A Boundary Value Problem in Extended

A Boundary Value Problem in Extended Thermodynamics – One-Dimensional Steady Flows with Heat Conduction

I-Shih Liu and M. A. Rincon Instituto de Matem´atica Universidade Federal do Rio de Janeiro 21945-970, Rio de Janeiro, Brazil Dedicated to Professor Ingo M¨ uller on the occasion of his 65th birthday Abstract One-dimensional steady flows with heat conduction, treated by the 13-moment theory of extended thermodynamics are considered. The usual well-posed boundary conditions for the corresponding problems in the Navier-Stokes-Fourier (NSF) theory are insufficient to give unique solutions. In order to have unique physically sensible solutions, minimization of the deviation of iterative approximations from the exact solutions, proposed in [1], is used as a criterion. Moreover, the solutions are shown to be invariant with respect to change of consistent boundary conditions – a requirement abide by our physical intuition. In the problems of plane shearing flow and Couette flow, the minimization with respect to two uncontrollable parameters is involved. The examples are carried out numerically, and the results are compared with the classical results of NSF theory.

1

Introduction

Boundary value problems in extended thermodynamics have become a subject of interest recently, because they usually required some extra boundary conditions which are not permitted in the corresponding problems in the classical theory. We refer to the boundary values of such conditions as uncontrollable parameters. Within the phenomenological theory of extended thermodynamics, it is then the problem of how to determine those uncontrollable parameters so as to have unique solutions consistent with the well-posed boundary conditions of the classical theory. A minimum entropy production criterion, known as the minimax principle, has been suggested in [2] for the determination of the uncontrollable parameters, however, as being pointed out in [1], it does not lead to sensible solutions consistent with our “physical intuition”. More precisely, in a onedimensional heat conduction problem, after the solution has been obtained with the criterion for given boundary values of temperature at both sides, one can also consider a complementary problem with a different set of well-posed boundary conditions, say, by prescribing the values of the temperature on one side and of the heat flux on the other side from the known solution. Intuitively, since the solution is unique for well-posed boundary conditions, if the criterion is sensible, then the solution of the complementary problem obtained from it should be the same as the original one – we refer to this as an invariance requirement with respect to change of consistent boundary conditions, or simply, as the consistency requirement. The minimax principle does not comply with the consistency requirement. A different criterion, which complies with the consistency requirement, has been proposed in [1] to study stationary heat conduction in extended thermodynamics of 14-moments – this is the simplest 1

interesting case for heat conduction, with only one uncontrollable parameter. The purpose of the present paper is to show the viability of the proposed criterion to a different problem in the theory of extended thermodynamics. Steady flows with heat conduction will be considered and for simplicity, we choose to work with the 13-moment theory, for which the problem involves two uncontrollable parameters. Examples are given for steady flows between two parallel plates kept at constant temperatures and moving with constant velocities. The velocity field is assumed to be parallel to the plate, so that in the NSF theory, the normal viscous stress and the heat flux in the direction of flow must vanish. However, the numerical results show that they do not vanish. The existence of the non-vanishing parallel heat flux and normal viscous stresses distinguishes the theory of extended thermodynamics and the classical theory. These features are well-known for plane Couette flows in the literature, and more recently, in molecular dynamics simulation [3], in direct Monte Carlo simulation [4] and in extended thermodynamics with minimax principle [5], to name some of them.

2

The Thirteen-Moment Theory

We consider a field theory in extended thermodynamics to account for the effect of viscosity and heat conduction. Such theories can be formulated with a system of equations for moments with increasing order for monatomic ideal gases ([6, 7, 8]). The first at hand is the 13-moment theory which takes as basic state variables, the mass density ρ, the velocity vi , the stress tensor Tij , and the heat flux qi .

