shock structure and temperature overshoot in macroscopic multi-temperature model of mixtures
TRANSCRIPT
Shock structure and temperature overshoot in macroscopic multi-temperature model ofmixturesDamir Madjarević, Tommaso Ruggeri, and Srboljub Simić Citation: Physics of Fluids (1994-present) 26, 106102 (2014); doi: 10.1063/1.4900517 View online: http://dx.doi.org/10.1063/1.4900517 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/26/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Multicomponent gas mixture air bearing modeling via lattice Boltzmann method J. Appl. Phys. 109, 07B759 (2011); 10.1063/1.3564945 Kinetic solution of the structure of a shock wave in a nonreactive gas mixture Phys. Fluids 23, 017101 (2011); 10.1063/1.3541815 Shock structure analysis in chemically reacting gas mixtures by a relaxation-time kinetic model Phys. Fluids 20, 117103 (2008); 10.1063/1.3013637 Shock structure in low density gas mixture flows over cylinders and plates Phys. Fluids 19, 106102 (2007); 10.1063/1.2786004 Macroscopic equations for a binary gas mixture AIP Conf. Proc. 585, 297 (2001); 10.1063/1.1407575
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PHYSICS OF FLUIDS 26, 106102 (2014)
Shock structure and temperature overshoot in macroscopicmulti-temperature model of mixtures
Damir Madjarevic,1,a) Tommaso Ruggeri,2,b) and Srboljub Simic1,c)
1Department of Mechanics, Faculty of Technical Sciences, University of Novi Sad,Trg Dositeja Obradovica 6, 21000 Novi Sad, Serbia2Department of Mathematics and Research Center of Applied Mathematics,University of Bologna, Via Saragozza 8, 40123 Bologna, Italy
(Received 22 August 2014; accepted 15 October 2014; published online 29 October 2014)
The paper discusses the shock structure in macroscopic multi-temperature model ofgaseous mixtures, recently established within the framework of extended thermody-namics. The study is restricted to weak and moderate shocks in a binary mixture ofideal gases with negligible viscosity and heat conductivity. The model predicts theexistence of temperature overshoot of heavier constituent, like more sophisticatedapproaches, but also puts in evidence its non-monotonic behavior not documentedin other studies. This phenomenon is explained as a consequence of weak energyexchange between the constituents, either due to large mass difference, or large rar-efaction of the mixture. In the range of small Mach number it is also shown thatshock thickness (or equivalently, the inverse of Knudsen number) decreases withthe increase of Mach number, as well as when the mixture tends to behave like asingle-component gas (small mass difference and/or presence of one constituent intraces). C© 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4900517]
I. INTRODUCTION
Macroscopic models of multicomponent flows seem to be a well established part of contempo-rary mixture theory. They are able to capture diverse phenomena, diffusion being the most prominentone, and to include different processes of thermo-chemical nature.1 A systematic approach to themixture problem, which includes verification of compatibility with an entropy inequality, could betraced back to thermodynamics of irreversible processes (TIP).2 Although macroscopic models ob-tained in this framework are widely appreciated, they also have some inherent drawbacks. Physically,they suffer from the paradox of infinite speed of propagation of disturbances, and their predictionsare relevant only for a class of processes which occur in the neighborhood of local equilibriumstate. This restriction appeared to be of crucial importance in irreversible processes with strongdeparture from local equilibrium, such as shock structure problem. Classical linear models of TIPfail to provide accurate predictions in such situations.3
Parallel to macroscopic approach, there is also a kinetic theory of mixtures which treats similarphenomena on a much smaller scale. It has a great potential for description of non-equilibriumprocesses by means of numerical solution of the system of Boltzmann equations.
Within these two approaches lies extended thermodynamics (ET). It is a macroscopic theorywhich tends to fill the gap between the classical TIP, on one side, and kinetic theory on the other side.Bridging the gap between two theories is, actually, bridging the gap between two scales—macroand meso scale. The way in which ET achieves that goal is through extension of the set of statevariables needed for description of non-equilibrium processes. This calls for introduction of theadditional set of balance laws for new state variables, and treatment of the entropy inequality in a
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1070-6631/2014/26(10)/106102/19/$30.00 C©2014 AIP Publishing LLC26, 106102-1
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106102-2 Madjarevic, Ruggeri, and Simic Phys. Fluids 26, 106102 (2014)
broader sense. This approach, thoroughly described in the book of Muller and Ruggeri,4 will be theone we shall stick to in our study. It provides a systematic way for derivation of extended set of fieldequations for non-equilibrium processes, which is in accordance with the basic physical principles—Galilean invariance and the entropy principle. One of the main features of these models, in view ofmathematical structure, is their hyperbolicity, i.e., finite speed of propagation of disturbances.
A long term intention to remove the paradox of infinite speed of diffusion flourished in Muller’shyperbolic model of homogeneous gaseous mixtures,5 where velocity fields for each constituent wereintroduced as state variables. This required additional set of evolution equations—the momentumbalance laws for each constituent—whereas closure of the problem was achieved through the entropyprinciple. Recently, this model was extended even more by adjoining to each constituent its owntemperature.6 Motivation for the multi-temperature (MT) model can be found in the study of ionizedgases: as a consequence of huge difference of masses, different temperatures of ions and electronscan be observed.7 On the other hand, development of the macroscopic mixture models witnessederroneous predictions of the shock structure when temperature difference between the constituentswas disregarded.8 The idea of multiple temperatures thus reflects the physically justified intention toget a deeper insight into non-equilibrium processes in mixtures, but this concept seems to be mostlyoverlooked in the context of macroscopic theories. Nevertheless, it was appreciated and naturallyembedded in the kinetic theory of gases, which is perfectly designed to monitor the processes far fromequilibrium. For example, it appeared as efficient tool in non-equilibrium flow computations9 whenequipartition of translational energy is assumed to be violated and several translational temperaturesappear. It was also used to keep track of vibrational non-equilibrium in mixtures10, 11 where discreteenergy levels are used to mimic non-translational degrees of freedom, and vibrational temperaturesfor species are introduced. In view of this fact, we shall adopt the notion of the constituent temperaturein a kinetic sense—as a measure of its mean kinetic energy.
The aim of this paper is to give a systematic analysis of the shock structure problem in binaryMT mixture of Euler fluids. Contemporary studies within the framework of kinetic theory repre-sent collections of valuable, yet particular results, mainly devoted to development of deterministicnumerical schemes for the solution of Boltzmann equations.12, 13 Similar observations can be givenfor DSMC (direct simulation Monte Carlo method).14 However, both approaches revealed certainimportant features of the shock structure in mixtures: velocities and temperatures of the constituentshave different values within the profile of the shock wave. Moreover, for certain values of parameters(Mach number, mass fraction, mass ratio, and diameter ratio of the atoms) there appears a temper-ature overshoot of a heavier constituent—a region within a profile where the temperature increasesabove the terminal temperature of the mixture. The complexity of numerical schemes preventedmassive calculations and detailed study of this aspect of the shock structure.
