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The elementary wave interactions for the Chapman-Jouguet model with combustion are investigated. We obtain the unique solution of the initial value problem under the global entropy conditions. We analyze the elementary wave interactions in the phase plane and construct uniquely the solution of this initial value problem. It is found that the combustion wave solution of the corresponding Riemann may be extinguished after perturbation which shows that the unburnt gas is unstable.

In this paper, we study the following equations

{ ρ t + ( ρ u ) x = 0 , ( ρ u ) t + ( ρ u 2 + p ) x = 0 , ( ρ E ) t + ( ρ u E + p u ) x = 0 , q ( x , t ) = { 0 if sup 0 ≤ y ≤ t T ( x , y ) > T i ; q ( x , 0 ) otherwise , (1)

where ρ , u , p < 0 , T are the density, velocity, pressure, temperature respectively. T i is the ignition temperature and q is the chemical binding energy. The total

energy E = u 2 2 + e + q , where e is the internal energy. The state equation is given by p = − 1 ρ and e = − p 2 ρ . The process of combustion is exothermic [

In [

In [

The authors [

In [

The authors [

We use usually two mathematical models to investigate the combustion phenomena. One is the Chapman-Jouguet (CJ) model, and the other is the Zeldovich-von Neumann-Döring (ZND) model [

{ u t + p x = 0 , τ t − u x = 0 , E t + ( u p ) x = 0 , q ( x , t ) = { 0 if sup 0 ≤ y ≤ t T ( x , y ) > T i ; q ( x , 0 ) otherwise , (2)

here τ = 1 ρ and τ is the specific volume, p > 0 . It is followed that there is

unique solution for (2). In [

In [

{ u t + p x = 0 , τ t − u x = 0 , E t + ( u p ) x = 0 , q ( x , t ) = − k t φ ( T ) q . (3)

and constructed its unique solution under some hypothesis. In [

In [

( τ , u , p , q ) = { ( τ − , u − , p − , q − ) when x < 0 , ( τ + , u + , p + , q + ) when x > 0. (4)

In this paper, we investigate the initial value problem for (1) with the following initial values

( τ , u , p ) ( x , 0 ) = { ( τ l , u l , p l ) , − ∞ < x ≤ x 1 , ( τ m , u m , p m ) , x 1 < x ≤ x 2 , ( τ r , u r , p r ) , x 2 < x < ∞ . (5)

In [

It is shown that the structures of the Riemann solutions can retain their forms after perturbation for most of the cases, while for some other cases, the perturbation can make a big difference. It is observed that although there is no combustion wave in the corresponding Riemann solution, the combustion wave may occur after perturbation. It shows the instability of the unburnt gas.

The paper is arranged as follows. In Section 2, we list the preliminaries. We obtain the unique solution for the initial value problem (1) and (5) in Section 3. Section 4 gives our main result.

In this section we give some preliminaries [

The characteristic roots for (1) are given by

μ 1 , 3 = u ± − p ρ , μ 2 = u , (6)

it follows that (1.1) is strictly hyperbolic. The right characteristic vector of λ i ( i = 1 , 2 , 3 ) is

r → 1,3 = ( 1, ± 1 ρ − p ρ , − p ρ ) Τ , r → 2 = ( 1,0,0 ) Τ .

From ∇ λ i ⋅ r → i ≡ 0, i = 1,2,3 , all characteristic fields are linearly degenerate.

