As we mentioned previously, the characteristic speeds for the Riemann prob-lem (4.1)-(4.3) can coincide with each other. Namely, each of the gas eigenvalues
ub−cb, ub, ub+cb can coincide with either of the solid eigenvalues
ua−ca, ua, ua+ca.
However, the solid parameters do not change across the gas waves, and the gas parameters can change only across the solid contact (and of course across the gas waves), see Section2.5. Therefore, the potentially interesting cases arise only when the solid contact, propagating with speed ua, coincides with either ub±cb or ub. Since the characteristic fields ub±cb can be either rarefactions or shock waves, and ub is the contact discontinuity, we will consider these three cases separately.
4.5. Coinciding waves 83
4.5.1 Coinciding contacts
Let us fixαb0,ρb0,pb0 to the left of the solid contact Σ as well as αb1 to the right of it. Further let the gas velocity approachua from the left, i.e.,
ub0+=ua, →+0. (4.40)
Under these conditions, we are interested in the change of the state to the right of the solid contact.
Consider the system of Riemann invariants (4.4) across the solid contact Σ.
Withηb = pb+πb
ργbb = const, we can rewrite (4.4c) as
αb0(pb0+πb)1/γb(ub0−ua) = αb1(pb1+πb)1/γb(ub1−ua). (4.41) Usingc2b = γb(pρb+πb)
b in (4.4e), we get (ub0−ua)2
2 +γbηb1/γb(pb0+πb)1−1/γb
γb−1 = (ub1−ua)2
2 +γbηb1/γb(pb1+πb)1−1/γb γb−1 .
(4.42) Let us show that pb1 remains bounded, i.e. pb1 6→ ∞. Indeed, assume that it is not true. Then pb1 must exceed pb0, pb1 > pb0. However, since the left-hand side of (4.42) is bounded, the following estimate
(ub1−ua)2
2 + γbη1/γb b(pb1+πb)1−1/γb
γb −1 > γbηb1/γb(pb1+πb)1−1/γb γb−1
implies that pb1 < const, since γb, ηb, πb are constants. Using this in (4.41), we getub1 →ua. But then according to (4.42) ispb1 →pb0. Finally using again that the entropy ηb = pb+πb
ργbb = const across the solid contact, we get also ρb1 →ρb0. For the case of coinciding contacts, i.e.ub0 =ua, we can easily find the solution of the system (4.4). Indeed, from (4.4c) it follows that ub1 = ua. Then (4.4b) and (4.4e) imply that ρb0 =ρb1 and pb0 =pb1.
Thus, we have proved the following theorem.
Theorem 4.11. Let us fix αb0, ρb0, pb0 to the left of the solid contact Σ as well as αb1 to the right of it. Then, as the gas velocity ub0 approaches ua, the gas variables to the right of Σ approach the ones to the left,
ρb1 →ρb0, ub1 →ua, pb1 →pb0. For the case ub0 =ua,
ρb1 =ρb0, ub1 =ua, pb1 =pb0.
Remark 4.12. Since the system of Riemann invariants (4.4) is symmetric with respect to subscripts 0 and 1, we can reverse the statement of Theorem 4.11.
Namely, for fixedαb1,ρb1,pb1 to the right of the solid contact Σ as well asαb0 to the left of it, and for ub1+ = ua, → −0, the gas variables to the left of the solid contact
ρb0 →ρb1, ub0 →ua, pb0 →pb1.
4.5.2 Sonic state attached to the solid contact
Let us consider the critical case when
(ub −ua)2 →c2b (4.43)
from either side of the solid contact Σ. At the limit, i.e. when (ub−ua)2 = c2b, the system of governing equations (4.1) becomes parabolic degenerate, it is not hyperbolic anymore. Since the notion of an evolutionary discontinuity is defined only for a hyperbolic system, see Section 4.2, it is not applicable here.
Using the gas dynamics analogy of preceeding section, it is easy to establish the physical meaning for this phenomena. Remember that the difference of αb0− αb1 determines a pore in the solid contact, which is a converging or diverging nozzle, see Fig. 4.6. The flow in this pore is governed by the equation (4.39).
