Elementary Wave Interactions for the Simplified Combustion Model in Magnetogasdynamics

We investigate the elementary wave interactions for the simplified combustion model in magnetogasdynamics. Under the modified entropy conditions, we construct the unique solution and observe some interesting phenomena; such as, the combustion wave may be extinguished by the contact discontinuity or the shock wave. Especially, the transition between the detonation wave and the deflagration wave is also captured.


Introduction
Magnetogasdynamics is very important in studying engineering physics [1]- [9].It is difficult to investigate the governing equations of Magnetogasdynamics flows; the corresponding results are less than the conventional gas dynamics.When the velocity field and the magnetic field are everywhere orthogonal, the magnetogasdynamics flow is still important.
In [3], Helliwell discussed the non-conducting inviscid gas at rest when ionization of the gas takes place across a shock wave and obtained that the magnetogasdynamic combustion wave has similar properties with the conventional gas dynamics.In [6], Mareev investigated the above problem further and they could use the obtained results to study the hypersonic gas flow around a thin wedge in an axial magnetic field.
In [4], Hu and Sheng constructed the unique Riemann solution of the one-dimensional inviscid flow under the assumption B kρ = , where 0 τ > , 0 p ≥ , u, 0 B ≥ and µ are respectively the specific volume, pressure, velocity, transverse magnetic field and magnetic permeability.is the specific total energy.and e is the specific internal energy.The Riemann problem of the conventional gas dynamics combustion models was investigated by many people ( [10] [11] [12] [13], etc.).Zhang and Zheng [13] studied the Riemann problem of the conventional gas dynamics flow of combustible ideal gases ( ) with an infinite rate of reaction which is described by Under the proposed global entropy conditions, they constructed uniquely the Riemann solutions by the characteristic analysis.In [11], we modified the above global entropy conditions and constructed the unique solution of the generalized Riemann problem for (2) and (3).
In [5], we obtained uniquely the Riemann solutions for (1) and (3) under the modified entropy conditions in [11] with the following initial data ( )( ) ( ) , , , , 0 , , , , 0, p u q x p u q x τ τ ± ± ± ± = ± > where 0, , p u τ ± ± ± > are arbitrary constants, + is the specific total energy, where q is the chemical binding energy.The temperature T satisfies Boyle and Gay-Lussac's law: p RT τ = . i T is the ignition temperature.For polytropic gases, we know ( ) e e T = and 2 2 1 , where 1 γ > is the adiabatic exponent.For simplicity, we usually assume that R and γ remain unchanged during the reaction.We also assume that the combustion process is exothermic, i.e. the energy used up in recombing the atoms to form the new molecules is smaller for the burnt gas than the binding energy of the unburnt gas [10].Journal of Applied Mathematics and Physics In the present paper, we are concerned with the wave interactions of the elementary waves of the Chapman-Jouguet (CJ) model ( 1) and (3).we can capture some interesting combustion phenomena by investigating the elementary wave interactions.For example, in most cases expect for Case 5.In Section 3, the combustion wave may be extinguished by the contact discontinuity or the shock wave.Especially, a detonation wave may be transformed into a deflagration wave by the contact discontinuity (see Case 1 and Case 2 in Section 3) and a deflagration wave may be transformed into a detonation wave by the contact discontinuity (see Case 4 in Section 3) or by the shock wave (see Case 6 in Section 3) which shows the transition between the detonation wave and the deflagration wave.In the case that when combustion waves interact with the contact discontinuity, our results are very different from the conventional gas dynamics combustion model ( 2) and ( 3) where there is no transition from the detonation (deflagration) wave to the deflagration (detonation) wave and the combustion wave can not be extinguished by the contact discontinuity.This paper is organized as follows.In Section 2, we present the results of the Riemann problem for the CJ model ( 1), (3) with the initial values (4).In Section 3, the elementary wave interactions are considered case by case under the modified entropy conditions in [11].

Preliminaries
As a preparation, we study the Riemann problem for the CJ model ( 1), (3) with the initial data (4) and we refer the detailed discussions to [4] [5].
There are three eigenvalues of (1) which are λ are genuinely nonlinear and 2 λ is linearly degenerate.
Considering the self-similar solution ( )( ) The forward or backward rarefaction waves R   passing through the point ( ) The Rankine-Hugoniot jump conditions at ζ σ = are as follows where [ ] r l The contact discontinuity J is given by and J is a curve in the ( ) , , p u τ space and the projection on the ( ) If [ ] 0 q = in (7), we get the forward or backward shock waves S   passing through the point ( ) where 2  1 1 If [ ] 0 q ≠ in (7), we obtain the combustion wave curve in the ( ) Draw two straight lines from ( ) 0 0 , p τ and they are tangent to the above curve.We call the tangent points A with respectively.From the RH condition (7), we should disregard the curve between the points C and D. We call the curve between C and A weak detonation (WDT) and the curve above A strong detonation (SDT), the curve between D and B weak deflagration (WDF) and the curve below B strong deflagration (SDF), respectively (see Figure 1).Journal of Applied Mathematics and Physics From the known Jouguet's rule in [13], there are at most three different kinds of wave series that can be linked to the state ( ) ( ) , , , 2) ( ) ( ) where is the state at ( ) S l with the ignition temperature i T , and the symbol "+" means "followed by".Notice that we let the temperature behind the pre-compressive shock wave which connects the state (l) and the ignition point (i) be the ignition point i T , we just need to construct the deflagration wave curve which is the successor to the pre-compressive shock wave from the point (i).
In the ( ) W l q > or both of them, and Now we study the combustion wave curves in the ( ) , u p plane and construct the backward combustion wave curve ( ) From ( 9), for the backward wave and from (10), we know thus we obtain the backward combustion wave curve in the ( ) ) Similarly, we can construct the forward wave curve that can be linked to the state ( ) ( ) , , , r r r r r p u q τ = .
Since the image of J in ( ) , , p u τ is a straight line which parallels to the τ-axis and the projection on the plane ( ) , , p u τ and the projection on the plane ( ) , u p is a straight line which parallels to the p-axis.Thus the Riemann prblem for (1) is much more complicated than that of the conventional gas dynamics.
When 0 l r q q = = , the gas on both sides are burnt, no combustion wave will occur.
When l q and r q are not both zero, there may exist more than one intersection points of ( ) to a unique Riemann solution.When the intersection point is unique, the solution is also unique, otherwise, in order to obtain the unique solution we select it under the following modified global entropy conditions (MGEC) ( [11]): We select the unique solution from the nine intersection points (at most) of the forward wave curves connecting (r) and the backward wave curves connecting (l) in the following order: 1) the solution with the propagating speed of combustion wave as low as possible; 2) the solution with the parameter β as small as possible, where β is defined as oscillation frequency of ( ) and the set 3) the solution containing as many combustion wave as possible.
Case 1. 0, 0 l r q q > = .In this case, the gas is unburnt on the left side, the gas is burnt on the right side, i.e.