The rarefaction wave curves R → ( l ) (or R ← ( l ) ) are

{ p ρ = p l ρ l , u = u l ± p − p l − p l ρ l , ( p > p l , or p < p l ) , (7)

and the shock wave curves S → ( l ) (or S ← ( l ) ) are

{ p ρ = p l ρ l , u − u l p − p l = ± − 1 p l ρ l , ( p l > p , or p l < p ) . (8)

The contact discontinuity J is

{ [ u ] = [ p ] = 0 , ρ l ≠ ρ r . (9)

Suppose

ρ = ρ 0 + ω ( t ) δ ( x − x ( t ) ) , ρ 0 = { ρ l , x < x ( t ) , ρ r , x > x ( t ) , (10)

u ( x , t ) = { u l , x < x ( t ) , u δ , x = x ( t ) , u r , x > x ( t ) , (11)

p ( x , t ) = { p l , x < x ( t ) , 0 , x = x ( t ) , p r , x > x ( t ) . (12)

When ρ r ≠ ρ l ,

ω ( t ) = ρ l ρ r ( u r − u l ) 2 − ( ρ r − ρ l ) ( p r − p l ) t

u δ = ρ r u r − ρ l u l + d ω ( t ) d t ρ r − ρ l ,

when ρ r = ρ l ,

ω ( t ) = ( ρ l u l − ρ r u r ) t

u δ = 1 2 ( u r + u l ) .

Further, the delta shock S δ satisfies the entropy condition

u r + − p r ρ r < d x ( t ) d t < u l − − p l ρ l . (13)

The non-combustion wave curves can be shown in the phase plane ( u , p ) (

From the R-H condition

{ η [ u ] = [ p ] , η [ τ ] = − [ u ] , η [ E ] = [ u p ] ,

we find that

− τ r p + p r τ = 2 q 0 > 0.

The forward combustion wave curve in ( u , p ) (

D → ( r ) : u − u r p − p r = − 2 q 0 + τ r ( p − p r ) p r ( p − p r ) , (14)

where p r < p < 0 or p < p r − 2 q 0 τ r . When p r < p < 0 , it is the detonation wave curve and when p < p r − 2 q 0 ρ r , it is the deflagration wave curve [

When q + > 0 , the forward wave curve W → ( + ) is

W → ( + ) ≐ W → S ( + ) ∪ S δ ( + ) ∪ D F ← ( + ) ∪ D T ← ( + ) ,

here

W → S ( + ) ≐ R → ( + ) ∪ S → ( + ) .

When q + = 0 , q − = 0 , it is studied in [

Case 2.1. When q − = 0 , q + = q 0 > 0 .

The backward wave curve is W ← ( − ) = W ← S ( − ) ∪ S δ ( − ) , and the forward wave curve is W → ( + ) = W → S ( + ) ∪ S δ ( + ) ∪ D F ← ( + ) ∪ D T ← ( + ) (

Subcase 2.1.1 u − − u + < − p − ρ − + − p + ρ + .

To guarantee the unique solution of the Riemann problem, we give the global entropy conditions:

We select the unique solution from the four intersection points (at most) in the following order:

1) the solution with the parameter δ as small as possible, where δ is defined as oscillation frequency of T ( ζ ) between the set { ζ ∈ R 1 : T ( ζ ) ≤ T i } and the set { ζ ∈ R 1 : T ( ζ ) > T i } ;

2) the solution containing the combustion wave as many as possible.

For simplicity, ∗ 1 is the intersection point of W ← S ( − ) and W → S ( + ) and ∗ 2 is the intersection point of W ← S ( − ) and D F ← ( + ) or D F ← ( + ) The temperature is respectively T 1 , T 2 at the point ∗ 1 , ∗ 2 (

1) When T − > T i , T 2 > T i , then δ ( ∗ 1 ) = 1 , δ ( ∗ 2 ) = 1 , due to (ii), we pick out ∗ 2 and get the combustion wave solution DF or DT (

2) When T − > T i , T 2 ≤ T i ( ⇒ T 1 ≤ T i ), then δ ( ∗ 1 ) = 1 , δ ( ∗ 2 ) = 3 , due to (i), we pick out ∗ 1 and get the noncombustion wave solution (

3) When T − ≤ T i , T 1 ≤ T i , then δ ( ∗ 1 ) = 0 , δ ( ∗ 2 ) = 2 , due to (i), we pick out ∗ 1 and get the noncombustion wave solution (

4) When T − ≤ T i , T 1 > T i ( ⇒ T 2 > T i ), then δ ( ∗ 1 ) = 2 , δ ( ∗ 2 ) = 2 , due to (ii), we pick out ∗ 2 and get the combustion wave solution DF or DT (

Subcase 2.1.2 u − − u + ≥ − p − ρ − + − p + ρ + .