Assume for simplicity that the gas relative speed is positive, v = ub −ua > 0, so the gas flow from the left to the right. If the condition (4.43) holds at the cross-section αb0, i.e. at the state u0 to the left of Σ, then the relative gas flow is almost sonic there,
M0 = ub0−ua
cb0 = 1 +, → ±0,
where M0 is the signed relative Mach number at the stateu0. Depending on the sign ofand on the differenceαb1−αb0, the gas flow in the pore either accelerates and expands, or decelerates and compresses, leading to very different states u1 to the right of Σ. Thus, such a sonic configuration is not stable.
We can also give an interpretation to the case in which the equation (4.11) has no roots. As it was mentioned in Section 4.1, it happens when αb1 becomes sufficiently small, cf. (4.10). According to the results of Section 4.4, the change of the gas parameters to the right of the solid contact is determined by whether the relative gas speed ub −ua is subsonic or supersonic. When it is subsonic, then the gas flow accelerates and expands, so that the sound speed decreases and the relative Mach number M = |ub1 −ua|/cb1 increases. As soon as the gas volume fraction falls below its critical value, where M = 1, a shock intervenes and isentropic flow does not exist anymore. On the other hand, when the relative gas flow is supersonic, the relative Mach number decreases. Again, for the gas volume fraction below its critical value, an isentropic flow does not exist. Geometrically, we can illustrate the possible scenarios by considering the different cases in the proof of Theorem 4.7, see Fig. 4.7.
It is an easy matter to check that the solid contact can lie inside of the gas rarefaction only in the trivial case when u0 = u1, where u is given by (4.2).
Then in particular αb0 = αb1, so that the phases are decoupled. For simplicity, consider the gas 4-rarefaction (the case of 6-rarefaction can be done analogously).
In the rarefaction wave, ub−cb varies monotonically across the wave, so that the
4.5. Coinciding waves 85 PSfrag replacements PSfrag replacements
PSfrag replacements PSfrag replacements
Fig. 4.7. Gas characteristics around the solid contact. As the char-acteristic ub0 ±cb0 moves in the arrow direction, the corresponding ub1±cb1 becomes equal to ua (one root). Further change inub0±cb0 leads to no roots in (4.11).
following equality should hold true,
ub0 −cb0 =ua =ub1−cb1. But using this in (4.4) implies
u0 =u1.
4.5.3 Shock vs. solid contact
Finally, let us describe the behaviour of the solution, when the (admissible) gas shock wave approaches the solid contact. For simplicity, we consider the shock at the left of the solid contact, i.e.,σ →ua, σ < ua, see Fig. 4.8.
Across the shock, the Rankine–Hugoniot conditions hold, and across the solid contact — the relations (4.4). Combining them, we get
ρb(ub −σ) = ρb0(ub0−σ)→ρb1
αb1
αb0(ub1−ub0) = M
αb0 =:M1 (4.44a) ρb(ub −σ)2+pb =ρb0(ub0−σ)2+pb0 → 1
αa1
(P −αa0pa0) =: P1 (4.44b) (ub−σ)2
2 + c2b
γb−1 = (ub0−σ)2
2 + c2b0
γb−1 → (ub1−ua)2
2 + c2b1
γb −1 =E (4.44c) asσ→ua. Multiply (4.44c) by 2ρb, and substract the result from (4.44b). Then
pbγb+ 1
1−γb − 2γbπb
γb−1 →P1−2ρbE. (4.45)
PSfrag replacements
u1 u0 u
shock Σ
x t
t t t t t t t t t
Fig. 4.8. Gas shock approaches the solid contact Σ from the left.
From (4.44a) and (4.44b)
pb →P1− M12 ρb
.
Substituting this into (4.45), we get the following relation for ρb, (P1− M12
ρb )γb+ 1
1−γb − 2γbπb
γb−1 →P1−2ρbE.
In the limit it reveals the quadratic equation for ρb, (P1− M12
ρb )γb+ 1
1−γb − 2γbπb
γb−1 =P1−2ρbE,
with ρb 6= 0. This equation has two roots. One of them is always ρb0, and corresponds to the state u ≡ u0. The other is necessarily admissible, since we consider the admissible shock wave. Using this root, we can find the state behind the shock
ub = M12 ρb +σ pb =P1− M12
ρb .
(4.46)
Thus, we see that the state behind the shock approaches some finite state which is given by (4.46), as the shock approaches the solid contact.