Recently the hyperbolic MT model was tested against experimentally determined shock structurein helium-argon mixture.15 Even though the model can be regarded as simplified, since viscosity andheat conductivity were neglected, a good agreement was obtained for available experimental data.16
This put the macroscopic MT model at the same level of accuracy as more refined models of kinetictheory,17 or the results of DSMC,18 at least for the weak shocks. The main advantage of the MTmodel is its applicability with only moderate numerical efforts and the use of standard numericalpackages. These facts stimulated further systematic study of the problem.
The main results will be concerned with the systematic analysis of the shock thickness andtemperature overshoot of heavier constituent in terms of three parameters—mass ratio, Mach number,and mass concentration. It will be shown that as long as mixture resembles the single-componentgas, in one way or another, the shock thickness will be smaller. However, the principal noveltyis concerned is with the temperature overshoot—it varies non-monotonically with the mass ratio.On one side of the scale, larger discrepancy of the masses produces larger overshoot. On the otherside, temperature overshoot can increase even when the masses are similar, but the temperatures aredriven apart because of larger rarefaction. Beneath both macroscopic observations lies small amountof exchange of internal energy between the constituents during the collisions as a main physicaldriving agent.
The paper is organized as follows. In Sec. II we give a description of the MT model forbinary mixtures of gases. Section III contains the formulation of the shock structure problem in
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106102-3 Madjarevic, Ruggeri, and Simic Phys. Fluids 26, 106102 (2014)
dimensionless form, description of numerical procedure used for its solution and observation aboutthe existence of continuous shock structure. A first look on the shock structure problem, containingparticular shock profiles, is given in Sec. IV. The systematic study of the shock structure is givenin Secs. V and VI. There we provide a detailed analysis of the shock thickness and temperatureovershoot in terms of the parameters present in the model. It is followed by Sec. VII where anobservation about different definitions of the average temperature and comparison with a single-temperature (ST) model are given. The paper is closed with conclusions.
II. THE MODEL
Basic hyperbolic model of gaseous mixtures5 was derived in the spirit of extendedthermodynamics,4 but also perfectly fitted into the framework of Truesdell’s rationalthermodynamics.19 Its basic postulates, so-called metaphysical principles, assume that we maydescribe the behavior of each constituent by the same balance laws as in the case of a single fluid,if we allow for proper actions of the constituents upon it. This assumption supports not only thehypothesis that each constituent can be described by its own velocity field, but we can also ascribethe internal energy density, i.e., its own temperature. A renewed interest in non-equilibrium pro-cesses in gaseous mixtures yielded an extended thermodynamic model of MT mixtures. Theoreticalframework was developed6 and relation to coarser (classical) theories like TIP was discussed.20 Itscounterpart within the framework of Boltzmann’s equations for mixtures of monatomic gases wasalso recently studied.21
A. Balance laws and hyperbolicity
Starting with the hyperbolic MT model6 we shall restrict the analysis to a non-reacting binarymixture of ideal gases which are neither viscous, nor heat conducting. The neglect of viscousdissipation and heat conduction has twofold motivation; for the processes not far from the localequilibrium state it is justifiable to neglect these effects; dissipation will enter through relaxationprocesses due to mutual exchange of momentum and internal energy between the constituents;moreover, it was this model for which a good agreement with experimental data was obtained.15
Following the principles stated above, the general balance laws for the constituents in n-componentmixture can be written as for a single fluid,
∂tρα + div(ραvα) = τα,
∂t (ραvα) + div(ραvα ⊗ vα − tα) = mα, (1)
∂t
(1
2ραv2
α + ραεα
)+ div
{(1
2ραv2
α + ραεα
)vα − tαvα + qα
}= eα,
where α = 1, . . . , n, and τα , mα , and eα describe the actions of other constituents upon the onelabeled α. In (1) ρα , vα , εα , tα , and qα denote, respectively, mass density, velocity, internal energydensity, stress tensor, and flux of internal energy of the constituent α. An equally important principlefor the mixtures states that all the properties of the mixture are mathematical consequences of theproperties of the constituents. To recover the conservation laws for the whole mixture one oughtto define properly the state variables and the fluxes of the mixture in terms of the properties of theconstituents,
ρ =n∑
α=1
ρα, v = 1
ρ
n∑α=1
ραvα, uα = vα − v,
εI = 1
ρ
n∑α=1
ραεα, ε = εI + 1
2ρ
n∑α=1
ραu2α, (2)
t =n∑
α=1
(tα − ραuα ⊗ uα) , q =n∑
α=1
{qα + ρα
(εα + 1
2u2
α
)uα − tαuα
},
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106102-4 Madjarevic, Ruggeri, and Simic Phys. Fluids 26, 106102 (2014)
where uα is the diffusion velocity of the constituent, and εI is the intrinsic part of internal energydensity. Moreover, the source terms must obey the following axioms:
n∑α=1
τα = 0,
n∑α=1
mα = 0,
n∑α=1
eα = 0,
which lead to reconstruction of the conservation laws for the mixture,
∂tρ + div(ρv) = 0,
∂t (ρv) + div(ρv ⊗ v − t) = 0, (3)
∂t
(1
2ρv2 + ρε
)+ div
{(1
2ρv2 + ρε
)v − tv + q
}= 0.
Turning back to the main assumptions about the constituents, the following restrictions willbe introduced: (a) since there is no chemical reactions, there will be no mass exchange betweenconstituents, and consequently
τα = 0,
(b) since the constituents are neither viscous, nor heat conducting, the stress tensor will have diagonalform, while the heat flux will vanish,
tα = −pαI, qα = 0,
where pα are partial pressures and I identity tensor. To complete the description of the constituents,assumed to be ideal gases, thermal and caloric equations of state will be used in the following form,
pα = ρα
kB
mα
Tα, εα = kBTα
mα(γα − 1)= cVα
Tα, (4)
where kB is the Boltzmann constant and mα , Tα , cVα, and γ α are atomic mass, temperature, constant
volume specific heat, and the ratio of specific heats for the constituent, respectively.Hyperbolicity of the model (1) trivially comes from its differential part. Due to simplifying
assumptions about the constituents and the structure of balance laws, characteristic speeds of themodel comprise all the characteristic speeds of the constituents, as if they were regarded as singlefluids,
λ(1)α = vαn − aα, λ(2,3,4)
α = vαn, λ(5)α = vαn + aα, (5)
vαn = vα · n, aα = {γα(kB/mα)Tα}1/2 ,
where vαn is the normal component to the wave front of the particle velocity of constituent α, nbeing its unit normal, and aα is the local speed of sound.
B. State variables
Behaviour of the constituents is described by Eq. (1), which comprise 5 scalar balance laws foreach one of them. Therefore, in the case of binary mixture (n = 2), which will be discussed in therest of this paper, we need a total of 10 state variables to completely determine the process. On theother hand, any thermodynamic process in the mixture must satisfy also the conservation laws (3).Thus, it is usual to replace the set of balance laws for the one constituent, say 2, with conservationlaws for the mixture. This choice of governing equations immediately suggests the following choiceof state variables (ρ, v, T, ρ1, v1, T1). It is obvious that it requires proper definition of the averagetemperature T of the mixture.