It is shown that we can get the delta shock wave solution S δ [

Case 2.2. When q − > 0 , q + > 0 . We go on as follows.

We know that W ← ( − ) = W ← S ( − ) ∪ S δ ( − ) ∪ D T → ( − ) ∪ D F → ( − ) , and W → ( + ) = W → S ( + ) ∪ S δ ( + ) ∪ D T ← ( + ) ∪ D F ← ( + ) (

Subcase 2.2.1 When u − − u + < − p − ρ − + − p + ρ + (

Since we have δ = 0 for Case (1), δ = 2 for Case (2)-(4), we pick out the intersection point a and get the non-combustion wave solution.

Subcase 2.2.2 When u − − u + ≥ − p − ρ − + − p + ρ + . We get the delta shock wave solution for this case similarly.

Theorem 2.1. Under the given global entropy conditions, we obtain uniquely the Riemann solution for (1) and (4).

For simplicity, we just consider the case that the delta shock wave does not appear. We consider the four kinds of wave interactions:

A combustion wave D T → and a shock wave S ← , a combustion wave D F → and a shock wave S ← , a combustion wave D T → and a rarefaction wave R ← , a combustion wave D F → and a rarefaction wave R ← .

Case 3.1. D T → and S ← (

We know μ m < η 1 < μ l for D T → , and μ r < η 2 < μ m for S ← , where η 1 and η 2 are the velocity of D T → and S ← respectively. ( m ) ∈ W ← D T ( l ) , ( r ) ∈ S ← ( m ) . From the global entropy conditions, we know that the solution for this case is

D T → + S → → S → + J + S → .

Theorem 3.1. For the wave interaction between the shock wave and the detonation wave, we find that it may extinguish the combustion wave which shows that the unburnt gas is unstable. And after the elementary wave interaction process, we observe that the contact discontinuity may emerge.

Case 3.2. D F → and S ←

In this case (

D F → + S ← → R ← or S ← + J + S → ,

Theorem 3.2. In this situation, the result of the wave interaction between the shock wave and the deflagration wave is that the deflagration combustion wave may be extinguished. Furthermore, we find that the contact discontinuity may emerge after the elementary wave interaction process.

Case 3.3. D T → and R ←

For this situation (

D T → + R ← → S ← or R ← + J + S → .

Theorem 3.3. In this situation, the result of the wave interaction between the rarefaction wave and the detonation wave is that it may extinguish the combustion wave. It shows that the unburnt gas is unstable. Further, we find that the contact discontinuity may emerge after the elementary wave interaction process.

Case 3.4. D F → and R ←

In this case (

D F → + R ← → R ← + J + S → .

Theorem 3.4. In this situation, the result of the wave interaction between the rarefaction wave and the deflagration wave is the deflagration wave may be extinguished. Furthermore, we find that the contact discontinuity may emerge after the elementary wave interaction process.

Based on the above analysis, we conclude our main results.

Theorem 4.1. There exists unique solution for the initial value problem (1) and (5). By studying the concrete elementary wave interactions, we observe interesting combustion phenomenon. The combustion process may be extinguished which reveals that the unburnt gas is unstable.

For simplicity, we suppose that the reaction rate in the combustion model (1) is infinite, Of course, it is still an important model to study the combustion phenomena in our many applications in every way. In our further study, we will study the self-similar ZND model which has a finite reaction rate.

Supported by the Foundation for Young Scholars of Shandong University of Technology (No. 115024).

The author declares no conflicts of interest regarding the publication of this paper.

Liu, Y.J. (2021) Wave Interactions for Chaplygin Gas with Combustion. Journal of Applied Mathematics and Physics, 9, 683-693. https://doi.org/10.4236/jamp.2021.94049