ideal parameter b. It is observed that both ζ and η decrease and tend to zero as t-1. It is also observed that the parameter ζ increases with an increase in τ or b and decreases with an increase in c or m. The parameter η increases with an increase in any of the parameters τ, b or m and decreases with an increase in c.

3. The acceleration wave The transport equation for the acceleration wave across the ith characteristic of a hyperbolic system of n equations of the type (1) is given by (see [32]) Lði;kÞ b

  tr dU d Λi ðiÞ b þ Lði;kÞ ðU bx þ Λi Þð∇λ Þb Λi þ ∇Lði;kÞ Λi b b dt dt ðiÞ

ðiÞ

Λi Þðð∇λ Þb U bx þ λbx Þ ðð∇ðLði;kÞ f Þb ÞΛi Þ ¼ 0; þ ðLði;kÞ b

ð12Þ Fig. 3. Variation of ζ with m; c ¼ 0:1; τ ¼ 5; b ¼ 0:0009.

ðiÞ

where λ ; i ¼ 1; 2; 3; …; p are distinct eigenvalues of the coefficient ðpÞ ðp  1Þ ð1Þ o ⋯ o λ with matrix A, assumed to be ordered as λ o λ Pp ði;kÞ and constant multiplicities mi such that i ¼ 1 mi ¼ n. Here, L Rði;kÞ ; k ¼ 1; 2; …; mi , denote the left and the right eigenvectors of A corresponding to the eigenvalue λi, and the subscript b refers to the state behind the characteristic shock and

Λi ¼

mi X k¼1

α

ðiÞ ðtÞRði;kÞ ; k b

ð13Þ

being is the jump in Ux across the ith characteristic curve with αðiÞ k ðiÞ

the amplitude of the wave propagating along dx ¼ λ . We consider dt the acceleration wave with amplitude αð1Þ emerges from the point ðx0 ; t 0 Þ behind the characteristic shock that originates from the point ðx1 ; t 0 Þ where x1 4 x0 . Using (1), (2), (3) and (13) in (12), we obtain the transport equation for the incident wave amplitude in terms of the dimensionless variables: dαð1Þ γ þ 1 ð1Þ 2 þ ðα Þ þ ΘðtÞαð1Þ ¼ 0; 4 ρa dt where

Θ¼

γ þ 9  8bρ 5γ þ1 að1 2bρÞ ux þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipx  ρ 4ð1  bρÞ ρð1  bρÞ x 4 γ pρð1  bρÞ

ð14Þ

ρumb γ 1 þ ðσ 0  σ  cpb  cð1  bρ0 ÞÞ 2xð1  bρÞ 2a2 τ ma γ mu 1 cðγ  1Þð1  bρÞ þ þ : þ 2x 2xð1  bρÞ 2 τ

þ

The transport equation (14) is the Bernoulli type equation and it is observed in [26] that the discontinuities in the first order derivatives of the flow variables lead to such type of equations. Integration of equation (14) with respect to t leads to

αð1Þ ¼

αð1Þ 0 ΦðtÞ ; 1 þ αð1Þ 0 Ψ ðtÞ

where Ψ ðtÞ ¼

α0ð1Þ ¼ αð1Þ ðt 0 Þ.

Rt

ð15Þ

ðγ þ 1Þ t 0 ð1  bρ0 Þ



ΦðwÞ dw, ΦðtÞ ¼ exp 

Rt t0

ΘðyÞ dy



and

It follows from (15) that for α0ð1Þ 4 0 (i.e. an expansion wave), α ðtÞ-0 as t-1, which imply that the wave decays and dies out eventually. However, if αð1Þ 0 o 0 (i.e. a compression wave), above solution breaks down at some critical time t ¼ t c where 1 þ αð1Þ 0 Ψ ðtÞ ¼ 0. This indicates the appearance of a discontinuous wave at an instant t ¼ t c , i.e. a compression wave turns out to a breaking wave in a finite time only when the initial discontinuity associated with the wave exceeds a critical value. ð1Þ

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M. Saxena, J. Jena / International Journal of Non-Linear Mechanics 82 (2016) 17–23

Fig. 4. Variation of ζ with b; c ¼ 0:1; τ ¼ 5; m ¼ 1.