2.1

Field Equations

The field equations for the basic fields (ρ, vi , Tij , qi ) are given by ∂ρ ∂ (ρvk ) = 0, + ∂t ∂xk ∂ ∂ (ρvi ) + (ρvi vk − Tik ) = 0, ∂t ∂xk ∂ ∂ (ρvi vj − Tij ) + (ρvi vj vk − 3T(ij vk) + ρijk ) = πij , ∂t ∂xk ∂ (ρvi vj vj − 3T(ij vj) + 2qi ) ∂t  ∂  ρvi vj vj vk − 6T(ij vj vk) + 4q(i vk) + 2ρijk vj + ρijjk = 3π(ij vj) + πijj . + ∂xk

(2.1)

The underlined terms are constitutive quantities depend on the field variables. We require the production density πij to be traceless, i.e., πii = 0, so that (2.1)1,2 together with the trace of the equation (2.1)3 ensure the conservation of mass, momentum, and energy. We have used the Cartesian tensor notation with summation index convention. The round brackets indicate symmetrization and the angular brackets indicate traceless symmetrization. The stress tensor can be written as Tij = −p δij + Thiji , where Thiji is the deviatoric stress satisfying Thiii = 0, and p is the (equilibrium) pressure giving by p=

k ρθ, m

for monatomic ideal gases, for which k is the Boltzmann constant, m is the mass of the gas molecule, and θ is the absolute temperature. One may replace (ρ, Tij ) with (p, θ, Thiji ) as field variables. 2

From [8], linear constitutive equations, for the underlined quantities in (2.1), are given by 2 (qi δjk + qj δki + qk δij ), 5 k ρijjk = θ(5p δik − 7 Thiki ), m 1 2 πhiji = Thiji , πijj = − qi . τ τ ρijk =

(2.2)

In the last two expressions for the production densities, only one relaxation time τ associated with both viscous and thermal properties is assumed. This corresponds to the BGK model [9] of the force interaction model in the kinetic theory. The choice is for convenience in the present paper, other more realistic models could be adopted, since solutions are mainly obtained by numerical methods. Insertion of (2.2) into (2.1) leads to a system of first-order partial differential equations, and we now proceed to solve them for one-dimensional steady flows with heat conduction.

2.2

One-Dimensional Flows with Heat Conduction

We shall be interested in steady processes, in which all fields depend only on y, and the velocity field is in the x–direction, i.e., v = (vx (y), 0, 0). In this case, the equation (2.1)1 is identically satisfied and the remaining 12 equations can be split into two sets:

and

dThxyi d d = 0, (p − Thyyi ) = 0, (qy − Thxyi vx ) = 0, dy dy dy   d 2 1 qx + (p − Thyyi )vx = Thxyi , dy 5 τ  1 d4 4 qy + Thxyi vx = − Thxxi , dy 15 3 τ  1 2 d8 qy + Thxyi vx = Thyyi , dy 15 3 τ  2 d k 14 7 θThxyi − qy vx + 3Thxyi vx2 = (qx − Thxxi vx ), dy m 5 τ  2 k 4 d k 5 p θ − 7 θThyyi + qx vx − Thyyi vx2 + p vx2 = − (qy − Thxyi vx ), dy m m 5 τ

(2.3)

dThyzi = 0, dy d 2  1 qz = Thyzi , dy 5 τ

(2.4)

d 1 (Thyzi vx ) = − Thxzi , dy τ  2 d k 7 θThyzi + Thyzi vx2 = (qz − Thxzi vx ). dy m τ

The second set of equations (2.4) are satisfied by Thxzi = 0,

Thyzi = 0,

qz = 0,

which we shall assume and hence we are left with a system of 8 equations (2.3) for the determination of 8 fields (p, vx , θ, Thxxi , Thyyi , Thxyi , qx , qy ). Part of the system (2.3) can be integrated easily, and the system can be simplified as Thxyi = S, p − Thyyi = P, 5 5 Sy + C1 , qx + P v x = 2 2τ 4 Thxxi = − Thyyi , 3

qy − Svx = Q, (2.5)