The average temperature of the MT mixture will be defined by means of the intrinsic (thermal)part of internal energy in equilibrium:20
(ρ1cV1 + ρ2cV2 )T = ρ1cV1 T1 + ρ2cV2 T2. (6)
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106102-5 Madjarevic, Ruggeri, and Simic Phys. Fluids 26, 106102 (2014)
By restricting the analysis to the binary mixture of monatomic gases, γ 1 = γ 2 = γ = 5/3, we mayexpress the total pressure and intrinsic part of internal energy in the same form as in the case of asingle component gas,
p = p1 + p2 = ρkB
mT, ρεI = ρ1ε1 + ρ2ε2 = ρ
kB
(γ − 1)mT,
provided we introduce the average mass m = m(c) and the average temperature T of the mixture inthe following form,
1
m(c)= c
m1+ 1 − c
m2, T = c
m(c)
m1T1 + (1 − c)
m(c)
m2T2, (7)
where c is the concentration variable related to the mass densities in the following way:
ρ1 = ρc, ρ2 = ρ(1 − c).
To make our study more feasible, we will introduce the difference of temperatures = T2 −T1, so-called diffusion temperature, and the ratio of masses of the constituents,
μ = m1
m2, 0 < μ ≤ 1,
where we assumed m1 ≤ m2. The temperatures of the constituents can now be expressed in terms ofnew variables T, , and c, using μ as a parameter,
T1 = T − f (c), T2 = T + (1 − f (c)),
where auxiliary function f(c) has the following form:
f (c) = μ(1 − c)
c + μ(1 − c).
Finally, we will also use the diffusion flux J of the constituent, instead of its velocity v1. Sincerelative velocities obey the relation ρ1u1 + ρ2u2 = 0, due to (2), the diffusion flux of constituent 1is defined as
J = ρ1u1 = −ρ2u2.
Therefore, we take the change of variables,
(ρ, v, T, ρ1, v1, T1) → (ρ, v, T, c, J,),
and rewrite the system of governing equations in terms of them. For the sake of brevity we will skipthis mostly technical result, since Sec. III contains the explicit form of the shock structure equations.
C. Source terms
Source terms mα and eα describe what metaphysical principles state as the action of otherconstituents upon the one for which balance laws are written. However, they do not provide anymean for their determination. To resolve this problem ET relies on the assumption that source termscould depend on state variables only through their local values (at point) and on two fundamen-tal principles—relativity and entropy principle. The first one determines the velocity dependencethrough Galilean invariance:4, 6
m1 = m1, e1 = e1 + m1 · v. (8)
The second one determines the structure of velocity independent parts by their compatibility with anentropy inequality, i.e., by securing the non-negativity of entropy production. In such a way sourceterms bring the dissipation into system. In the first approximation they have the following form:6
m1 = −ψ11
(u1
T1− u2
T2
), e1 = −θ11
(− 1
T1+ 1
T2
), (9)
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106102-6 Madjarevic, Ruggeri, and Simic Phys. Fluids 26, 106102 (2014)
where ψ11 and θ11 are the phenomenological coefficients. They can be related to state variables andrelaxation times for diffusion τD and temperature τ T:15
ψ11 = 1
τD
ρ1ρ2
ρT, θ11 = 1
τT
ρ1cV1ρ2cV2
ρ1cV1 + ρ2cV2
T 2. (10)
ET does not possess an inherent method to determine the phenomenological coefficients com-pletely. However, by the methods of kinetic theory of gases one can relate the relaxation times τD
and τ T to the diffusivity D12 of binary mixture of monatomic gases:14, 22
τD = cm2 + (1 − c)m1
kBTD12, τT = m1 + m2
kBTD12. (11)
In fact, kinetic theory establishes relation (11)1 between τD and D12, while the other one comes asa consequence of the ratio between the relaxation times in the case of monatomic gases:7, 15, 20
τT
τD= m1 + m2
c m2 + (1 − c) m1> 1. (12)
Diffusivity of the binary mixture, derived from the model of hard spheres, is determined as14, 22
D12 = 3
8nd212
(kBT
2π
m1 + m2
m1m2
)1/2
, (13)
where n = ρ/m is the mixture number density, n = n1 + n2 = ρ1/m1 + ρ2/m2, and d12 = (d1 + d2)/2is the average atomic diameter of the mixture constituents whose diameters are d1 and d2.
III. THE SHOCK STRUCTURE PROBLEM
The shock structure is a continuous solution with steep gradients of state variables in theneighborhood of singular surface—the shock wave—which is diffused due to dissipative mechanismstaken into account. Our attention will be restricted to the shock structure related to plane shocks,moving at constant speed s in direction orthogonal to the singular surface. Consequently, one spacevariable, say x, will suffice for description of the problem, and the shock structure will be described asa traveling wave solution, depending on a single variable ξ = x − st, which asymptotically connectsequilibrium states in front and behind the shock wave. These assumptions transform the model intoa set of ordinary differential equations where the velocity, the diffusion flux, and the source term(momentum exchange) are described by a single component, i.e., v = (v, 0, 0), J = (J, 0, 0), andm1 = (m1, 0, 0), in Cartesian coordinates.
A. Dimensionless shock structure equations
The problem of the shock structure will be studied in dimensionless form. For that purpose, weshall introduce the dimensionless variables by scaling the state variables and independent variableξ with appropriate upstream equilibrium values, indicated by the subscript 0
ρ = ρ
ρ0, u = u
a0, T = T
T0, J = J
ρ0a0, =
T0, ξ = ξ
l0, M0 = u0
a0, (14)
where u = v − s is the relative mixture velocity with respect to the shock wave, l0 is the upstreamreference length, and a0 = {γ (kB/m0)T0}1/2 is the upstream speed of sound; m0 = m(c0) is theequilibrium average mass of the mixture and M0 is the upstream Mach number. For the sake ofsimplicity, tilde will be dropped in the sequel.
The upstream reference length l0 is usually taken as the mean free path of the atoms. The averagemean free path in the mixture will be expressed in terms of other more primitive properties of the
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106102-7 Madjarevic, Ruggeri, and Simic Phys. Fluids 26, 106102 (2014)
constituents,14
l0 = n1
n(l1)0 + n2
n(l2)0, (15)
(lα)0 = 1
πd212
[n1
(1 + mα
m1
)1/2
+ n2
(1 + mα
m2
)1/2]−1
,
where nα are number densities of the constituents, and n = n1 + n2 is the mixture number density.Using the scaled variables (14) we obtain the following set of dimensionless equations:
d
dξ(ρu) = 0,
d
dξ
(ρu2 + 1
γ
m0
mρT + J 2
ρc(1 − c)
)= 0,
d
dξ
{(1
2ρu2 + 1
γ − 1
m0
mρT + J 2
2ρc(1 − c)
)u +
(u J
ρc(1 − c)+ 1
β
)J
}= 0,
d
dξ(ρcu + J ) = 0, (16)
d
dξ
{ρcu2 + J 2
ρc+ 2u J + 1
γ
m0
m1ρc (T − f (c))
}= − l0
τDa0mμ(T, c,)J,
d
dξ
{(1
2ρc
(u + J
ρc
)2
+ 1
γ − 1
m0
m1ρc (T − f (c))
) (u + J
ρc
)}
= − l0
τDa0mμ(T, c,)Ju + l0
τT a0eμ(ρ, T, c,).