Fig. 7. Variation of η with m; c ¼ 0:1; τ ¼ 5; b ¼ 0:0009.

Fig. 5. Variation of η with c; τ ¼ 5; m ¼ 1; b ¼ 0:0009.

Fig. 8. Variation of η with b; c ¼ 0:1; τ ¼ 5; m ¼ 1.

Fig. 6. Variation of η with τ; c ¼ 0:1; m ¼ 1; b ¼ 0:0009.

Fig. 9. Variation of αð1Þ with c; τ ¼ 5; m ¼ 1; b ¼ 0:0009, αð1Þ 0 ¼ 0:001.

M. Saxena, J. Jena / International Journal of Non-Linear Mechanics 82 (2016) 17–23

21

Fig. 12. Variation of αð1Þ with b; c ¼ 0:1; τ ¼ 5; m ¼ 1, αð1Þ 0 ¼ 0:001.

Fig. 10. Variation of αð1Þ with τ; c ¼ 0:1; m ¼ 1; b ¼ 0:0009, αð1Þ 0 ¼ 0:001.

Fig. 11. Variation of αð1Þ with m; c ¼ 0:1; τ ¼ 5; b ¼ 0:0009, αð1Þ 0 ¼ 0:001. Fig. 13. Variation of αð1Þ with c; b ¼ 0:0009; τ ¼ 5; m ¼ 1, αð1Þ 0 ¼  0:001.

Eq. (14) is solved numerically to study the behavior of the amplitude of the acceleration wave αð1Þ and the results are depicted in Figs. 9–16 for different values of flow geometry m (planar ðm ¼ 0Þ and radially symmetric ðm ¼ 1; 2Þ), relaxation parameters c and τ, and non-ideal parameter b. It is observed that as t increases, αð1Þ decreases and tend to zero as t-1. For αð1Þ 0 4 0, the amplitude of the acceleration wave αð1Þ decreases with increase in any of the variables c, m or b and increases with an increase in τ. However, for αð1Þ 0 o0, an opposite trend is observed i.e. the amplitude of the acceleration wave αð1Þ increases with increase in any of the variables c, m or b and decreases with an increase in τ.

 H¼



mρu mρu2 ρmu3 ðγ bρÞpmu ρmuσ ;  ρQ ð1  bρÞ   ; ρQ  ; x 2x x x ðγ  1Þx

:

In Section 2, we have considered U ¼ ðρ; u; p; σ Þtr and U n ¼ ðρn ; un ; pn ; σ n Þtr for the flow behind and the flow ahead of the shock, respectively. Let ϕðx; tÞ ¼ 0 be the equation of the characteristic passing through ðx0 ; t 0 Þ forming the initial wavefront and ð1Þ ¼ λ . Let Pðxp ; t p Þ be the point at is defined by the equation dx dt which the fastest discontinuity in U, moving along the characterð1Þ istic dx ¼ λ and originating from the point ðx0 ; t 0 Þ intersects the dt ¼ u ¼ V. The jumps in Ux across the characteristic shock with dx dt incident, reflected and transmitted waves at P, denoted by Λ1 ðPÞ; ΛRi ðPÞ and Λni ðTÞ ðPÞ, respectively, and are given by [32] m1 X

4. Collision of the acceleration wave with the characteristic shock

Λ1 ðPÞ ¼

To evaluate the amplitudes of the reflected and transmitted waves after interaction of the acceleration wave with the characteristic shock, we require the generalized conservation systems of the original system (1) and the same are written in the following forms in the regions behind and ahead of the shock

Λni ðTÞ ðPÞ ¼

 tr ð1  bρÞp ρu2 þ ; ρσ ; G ¼ ρ; ρu; ðγ  1Þ 2  tr uð1  bρÞp ρu3 þ pu þ ; ρuσ ; F ¼ ρu; ρu2 þ p; ðγ  1Þ 2

tr

αð1Þ ðt p ÞRð1;kÞ ; s k

k¼1 min X

ΛðRÞ i ðPÞ ¼

mi X k¼1

αðiÞ ðt p ÞRði;kÞ s ; k

nði;kÞ βðiÞ ; k ðt p ÞRs

ð17Þ

k¼1

where a subscript ‘s’ refers to the values evaluated at point P on the characteristic shock. The evolutionary equations to determine the jump in the shock acceleration j ½V_ j , and the amplitudes αðiÞ k ðiÞ

and βk of reflected and transmitted waves, respectively are given by the system of ‘n’ algebraic equations [15]