3

and

1 5 dvx = Thyyi , dy τ 6
 k dθ 2 dvx 12 1 5 1 (2.6) S − Q = C1 + Thyyi vx − P vx − Sy , m dy 5 dy τ 7 3 7 τ 2 k 4 k 2 Thyyi θ = 5 P θ + C1 vx − P vx2 + (Svx + Q)y − C2 . m m 5 τ Therefore, the general solution of this system contains 5 constants, P , S, Q, C 1 and C2 , as well as two more integration constants, say, v0 and θ0 , by further integration of the two remaining first order differential equations in (2.6)1,2 , after the substitution of Thyyi from (2.6)3 . S

Apparently, from a system of 8 differential equations of first order, there should be 8 integration constants, however, by the use of (2.3)3 , one of the equations (2.3)5,6 reduces to the simple relation (2.5)5 , without integration. Note that, all together, there are 7 parameters in the general solution as stated above.

2.3

Navier-Stokes-Fourier Theory

It is known that the classical Navier-Stokes-Fourier (NSF) theory may be regarded as the first order approximation of the extended field theories with higher moments. For comparison, let us also consider the one-dimensional flow in the classical theory. The constitutive equations are given by the following Navier-Stokes and Fourier laws for the stress and the heat flux: Thiji = 2µ

∂vhi , ∂xji

qi = −κ

∂θ . ∂xi

The viscosity coefficient µ and the thermal conductivity κ are related to the material parameters, in the BGK model of the extended thermodynamics for monatomic ideal gases, by µ = τ p,

κ=

5 k τ p. 2m

(2.7)

For one-dimensional flows with heat conduction considered in Sect. 2.2, since v = (v(y), 0, 0) and θ = θ(y), we have, for the NSF theory, Thxxi = Thyyi = Thxzi = Thyzi = 0,

qx = qz = 0.

The field equations consists of equations of balance of mass, momentum, and energy. The mass balance is identically satisfied, and the remaining equations now read  dv 2 d 2 vx d2 θ dp x = 0, µ = 0, κ + µ = 0, dy dy 2 dy 2 dy and the general solution is given by p(y) = P,

vx (y) =

S y + v0 , µ

θ(y) = −

1 S2 2 1 y − (Q + Sv0 )y + θ0 . 2 µκ κ

(2.8)

There are 5 integration constants: P , S, Q, v0 , and θ0 . These parameters can be determined from the boundary conditions.

3

Boundary Value Problems

Mathematically, the parameters in the general solution must be determined from boundary conditions. However, in practise, not only the assignment of boundary values must be realistic in the physical sense, arbitrary assignment of boundary conditions may also result in senseless solutions or no solution at all. 4

3.1

Physical Boundary Conditions

We consider a solid boundary with an exterior unit normal ni . There are two types of boundary conditions we can impose, namely, the jump conditions of mass, momentum, and energy, and the second type concerns the continuity of the temperature and the velocity fields at the boundary. Jump Conditions The gas is assumed not to penetrate a solid boundary, therefore, the normal velocity components must be continuous at the boundary. In this case, there is no jump of mass flux, while the jump conditions of momentum and energy, at the boundary, can be written as Tij nj = Fi ,

(qi − Tij vj )ni = Q.

(3.1)

Here, we have assumed that there is an energy flux Q as well as a force Fi acting on the boundary. They are controllable physical quantities at the boundary. Adherence Conditions The temperature of the boundary surface is assumed to be the same as the temperature of the gas attached to it. In other words, the temperature is assumed to be continuous across the boundary. We shall also assume that material particles adhere to the surface of a solid boundary. Therefore, not only the normal component but also the tangential component of the velocity are continuous across the boundary. Sometimes temperature jump and velocity slip at the wall are considered in rarefied gases (see [4]), no such features are taken into account in the present paper, because some specific assumptions on the jump and slip conditions would have to be postulated. However, such conditions, if incorporated, would not influence the proposed criterion.