Equations (16)1 − 3 represent conservation laws of mass, momentum, and energy of the mixture,while (16)4 − 6 are the balance laws of mass, momentum, and energy of the constituent 1. Auxiliaryfunctions in source terms read
mμ(T, c,) = T + [1 − c − f (c)]
[T − f (c)][T + (1 − f (c))]T,
eμ(ρ, T, c,) = 1
γ (γ − 1)
m0
m1
m
m2
ρc(1 − c)T 2
[T − f (c)][T + (1 − f (c))],
and ratios of masses can be expressed as
m0
m= c + μ(1 − c)
c0 + μ(1 − c0),
m0
m1= 1
c0 + μ(1 − c0),
m
m2= μ
c + μ(1 − c).
In (16) we also used abbreviation 1/β for the following expression:
1
β= 1
γ − 1
[m0
m1(1 − μ)T − m0
m1
m
m2
]+ J 2
2ρ2
[1
c2− 1
(1 − c)2
],
where β can be interpreted as thermal inertia.6 One may note that shock structure equations (16)comprise only two dimensionless groups:
�D = l0
τDa0, �T = l0
τT a0.
B. An outline of the numerical procedure
It was shown by Gilbarg and Paolucci,23 and exploited later on in numerous studies, that theshock structure problem can be approached using dynamical systems theory. The system (16) can
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106102-8 Madjarevic, Ruggeri, and Simic Phys. Fluids 26, 106102 (2014)
formally be written as
d
dξF(U) = P(U) ⇔ A(U)
dUdξ
= P(U), (17)
where U = (ρ, u, T, c, J,)T is the column vector of state variables, F and P are column vectorsof fluxes and source terms, and A(U) = DF is Jacobian matrix. Equilibrium states of the mediumcorrespond to the stationary points of the system, P(U) = 0. Thus, the shock structure can be regardedas a continuous solution—heteroclinic orbit—which tends to stationary points U0 �= U1 at infinity:
limξ→−∞
U(ξ ) = U0, limξ→∞
U(ξ ) = U1, limξ→±∞
U′(ξ ) = 0. (18)
These are actual boundary conditions adjoined to the system (17) as a part of the shock structureproblem.
Mere observation of the source terms in (16) yields that diffusion flux and diffusion temperaturevanish both in upstream and downstream stationary points, i.e., J0 = J1 = 0, 0 = 1 = 0. Thus,for a given upstream equilibrium state U0 one may determine the downstream equilibrium state U1
by integration of the conservative part of the system (16)1 − 4. Nontrivial solution in dimensionlessform reads
U0 =
⎡⎢⎢⎢⎢⎢⎢⎣
ρ0
u0
T0
c0
J0
0
⎤⎥⎥⎥⎥⎥⎥⎦
=
⎡⎢⎢⎢⎢⎢⎢⎣
1M0
1c0
00
⎤⎥⎥⎥⎥⎥⎥⎦
, U1 =
⎡⎢⎢⎢⎢⎢⎢⎣
ρ1
u1
T1
c1
J1
1
⎤⎥⎥⎥⎥⎥⎥⎦
=
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣
4M20
3+M20
3+M20
4M0
116
(14 − 3
M20
+ 5M20
)c0
00
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦
. (19)
Note that relations between mixture state variables, ρ1, u1, T1 and ρ0, u0, T0, correspond to the solu-tion of usual Rankine-Hugoniot equations between the state variables at the shock wave for a singlefluid. On the other hand, constituent-related state variables c, J, and have the same equilibriumvalues in front and behind the shock. Since diffusion flux J and diffusion temperature vanish inequilibrium, they can be regarded as genuine non-equilibrium variables. Also, concentration is thesame in both equilibrium states, c1 = c0, and in the sequel it will be termed equilibrium concentra-tion, without special regard to upstream or downstream state. Finally, downstream equilibrium canbe regarded as a one-parameter family of states parametrized by Mach number (i.e., shock speed),U1 = U1(U0, M0). In such a way, the third parameter—Mach number—is naturally introduced inthe model through the boundary conditions.
In the boundary-value problem (17)-(18) the number of boundary conditions is twice the numberof equations (condition (18)3 just secures that U0 and U1 are equilibrium states). Nevertheless, theproblem is well-posed since we seek for a particular solution—heteroclinic orbit. The problem canbe approached numerically in two different ways: as a boundary-value problem, or as an initial-valueproblem. We shall pursue in the latter direction.
The numerical procedure has few crucial points. First, it was shown28 that stationary pointschange their character when the shock parameter—Mach number—crosses certain critical value. Thecritical value, M0 = 1, corresponds to the highest characteristic speed of the equilibrium subsystem,here consisted of the conservation laws for the mixture. Actually, if the highest characteristicspeed is genuinely nonlinear (in the sense of theory of hyperbolic systems of conservation laws),it was shown29 that there is a single eigenvalue of the system (17), linearized at stationary point,which changes the sign. Furthermore, upstream equilibrium U0 is saddle point (in a generalizedsense), while downstream equilibrium U1 is stable node. Thus, one may “follow” the direction ofthe eigenvector corresponding to a positive eigenvalue in U0 and asymptotically reach the stablestationary point U1. Since the domain on which the heteroclinic orbit is defined is infinite, −∞ < ξ
< ∞, the integration should be performed on a truncated domain, ξ ∈ [ξ 0, ξ 1], sufficiently large tosecure that terminal values U(ξ0) and U(ξ1) lie in small neighborhoods of the stationary points U0
and U1, respectively.
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106102-9 Madjarevic, Ruggeri, and Simic Phys. Fluids 26, 106102 (2014)
C. On existence of the shock structure
The procedure described above can be safely applied if the solution of the shock structureequations exists. The existence of solution is analyzed, among others, by Gilbarg24 in the context ofcontinuum theory of fluids, and by Yong and Zumbrun25 for hyperbolic systems with relaxation. Inthe context of extended thermodynamics it was particularly studied by Weiss,26 while new insightin the problem for moment equations was recently given by Myong.27 As mentioned above, it wasrecently put in the context of stability and bifurcation analysis of stationary points by Simic.28, 29
However, it was indicated by Weiss,26 and proved by Boillat and Ruggeri,30 that continuous shockstructure does not exist if the shock speed is greater than the highest characteristic speed of the fullsystem in upstream equilibrium. In the case of mixtures the problem of non-existence becomes moredelicate because of several parameters which influence the solution.