ð16Þ

j ½V_ j ðG Gn Þs þð∇GÞs

p X

mi X

i ¼ pqþ1 k ¼ 1

ðiÞ αðiÞ ðV  λ Þ2 Rði;kÞ s k

22

M. Saxena, J. Jena / International Journal of Non-Linear Mechanics 82 (2016) 17–23

 ð∇n Gn Þs

mj q X X j¼1k¼1

¼  ð∇GÞs

mi X k¼1

ðjÞ 2 nðj;kÞ βðjÞ k ðV  λn Þ Rs

ð1Þ αð1Þ ðV  λ Þ2 Rð1;kÞ : s k

ð18Þ

At t ¼ t p , the eigenvalues on both sides of the shock are given by

λð1Þ ¼ u þ a; λð2Þ ¼ u; λð3Þ ¼ u  a; λð2Þ λð3Þ n ¼ u; n ¼ u  an :

λð1Þ n ¼ u þ an ; ð19Þ

Lax's evolutionary conditions for a physical shock for an integer l in interval 1 r l r p, are given by [15]

λðpÞ o λðp  1Þ o ⋯ o λðl þ 1Þ o V o λðlÞ o ⋯ o λð1Þ ; ðp  1Þ ðlÞ ðl  1Þ ð1Þ λðpÞ o⋯ o λn o V o λn o ⋯ o λn ; n o λn which for the given system are of the following form:

λð3Þ o λð2Þ ¼ u o λð1Þ ;

ð2Þ ð1Þ λð3Þ n o λn ¼ u o λn :

ð20Þ

Fig. 14. Variation of αð1Þ with τ; c ¼ 0:1; b ¼ 0:0009; m ¼ 1, αð1Þ 0 ¼  0:001.

The above-mentioned equation implies that when the incident ð1Þ wave with velocity λ at t ¼ t p interacts with the characteristic shock at the point P, it gives rise to one reflected wave with ð3Þ ð1Þ velocity λ and one transmitted wave with velocity λn along the characteristics issuing from the point of interaction. It follows from (18) that the amplitudes of reflected and transmitted waves αð3Þ ð1Þ and β , and the jump in shock acceleration j ½V_ j ¼ V_ tp þ  V_ tp  at the point P (i.e. at time t ¼ t p ) can be determined from the following system of algebraic equations in matrix form ð3Þ 2 ð3Þ ð1Þ ð1Þ  ð∇n Gn Þs Rsnð1Þ ðV  λns Þ2 β ðG  Gn Þs j ½V_ j þ ð∇GÞs Rð3Þ s ðV  λs Þ α ð1Þ

2 ð1Þ ¼  ð∇GÞs Rð1Þ s ðV  λs Þ α :

ð21Þ

In view of (3), (16) and (19), the above system of Eq. (21) can be written in the following system of algebraic equations with the ð1Þ unknowns j ½V_ j ; αð3Þ and β ð1Þ 2ζ j ½V_ j þ αð3Þ  β ¼  αð1Þ ; ð1Þ 2uζ j ½V_ j þðu aÞαð3Þ  ðu þ an Þβ ¼  ðu þ aÞαð1Þ ;



ð22Þ



     pbζ u2 ζ 1  bp u2 ua a2 1 bρ þ j ½V_ j þ þ  þ αð3Þ  2 γ 1 2 2 ðγ  1Þ 2 2 γ 1         1  bp u2 uan a2n 1  bρn 1  bp u2 þ þ þ þ  βð1Þ ¼  2 γ 1 2 2 γ 1 2 2 2 ðγ  1Þ   2 ua a 1  bρ αð1Þ ; þ þ 2 2 γ 1 ð1Þ 2ðρσ  ρ σ n Þj ½V_ j þ σαð3Þ  σ n β ¼  σαð1Þ :