S

y 6

Q -- vx (y) -

0 vx (0) = v0

vx (L) = vL

y 6 -x

0

θ(L) = θL - vx (y) -x

vx (0) = v0

θ(0) = θ0

θ(0) = θ0

Problem II

Problem I

Figure 1: One-dimensional flow with heat conduction

3.2

Essential Boundary Conditions and Uncontrollable Parameters

We shall consider one-dimensional flows between two movable plates, at y = 0 and y = L, as shown in Fig. 1. In this case, from the jump conditions at the boundary given by (3.1), the solution (2.5) implies that the parameter Q is the outward energy flux, while P is the normal pressure, and S is the shear force, per unit area, i.e., they are the normal and the tangential components of F i , respectively. The adherence conditions also permit us to prescribe the temperature and the velocity of the boundary plate.

5

With constant pressure P of the gas given, we consider two typical boundary value problems, shearing flow and Couette flow, by prescribing the following controllable boundary values (see Fig. 1): Problem I : at y = 0 : at y = L :

v x = v0 ,

θ = θ0 ;

(3.2)

Thxyi = S,

qy − Svx = Q.

v x = v0 , v x = vL ,

θ = θ0 ; θ = θL .

Problem II : at y = 0 : at y = L :

(3.3)

Note that either of these problems involves 5 parameters. Therefore, it is insufficient for the determination of unique solutions in the 13-moment theory given by (2.5) and (2.6). On the other hand it is well-known that, in the classical NSF theory, these boundary conditions are well-posed in the sense that they determine unique solutions. We shall call the boundary conditions essential if they are well-posed for the corresponding problem in the classical theory. In Navier-Stokes-Fourier Theory Indeed, in NSF theory, the solution for Prob. I is given by (2.8), S y, µ 1 S2 2 1 θI (y) = θ0 − y − (Q + Sv0 )y; 2 µκ κ vxI (y) = v0 +

(3.4)

and from it, one can easily obtain the solution of Prob. II, y vxII (y) = v0 + (vL − v0 ) , L  y2 1µ y y II (vL − v0 )2 2 − θ (y) = θ0 + (θL − θ0 ) − . L 2κ L L

(3.5)

We remark that although S, Q, vx , and θ are all physically controllable quantities on the boundary, they can not be arbitrary prescribed at the same boundary. In particular, it is ill-posed if S and v x , or Q and θ, are arbitrarily prescribed at y = L, since there may not have solution at all. Uncontrollable Parameters in the 13-Moment Theory The general solution of the 13-moment theory, given by (2.5) and (2.6), contains 7 parameters, say, P, S, Q, vx (0), θ(0), vx (L), θ(L). Hence, mathematically, besides the essential boundary conditions, we can obtain a unique solution by prescribing values to two more parameters, say, c1 and c2 — Note that, for Prob. I, we can take c1 = vx (L) and c2 = θ(L); and for Prob. II, c1 = S and c2 = Q. In general, there will be infinitely many solutions, depending on two parameters, satisfying the essential boundary conditions for the well-posedness of the classical solution. This, of course, does not meet our physical intuition, because, in particular, fair agreement with NSF theory is expected for flows with small Knudsen number – the ratio of the mean free path of the gas particles to the characteristic length of the problem.1 1 Rarefied gas flow are usually characterized by the Knudsen number, which could be defined, in the present context, pk as Kn = τ θ/L. m