( ) ( ) ( ) ( )
When there exists only one intersection point of ( ) ( ) W r  , we obtain the unique solution is a detonation wave solution DT R When there are three intersection points of ( )  From the condition A, we discard the possible detonation DT wave solution and find that the possible Riemann solution is R According to the modified global entropy conditions we obtain the unique Riemann solution as follows (see Figure 5).* and obtain a non-combustion wave solution (Figure 5(a)).Case 2. 0, 0 l r q q > = and there are two intersection points of ( )

4) When
( ) S W r  (see Figure 6).Figure 6.q l > 0, q r = 0 and there are two interactions.Case 3. 0, 0 l r q q > > and the gas on the both sides are unburnt.In this case ,we know that If the intersection point of ( ) From the condition A, we just need to consider the intersection points 1, 2, 3, 4. We should select the unique solution from the four possible solutions (see Figure 10).
It is obvious that 0 β = for (a), and it holds that

Wave Interactions for the Combustion Model (1) and (3)
In this section, we discuss the wave interactions of the elementary waves for our combustion model ( 1) and ( 3).Consider the equations ( 1) and ( 3) with the following initial data In order to capture the interesting combustion phenomena, in the present paper we investigate the following kinds of interactions: The interaction of a combustion wave SDT   or WDF  and a contact The interaction of a combustion wave SDT   or WDF  and a shock wave: SDT S   , WDF S  .
In what follows, we construct the solutions of the wave interactions case by case.

SDT
  will encounter each other at a finite time and a new Riemann problem is formed with (l) and (r) as its left-hand side state and right-hand side state, respectively (Figure 13).We solve this new Riemann problem in the ( ) From the analysis of the wave curves in the ( ) There are two possible cases: there is only one intersection point of where " → " means the result of the wave interaction.Journal of Applied Mathematics and Physics     Theorem 3.7.For this case, when a shock wave overtakes a weak deflagration combustion wave, the weak deflagration combustion wave may be extinguished or be transformed into a detonation wave.And the contact discontinuity may appear or not.

S R J R S J WDF DF J R S
After the discussions of the elementary wave interactions, we summarize the results as follows.
By investigating the kinds of wave interactions of the elementary waves, we can capture some interesting combustion phenomena.For example, the combustion wave may be extinguished by the contact discontinuity or by the shock wave.Especially, a detonation wave may be transformed into a weak deflagration wave coalescing with pre-compression shock wave by the contact , (1) is strictly hyperbolic.The characteristic fields 1,3

,
p u plane is a straight line parallel to the p-axis.Denote J by J detonation (CJDT) and Chapman-Jouguet deflagration (CJDF),

11 )
Now denote the backward DF and DT wave curve by( )

Figure 2 .
Figure 2. The combustion wave curve in the plane (u, p).
, from the condition B, we select S , from the condition C, we select DF * and obtain a combustion wave solution containing a DF (Figure 5(b)).
, from the condition B, we select S * and obtain a non-combustion wave solution S R are two possibilities: one is that there exists only one intersection point of and we obtain the unique Riemann solution is DT J R > + +   Journal of Applied Mathematics and Physics

1 .
subcases: one is that there is an intersection point of In the former subcase (see Figure9), we discuss it in the following

2 .,
2 β = for (b), (c) and (d).From the condition B, we select the intersection point 1 and obtain the unique non-combustion wave solution R  similar way as the above discussions in Subcase 3.1.1.,we obtain that the unique Riemann solution is still the non-combustion wave solution R  or In the latter subcase, there are only two possibilities: just need to consider the former.If the intersection point is unique, the solution is DT the condition A, the intersection point of ( ) should be discarded.We denote the intersection point of ( ) Journal of Applied Mathematics and Physics

Figure 9 .
Figure 9.There is an intersection point of

Figure 10 .
Figure 10.The possible solutions in Subcase 3.1.1.(a) solution corresponds to point 1; (b) solution corresponds to point 2; (c) solution corresponds to point 3; (d) solution corresponds to point 4.
similar way as the above discussions in Subcase 3.2.1.,we obtain that the unique Riemann solution is still the combustion wave solution DF DT +   .The only difference is that there exists the contact discontinuity in Subcase 3.2.1.Based on the above analysis, we have the following result.Theorem 2.1.There exists a unique piecewise smooth solution to the Riemann problem (1) and (3) with the initial data (4) under the modified global entropy conditions (MGEC).
the former section, we know the result is as follows:

Figure 13 .
Figure 13.The interaction of J > and SDT   .

Figure 14 .
Figure 14.The interaction of J < and SDT   .