Fig. 15. Variation of αð1Þ with m; c ¼ 0:1; τ ¼ 5; b ¼ 0:0009, αð1Þ 0 ¼  0:001.

the reflection coefficients and the jump in shock acceleration to increase in magnitude. The amplitudes of the reflected and ð1Þ transmitted coefficients αð3Þ and β , and the jump in shock acceleration j ½V_ j depend on ρ and ρn , i.e. the ambient density on both sides of the characteristic shock.

n

5. Results and conclusions It may be noted that the system of Eq. (22) is over determined. In fact, in case of characteristic shock, the system of equations to determine the amplitudes of the reflected and transmitted coefficients, and the jump in shock acceleration is always over determined [7]. On solving the system of Eq. (22), we get   2a2 ð1  bρÞ βð1Þ ¼ αð1Þ ; an ðð1  bρn Þan þ að1 bρÞÞ   a ð1  bρn Þ  að1  bρÞ ð1Þ αð3Þ ¼ n α ; ð23Þ an ð1  bρn Þ þ að1  bρÞ j ½V_ j ¼



  a2n ð1  bρn Þ þ a2 ð1  bρÞ αð1Þ ; ðρ  ρn Þðan ðan ð1  bρn Þ þ að1 bρÞÞÞ

where αð1Þ is determined from Eq. (14). It is evident from (23) that the amplitudes of reflected and transmitted waves and the jump in shock acceleration are proportional to the amplitude of the incident wave, and hence an increase in the magnitude of the incident acceleration wave causes

We considered a system of partial differential equations describing the one dimensional unsteady plane and radially symmetric flow of an inviscid vibrationally relaxing gas with van der Waals equation of state and studied the evolution of characteristic shock and its interaction with an acceleration wave. As for a characteristic shock there is no jump in the velocity and pressure, the jumps in the density ζ and vibrational energy η are evaluated. It is observed that both ζ and η decrease and tend to zero as t-1. It is also observed that the parameter ζ increases with an increase in τ or b and decreases with an increase in c or m. The parameter η increases with an increase in any of the parameters τ, b or m and decreases with an increase in c. The amplitude of the acceleration wave travelling on the fastest ð1Þ characteristics at the speed of λ and satisfying the Bernoulli type equation is evaluated. On solving, it is observed that a discontinuity wave appears at an instant t ¼ t c indicating that a compression wave culminates into a shock in a finite time only when the initial discontinuity associated with the wave exceeds a

M. Saxena, J. Jena / International Journal of Non-Linear Mechanics 82 (2016) 17–23

Fig. 16. Variation of αð1Þ with b; c ¼ 0:1; τ ¼ 5; m ¼ 1, αð1Þ 0 ¼  0:001.

critical value. It is also observed that as t increases, αð1Þ decreases and tend to zero as t-1. For αð1Þ 0 4 0, the amplitude of the acceleration wave αð1Þ decreases with increase in any of the variables c, m or b and increases with an increase in τ. However, for αð1Þ 0 o 0, an opposite trend is observed i.e. the amplitude of the acceleration wave αð1Þ increases with increase in any of the variables c, m or b and decreases with an increase in τ. The interaction of acceleration wave with the characteristic shock is studied and it is observed that there is a reflected wave ð3Þ along the characteristic line with velocity λ and a transmitted ð1Þ wave along the characteristic line with velocity λ . The amplið1Þ tudes of reflected and transmitted waves αð3Þ and β and the _ jump in the shock acceleration j ½V j after interaction of the acceleration wave through a characteristic shock are evaluated. In the absence of the incident wave, the jump in the shock acceleration vanishes and there are no reflected or transmitted waves. As the amplitudes of reflected and transmitted waves are proportional to the amplitude of the incident wave, an increase in the magnitude of the initial discontinuity of the incident wave causes the reflection coefficients and the jump in shock acceleration to increase in magnitude. The amplitudes of the reflected and ð1Þ transmitted coefficients αð3Þ and β , and the jump in shock acceleration j ½V_ j depend on ρ and ρn , i.e. the ambient density on both sides of the characteristic shock.

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