To analyze the existence of solution in binary mixture we have to determine the characteristicspeeds of the full system in equilibrium, where both constituents have common velocity v andtemperature T. They read (in increasing order)
λ(1) = v −√
γkB
m1T , λ(2) = v −
√γ
kB
m2T ,
λ(3) = λ(4) = 0,
λ(5) = v +√
γkB
m2T , λ(6) = v +
√γ
kB
m1T ,
while characteristic speeds of equilibrium subsystem are (in increasing order)
ν(1) = v −√
γkB
m0T , ν(2) = 0, ν(3) = v +
√γ
kB
m0T .
By Theorem 3.2 in Ref. 31, the subcharacteristic conditions hold
λ(1) < ν(1), ν(3) < λ(6).
Admissible equilibrium states (19) are determined by the Lax condition ν(1)1 < s < ν
(1)0 (or ν
(3)0 <
s < ν(3)1 ), which can be equivalently expressed as M0 > 1 (or M0 < −1). However, non-existence of
the smooth shock structure26, 30 is related to the condition s < λ(1)0 (or s > λ
(6)0 ).
Characteristic speeds of the full system correspond to the singularities of the matrix A(U), i.e.,det A(U) = 0. Therefore, critical values of Mach number (e.g., shock speed) for which smooth shockstructure ceases to exist can be determined by the condition det A(U0) = 0. Non-existence conditionthen reads
s < λ(1)0 ⇔ M0 > M (0)
0crit =√
1
c0 + (1 − c0)μ,
which corresponds to regions II and III in parameter space shown in Figure 1(a). However, singularityof A can be reached in a downstream equilibrium as well, det A(U1) = 0, for M0 < M (0)
0crit. Namely,the following situation may occur:
λ(1)0 < s < λ
(2)1 ⇔ M (0)
0crit > M0 > M (1)0crit =
√4μ + 3(1 − μ)c0
4μ − (1 − μ)c0,
which corresponds to region IV in Figure 1(a). In this case the singularity which appears in down-stream equilibrium moves into the domain as M0 is increased, but does not prevent the existence ofthe smooth shock structure since it is a regular singularity.26 The proof of this statement in the caseof mixtures is neither at our disposal at the moment, nor it can be proved numerically by the initialvalue strategy described above.
In this study we will restrict our attention to cases in which M0 < M0crit = min{M (0)0crit, M (1)
0crit}and numerical solution can be obtained as solution of initial value problem (Figure 1(a), region I;
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106102-10 Madjarevic, Ruggeri, and Simic Phys. Fluids 26, 106102 (2014)
0 0.2 0.4 0.6 0.8 11
1.5
2
c0
M0cr
it
μ = 0.1
I
II
III
IV
M(0)0critM
(1)0crit
0 0.2 0.4 0.6 0.8 11
1.2
1.4
1.6
1.8
2
c0
M0cri
t
μ
(a) (b)
FIG. 1. Critical values of Mach number M0 induced by the model. (a) Regions in parameter space (c0, M0crit) determined bythe singularities of the matrix A(U). (b) Regions with M0 < M0crit for increasing mass ratio μ. Mass ratio is increased fromμ = 0.1 with an increment 0.1. Arrow indicates graphs for increasing values of μ.
the dependence of this region on the value of mass ratio is shown in Figure 1(b)). This will give us apossibility to analyze systematically two main properties of the shock structure: the shock thicknessand the temperature overshoot of heavier constituent. To that end we created a database of shockprofiles obtained for 4394 different combinations of the parameters μ, c0, and M0. As far as we areconcerned, no such systematic analysis was performed so far in the study of binary mixtures.
IV. A FIRST LOOK ON THE SHOCK STRUCTURE
The aim of this study is to analyze the shock structure in terms of parameters which appear inthe model—mass ratio μ, upstream Mach number M0, and equilibrium concentration c0. To grasp abit of intricacy of the problem, we shall show the shock structures obtained by numerical solution ofthe model for two different sets of parameters, taking care about the restriction M0 < M0crit (M0crit
= 1.62 in case (a), M0crit = 1.86 in case (b)).In both cases a monotonicity of the mixture state variables u (Figure 2) and T (Figure 3) can
be observed. Also, there is a lag in velocity profile of heavier component in either case, and theshock thickness is obviously different. Perhaps the most striking phenomenon is appearance of thetemperature overshoot—existence of the profile region where the temperature of heavier constituentraises above the downstream equilibrium temperature (Figure 3(b)). This phenomenon was alreadyobserved in calculations based upon Boltzmann equations for mixtures,12, 13, 17, 32 as well as DSMC.18
Its occurrence is also anticipated, in a certain sense, in experimental studies.16
How does the shock structure, shock thickness, and temperature overshoot particularly, dependon the parameters μ, M0, and c0? Boltzmann equations and DSMC are doubtlessly accurate but time-consuming. The results they provide are informative but not conclusive since based upon collection
0 50 100 150 200 2500.9
1
1.1
1.2
ξ
uu1u2
0 20 40 600.8
1
1.2
1.4
1.6
ξ
uu1u2
(a) (b)
FIG. 2. Velocity profiles in the shock structure (u – average velocity of the mixture, u1 – velocity of the lighter constituent,u2 – velocity of the heavier constituent): (a) M0 = 1.2, c0 = 0.35, μ = 0.05; (b) M0 = 1.6, c0 = 0.21, μ = 0.1.
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106102-11 Madjarevic, Ruggeri, and Simic Phys. Fluids 26, 106102 (2014)
0 50 100 150 200 2501
1.05
1.1
1.15
1.2
ξ
TT1
T2
0 20 40 601
1.2
1.4
1.6
ξ
TT1
T2
(a) (b)
FIG. 3. Temperature profiles in the shock structure (T – average temperature of the mixture, T1 – temperature of thelighter constituent, T2 – temperature of the heavier constituent): (a) M0 = 1.2, c0 = 0.35, μ = 0.05; (b) M0 = 1.6,c0 = 0.21, μ = 0.1.
of particular cases. At the same time, the model proposed by ET proved itself as accurate enoughand easy for implementation. This makes it quite promising tool in the search for more completeanswers (Fig. 4).
V. SHOCK THICKNESS AND KNUDSEN NUMBER
One of the parameters which describe the shock structure globally is the shock thickness. It isusually defined as follows:
δ =∣∣∣∣ u1 − u0
(du/dξ )max
∣∣∣∣ ,where u is chosen state variable. Since velocity field is typically used for calculation of the shockthickness, we shall take the average velocity of the mixture as the chosen state variable. It is importantto notice that dimensionless shock thickness is equal to the reciprocal of the Knudsen number,
δ = δ
l0= 1
Kn, (20)
whose value helps to distinguish between different flow regimes. In view of (20), shock thicknesswill carry also the information about the flow regime. In this section we shall analyze the shockthickness (and Knudsen number) in terms of mass ratio, Mach number, and upstream concentration.
The dependence of Knudsen number on Mach number is monotonous for fixed mass ratioand upstream concentration (Figure 5). It increases with the increase of the Mach number, whichamounts to a decrease of the shock thickness. This is rather expected result which resonates with
0 50 100 150 200 250
-0.01
-0.005
0
0.005
ξ
JΘ
0 10 20 30 40 50 60
-0.1
-0.05
0
0.05
ξ
JΘ
(a) (b)
FIG. 4. Profiles of diffusion flux J and diffusion temperature in the shock structure: (a) M0 = 1.2, c0 = 0.35, μ = 0.05;(b) M0 = 1.6, c0 = 0.21, μ = 0.1.
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106102-12 Madjarevic, Ruggeri, and Simic Phys. Fluids 26, 106102 (2014)
0 0.1 0.2 0.3 0.40
0.05
0.1
0.15
0.2
μ
Kn
c0 = 0.21
M0
0 0.1 0.2 0.3 0.4 0.5 0.60
0.1
0.2
0.3
0.4
μ
Kn
c0 = 0.41
M0
0 0.1 0.2 0.3 0.4 0.50
0.05
0.1
0.15
0.2
0.25
μ
Kn
c0 = 0.59
M0 = 1.1M0 = 1.2
0 0.1 0.2 0.3 0.4 0.50
0.05
0.1
0.15
0.2
0.25
0.3
μ
Kn
c0 = 0.67
M0 = 1.1M0 = 1.2
(a) (b)
(c) (d)
FIG. 5. Dependence of Kn on mass ratio μ for fixed mass concentration c0. Mach number is increased from M0 = 1.1 withan increment 0.1. Arrow indicates graphs for increasing values of M0.
the behaviour of a single-component gas. However, experimental facts about shock structure in asingle fluid, as well as comparative study based upon Navies-Stokes-Fourier model, reveals that thistendency seem to be opposite for larger Mach numbers. Since our calculations are confined to smallMach number flows, at most M0 ≤ 2.0, the results obtained here are in agreement with the singlefluid model in this range.
Knudsen number increases monotonically also with the increase of the mass ratio, for fixed Machnumber and upstream concentration (Figure 5). In other words, the smaller the mass difference, thesmaller the shock thickness. This outcome provides a new insight on the shock structure in mixtures.Namely, for smaller mass ratio, e.g., μ ≤ 0.1, the shock structure is wider than for μ → 1 at fixed M0
and c0. The latter case corresponds to the constituents with similar masses, which therefore behavemuch like a single-component gas. Increase of c0 obviously makes the dependence of Knudsennumber on mass ratio steeper, but the tendency is preserved. On the other hand, in mixtures withlarger mass discrepancy between the constituents (smaller μ) the thickness is increased. These factswill be put in a broader context after the following paragraph.
The most significant, and perhaps the most intriguing result is obtained when Knudsen numberis calculated in terms of equilibrium concentration, with Mach number and mass ratio maintainedfixed (Figure 6). For different combinations of M0 and μ the same pattern can be observed: Knudsennumber increases when c0 tends to extreme values, i.e., to 0 or to 1, whereas it has a single localminimum in between. From another perspective one may say that shock thickness decreases whenone of the constituents dominates, i.e., when the mixture behaves like a single-component gas. Onthe other hand, when genuine mixture is at hand, the shock thickness is increased, i.e., the shockprofile becomes wider.
This result can be attributed to the fact that, for c0 far from lower or upper bound, cross-collisionsbetween the molecules of different species cause the intense exchange of momentum and energybetween the constituents, described by the source terms. This mechanism becomes equally importantas self-collisions among the molecules of the same specie, and causes the endurance of relaxation ofstate variables towards the equilibrium. Consequently, more time (and space) is needed in a travelingwave to attain the equilibrium state, which increases the thickness of the shock.
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106102-13 Madjarevic, Ruggeri, and Simic Phys. Fluids 26, 106102 (2014)
0 0.2 0.4 0.6 0.80
0.005
0.01
0.015
0.02
c0
Kn
μ = 0.01
M0
0 0.2 0.4 0.6 0.80
0.01
0.02
0.03
0.04
c0
Kn
μ = 0.03
M0
0 0.2 0.4 0.6 0.80.01
0.03
0.05
0.07
0.09
c0
Kn
μ = 0.1
M0
0 0.2 0.4 0.6 0.80.05
0.1
0.15
0.2
0.25
c0
Kn
μ = 0.3
M0 = 1.1
M0 = 1.2
M0 = 1.3
(a) (b)
(c) (d)
FIG. 6. Dependence of Kn on mass concentration c0 for fixed mass ratio μ. Mach number is increased from M0 = 1.1 withan increment 0.1. Arrow indicates graphs for increasing values of M0.
The behaviour of Knudsen number for large mass ratios, described in previous paragraph, canbe put in the same framework with its dependence on the concentration in the neighborhood ofits limiting values. In either case we observe a single-constituent-like behaviour of the mixture,albeit due to different causes: in the former case due to small difference between the masses, andin the latter case due to the presence of one constituent only in traces. Although the causes aredifferent, they have the same consequence—the increase of Knudsen number, i.e., the decrease ofshock thickness.
Careful analysis of the data obtained by numerical integration the shock structure equationsyields that for a narrow region of low Mach numbers, as well as for very low mass ratio μ, wehave Kn ≤ 0.01 and the flow falls into a hydrodynamic regime. For the most of the values of theparameters M0, μ, and c0 the flow belongs to slip flow and so-called transition regime.
VI. TEMPERATURE OVERSHOOT
Temperature overshoot is one of the peculiarities of the shock structure in mixtures whoseconstituents have disparate masses. It manifests through existence of the region of non-zero widthwhere the temperature of one constituent raises above the terminal, i.e., downstream equilibriumtemperature of the mixture. This phenomenon was observed in numerical calculations based uponBoltzmann equations for mixtures12, 13, 32 and DSMC.18 Available experimental data do not pro-vide enough evidence to support numerical simulations, although Harnet and Muntz16 regard theovershoot of the parallel temperature of argon as the onset of the overshoot of its mean temperature.
In the studies mentioned above it was emphasized that temperature overshoot is the mostsignificant in the case of small molar fraction of heavier component. Abe and Oguchi17 offeredphysical explanation of this phenomenon. They stated that in the case of vanishingly small molefraction of heavier component the main structure of the shock wave is determined by the lighterone. This causes the deceleration of heavier component and, at the same time, conversion of kineticinto thermal energy. However, dissipation through conduction is slow process which cannot diffuse
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106102-14 Madjarevic, Ruggeri, and Simic Phys. Fluids 26, 106102 (2014)
0 0.04 0.08 0.12 0.160
0.01
0.03
0.05
0.07
μ
TO
c0 = 0.09
M0
0 0.1 0.2 0.3 0.4 0.50
0.1
0.2
μ
TO
c0 = 0.21
M0
0 0.2 0.4 0.60
0.05
0.1
0.15
0.2
μ
TO
c0 = 0.39
M0
M0 = 1.1
c0 = 0.79
0 0.1 0.20
0.05
0.15
0.25
μ
TO
(a) (b)
(c) (d)
FIG. 7. Dependence of TO on mass ratio μ for fixed mass concentration c0. Mach number is increased from M0 = 1.1 withan increment 0.1. Arrow indicates graphs for increasing values of M0.
thermal energy gained by deceleration. As a consequence, internal energy (temperature) of heaviercomponent is raised above the terminal one. In our model momentum and energy transfer throughviscosity and heat conduction are neglected. We are focused on dissipation caused by mutualexchange of momentum and energy between the constituents, where the most prominent role isplayed by their mass ratio μ. Thus, we shall examine the temperature overshoot from this perspective,analyzing its dependence on the mass ratio μ, as well as upstream Mach number M0 and equilibriumconcentration c0.
The central result of our study is concerned with the analysis of temperature overshoot (TO)defined as
TO = T max2 − T1
T1 − T0,
where T max2 denotes the maximum temperature of the heavier constituent within the profile, whereas
T0 and T1 are upstream and downstream equilibrium temperatures of the mixture, respectively. TOwill be analyzed in terms of mass ratio, with fixed concentration c0. Vast amount of numericalsimulations revealed two typical patterns (Figure 7). The first one, which appears for low Machnumbers, is characterized by the existence of the minimal value of the mass ratio below whichthe temperature overshoot does not occur. For mass ratios above this value, temperature overshootincreases with the increase of μ.
Different pattern appears when the Mach number is increased. It has an outstanding feature, notreported in previous studies, that temperature overshoot varies non-monotonically with mass ratio.Namely, there exists a value μ* of the mass ratio which determines the local minimum of temperatureovershoot. Since other studies were based upon limited number of numerical simulations, whichprovided information on certain particular cases only, this phenomenon remained unobserved thusfar.
The outstanding feature of non-monotonic behaviour of the temperature overshoot can be givenan explanation within the simplicity of our model. However, it has to be analyzed in conjunctionwith the information about the shock thickness provided in Sec. V.
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106102-15 Madjarevic, Ruggeri, and Simic Phys. Fluids 26, 106102 (2014)
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
c0
TO
μ = 0.01
M0
0 0.2 0.4 0.6 0.80
0.1
0.2
0.3
c0
TO
μ = 0.03
M0
0 0.2 0.4 0.6 0.80
0.02
0.04
0.06
0.08
c0
TO
μ = 0.1
M0
0 0.2 0.4 0.6 0.80
0.02
0.04
0.06
c0
TO
M0 = 1.1
M0 = 1.2
M0 = 1.3 μ = 0.3
(a) (b)
(c) (d)
FIG. 8. Dependence of TO on mass concentration c0 for fixed mass ratio μ. Mach number is increased from M0 = 1.1 withan increment 0.1. Arrow indicates graphs for increasing values of M0.
a. μ < μ∗. For μ < μ∗ temperature overshoot is increased due to large mass difference and lowKn. The flow is between hydrodynamic and slip flow regime, but the mass ratio is too small to yieldsufficient exchange of energy between the constituents which could attenuate TO.
b. μ > μ∗. For μ > μ∗ Knudsen number is increased, which puts the flow into transition regime.Although the masses of the constituents become comparable, the exchange of energy is prevented byrarefaction of the mixture, i.e., small number of cross-collisions which could cause it. Consequently,the temperature of heavier constituent cannot be attenuated, and temperature overshoot is increased.
Therefore, in a simplified model of MT mixtures, where viscosity and heat conductivity areneglected, small mutual exchange of energy between the constituents can be pointed out as mainphysical reason for the increase of temperature overshoot. It can occur for two reasons: (a) eitherthere is a large mass discrepancy between the constituents (small μ), or (b) the mixture became morerarefied.
Temperature overshoot can also be analyzed in terms of equilibrium concentration c0, forfixed mass ratio μ (Fig. 8). It is obvious that TO increases with c0, which corresponds to lowfraction of heavier component. This is in sound agreement with the known results obtained by othermethods. Interestingly enough, there is also a region of low values of c0, i.e., high fraction of heaviercomponent, for which TO exhibits a slight increase. For low values of Mach number one can alsoobserve that temperature overshoot does not have significant variations for a broad range of c0.
VII. REMARKS ABOUT THE AVERAGE TEMPERATURE
Although our main concern was the temperature overshoot, certain attention should also be paidto the average temperature of the mixture. Two remarks will be given in the sequel. The first onedevoted to the very definition of the average temperature, used in our study, and its comparison withkinetic definition. The second one provides a rough comparison of MT model with a ST one.
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106102-16 Madjarevic, Ruggeri, and Simic Phys. Fluids 26, 106102 (2014)
A. Contribution of diffusion to the average temperature
In the previous discussion we introduced the average temperature of the mixture T (7)2 accordingto the definition of the mixture internal energy (2)5 by using only its intrinsic (thermal) part (2)4:
ρεI (ρβ, T ) =2∑
α=1
ραεα(ρα, T ) =2∑
α=1
ραεα(ρα, Tα).
This assumption is equivalent to the fact that intrinsic internal energy of the multi-component mixturemust correspond to the intrinsic internal energy of the mixture with one macroscopic temperature.20
In the kinetic theory21 average temperature of the mixture Tkin can also be based upon completedefinition of internal energy (2)5, where kinetic energy of diffusion is also taken into account. Indimensionless form it reads
Tkin = cm
m1T1 + (1 − c)
m
m2T2 + 1
2
γ − 1
γ
J 2
ρ2c(1 − c), (21)
where we can recognize the following structure:
Tkin = T + Tdiff = T
(1 + Tdiff
T
). (22)
T is given by (7)2 and Tdiff is the part of the mixture temperature corresponding to the kinetic energyof diffusion.
In Figure 9, the influence of the diffusion part Tdiff on the mixture temperature definition Tkin iscompared to its intrinsic part T. From example given in Figure 9(a) we can observe that the influence
0 100 200 3000
1
2
3·10−3
ξ
Tdiff
/T
M0 = 1.2M0 = 1.3
M0 = 1.4
M0 = 1.5
M0 = 1.6 c0 = 0.35μ = 0.05
0 20 40 60 80 100 1200
0.5
1
1.5
·10−3
ξ
Tdiff
/T
M0 = 1.4c0 = 0.35
μ = 0.05
μ = 0.1
μ = 0.15
μ = 0.2
60 80 100 120
0
0.1
0.2
0.3
0.4
0.5·10−3
ξ
Tdiff
/T
0.110.210.31
M0 = 1.3μ = 0.07
c0
0 50 100 150−30
−20
−10
0·10−3
ξ
J
0.110.210.31
M0 = 1.3μ = 0.07
c0
(a) (b)
(c) (d)
FIG. 9. Influence of the diffusion part Tdiff on the average temperature T: (a) different Mach numbers M0; (b) different massratios μ; (c) different concentrations c0; (d) diffusion flux J for different concentrations c0.
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106102-17 Madjarevic, Ruggeri, and Simic Phys. Fluids 26, 106102 (2014)
TABLE I. Comparison between single-temperature (ST) and multi-temperature (MT) model – Maximum absolute differenceof velocities of the components |(u1 − u2)max | (M0 = 1.2, c0 = 0.11).
μ ST MT Knsingle Knmulti
0.05 0.0395 0.0394 0.0182 0.01840.10 0.0529 0.0527 0.0439 0.04510.15 0.0675 0.0668 0.0798 0.08430.20 0.0833 0.0817 0.1273 0.1441
of the Tdiff increases with the increase of the strength of the shock wave (i.e., Mach number M0). Thediffusive part Tdiff increases also with the increase of the mass ratio (Figure 9(b)) and equilibriumconcentration (Figure 9(c)). However, in the case of weak shock waves the fraction of the diffusionpart in the mixture temperature remains less then 0.3%. This result shows that relative contributionof the kinetic energy of diffusion to the average temperature is small, under conditions stated above,and permits determination of the average mixture temperature from the intrinsic part of internalenergy.
Increase of the diffusion part of the average temperature with the increase of the shockstrength (Mach number) can be explained through diffusion flux increase within the shock structure(Figure 9(d)). With the increase of the shock strength the shock thickness is reduced, and Knudsennumber is increased. The mixture becomes more rarefied and diffusion becomes important dissipa-tion mechanism. With the increase of the mass ratio influence of the diffusion increases, but is lesspronounced compared to the case when Mach number increases.
B. Comparison of the single and multi-temperature model
We will compare the results of our study with a single-temperature model of mixtures. Single-temperature assumption excludes exchange of internal energy as a relaxation mechanism.
The influence of this simplification is analyzed by velocity and (average) temperature differencecomparison within the shock structure in some particular cases. It appears that ST model provideslarger maximum differences of the velocities of the constituents, which can be regarded as a con-sequence of neglecting the internal energy exchange (Table I). This behavior becomes even moreprominent with large values of mass ratio, which can also be related to rarefaction of the mixture,i.e., the rise of Knudsen number. However, we have to bear in mind that for given values of theparameters this difference is less than 10−3.
In Figure 10 the average temperature of the MT model is compared with a mixture temperaturein the ST model for one set of parameters. It can be observed that there is no obvious differencebetween the two cases. However, this result does not imply that the temperatures of the constituents
50 60 70 80 90 100 110 120 1301
1.05
1.1
1.15
1.2
ξ
T (ST)T (MT)
c0 = 0.11μ = 0.1
M0 = 1.2
FIG. 10. Single-temperature and multi-temperature model – comparison of the mixture temperatures.
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106102-18 Madjarevic, Ruggeri, and Simic Phys. Fluids 26, 106102 (2014)
are irrelevant—it only shows that predictions about the mixture temperatures in these two modelsare almost indistinguishable. Nevertheless, if one wants to capture the effects of mutual exchangeof internal energy between the constituents as a relevant relaxation mechanism, the use of multi-temperature assumption is indispensable.
VIII. CONCLUSIONS
This paper studied the shock structure in macroscopic multi-temperature model of mixtures. Themodel itself was developed within the framework of extended thermodynamics. However, inherentlimitations of the theory dictated determination of the phenomenological coefficients, i.e., relaxationtimes, using the results of kinetic theory of gases. Previous results, concerned with good agreementwith experimental data, as well as lack of extensive analysis by other methods such as kinetic theoryand DSMC, motivated systematic study of the shock structure in binary mixture of ideal gases withnegligible viscosity and heat conductivity.
Two main features of the shock structure were analyzed: shock thickness (or the inverse ofKnudsen number) and temperature overshoot of heavier constituent. It was observed that shockthickness decreases with the increase of Mach number up to M0 = 2.0. This is in accordance withthe results for single-component gases. New results obtained in this study are consequences of themixture properties. Namely, for a fixed value of Mach number it was observed that shock thicknessdecreases under the following circumstances: (a) mass ratio μ = m1/m2 < 1 tends to its maximumvalue (μ → 1), or (b) concentration of one of the constituents is small, i.e., c0 → 0 or c0 → 1.In other words, as long as mixture behaves like a single-component gas, for one reason or another,the dissipation is weaker and the changes in state variables are more abrupt. On the contrary, when(a) the mass ratio is small (μ � 1), or (b) both constituents have non-negligible mass concentrations,the shock thickness is increased—the gas behaves like a genuine mixture and the dissipation is morepronounced.
Another feature—the temperature overshoot—has even more intricate behavior. It was observedthat, for certain values of Mach number and equilibrium mass concentration, there exists a criticalvalue of mass ratio μcrit below which there is no temperature overshoot. On the other hand, if itexists for all calculated values of μ, there exists a value μ* which determines the local minimum ofthe temperature overshoot. This outstanding feature has not been observed so far, mainly becauseof lack of extensive numerical evidence based upon other methods. On one hand, increase of thetemperature overshoot with the decrease of μ, from μ* towards 0, is intuitively expected for it isrelated to large mass discrepancy of the constituents. On the other hand, increase of temperatureovershoot with the increase of mass ratio for μ > μ* seems to be counterintuitive and reveals thatthere exists another physical reason, apart from large mass discrepancy, which is responsible for that.It was revealed through the simultaneous study of Knudsen number that the increase of mass ratioleads to a greater rarefaction of the mixture. This led us to a conclusion that either great discrepancybetween the masses of the constituents, or rarefaction of the mixture, causes smaller exchange ofinternal energy between the constituents due to small influence of cross-collisions. Consequently,slower relaxation occurs which prevents attenuation of the temperature of heavier constituent andcreates an overshoot. This is the main result of our systematic study, not reported in other studiesdevoted to the same subject.
The paper is concluded with remarks related to the contribution of kinetic energy of diffusionto the average temperature in the sense of kinetic theory and comparison with a single-temperaturemodel. This opens the problem of comparison of the model with more sophisticated ones. On onehand, the model was closed, in the sense of phenomenological coefficients, by the help of kinetictheory. On the other hand, for a reliable comparison with available results obtained by numericalsolution of the Boltzmann equations one has to take into account viscosity and heat conductivity—they are obtained for much larger values of Mach number than allowed by our inviscid approximation.Moreover, it is a long-term intention to make a deeper comparative study with corresponding MTmodel developed within kinetic theory,21 as well as to make a proper generalization to the MT mixtureof polyatomic gases, whose single-temperature counterpart was developed by Desvillettes, Monaco,and Salvarani.33 This delineates possible future research activity related to multi-temperature models.
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106102-19 Madjarevic, Ruggeri, and Simic Phys. Fluids 26, 106102 (2014)
ACKNOWLEDGMENTS
This work was supported (D. Madjarevic and S. Simic) by the Ministry of Education, Scienceand Technological Development, Republic of Serbia, through the project “Mechanics of nonlinearand dissipative systems – contemporary models, analysis, and applications,” Project No. ON174016,and by the Gruppo Nazionale per la Fisica Matematica (GNFM) of INdAM (T. Ruggeri). Part of theresearch was done during the stay of S. Simic at the University of Bologna, Italy, supported by the“JoinEU-SEE IV” scholarship scheme of Erasmus Mundus Partnerships Action 2.
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