6

Therefore, the values of c1 and c2 , should not be imposed and we call them uncontrollable parameters. To solve the boundary value problem, it remains to find a criterion to determine the uncontrollable parameters, so that the essential boundary conditions will also be “well-posed” for the 13-moment theory. Consistency of Complementary Problem Suppose we have solved Prob. I, and the solution is denoted by vxI (y) and θ I (y), then we can consider a complementary boundary value problem with different essential boundary conditions consistent with the solution. For example, we can prescribe the values, vx = vxI (L) and θ = θ I (L) at y = L. This complementary problem is a type II problem and the solution will be denoted by v xII (y) and θ II (y). For NSF theory, from (3.4) and (3.5), it follows immediately that vxI (y) = vxII (y) and θ I (y) = θ II (y). Of course, this is also true for a different complementary problem with other type of consistent essential boundary conditions. Similarly, for Prob. II, we can consider a complementary problem by prescribing S = S II and Q = QII at y = L, where S II and QII are determined from the condition (3.1) for the solution of Prob. II. Then the complementary problem will be consistent with the original problem in NSF theory, since essential boundary conditions are well-posed. In other words, the solution of the NSF theory is invariant with respect to change of consistent boundary conditions. However, for the 13-moment theory, since essential boundary conditions do not uniquely determine the solution, the consistency will be regarded as an ultimate requirement, in establishing a sensible criterion for the determination of uncontrollable parameters. We refer to this as an invariance requirement with respect to change of consistent boundary conditions, or simply, as the consistency requirement.

4

Determination of Uncontrollable Parameters

A criterion, known as minimax principle, was proposed in [2] to determine the uncontrollable parameters that minimize the entropy production. In was pointed out in [1] that the minimax principle has a serious drawback, since it does not comply with the consistency requirement. A different procedure that takes into account the consistency requirement is proposed in [1] that determines the uncontrollable parameters by minimizing the deviation of approximated solutions, based on a differential iterative scheme akin to the Maxwellian iteration in the kinetic theory (see [10]).

4.1

Iterative Approximation

The iterative scheme, for the non-equilibrium state variables (Thxyi , Thxxi , Thyyi , qx , qy ), are based on the relations (2.3). They read (n)

2 (n−1) d P vx + qx , dy 5  (n) d  4 (n−1) 4 (n−1) qy + T hxxi = −τ T hxyi vx , dy 15 3   (n) d 8 (n−1) 2 (n−1) T hyyi = τ qy + T hxyi vx , dy 15 3  (n) (n) 3 (n−1) 2 7 (n−1)  d 7 k (n−1) T hxyi vx − q x − T hxxi vx = τ θ T hxyi + q y vx , dy 2 m 2 5 (n) (n) 1 k (n−1) 2 (n−1)  d 5 k P θ + P vx2 − θ T hyyi + q x vx , q y − T hxyi vx = −τ dy 2 m 2 m 5 T hxyi = τ

7

(4.1)

for n = 1, 2, 3, · · ·, with initialization: (0)

(0)

(0)

(0) qx

Thxyi = Thxxi = Thyyi = 0,

(0)

= q y = 0.

The first two iterates can be carried out easily. They read (1)

(1)

Thxyi = P τ

Thxxi = Thyyi = 0,

dvx , dy (1) k dθ 5 , qy = − P τ 2 m dy and

(1)

(1) qx

(4.2) = 0;

(2)

dvx , dy  k d2 θ (2) dvx dvx d 2 vx  2 − 2 − 2v , Thxxi = P τ 2 x 3 m dy 2 dy dy dy 2  k d2 θ (2) d 2 vx  dvx dvx 2 − − v Thyyi = − P τ 2 2 , x 3 m dy 2 dy dy dy 2  k dθ dv (2) 25 k 5 dvx dvx 7 k d 2 vx d2 θ 1 d 2 vx  x + θ 2 + vx 2 + vx + vx2 2 , qx = P τ 2 7 m dy dy 2 m dy 6 m dy 3 dy dy 6 dy (2) 5 k dθ qy = − Pτ . 2m dy Thxyi = P τ

(4.3)

Higher iterates become increasingly complicated, but they can be easily obtained by using an available computer software (e.g., MAPLE, used by the authors). Note that the first iterate leads to the laws of Navier-Stokes and Fourier. However, we shall not regard the iterates as approximate constitutive equations, as the Maxwellian-iteration procedure does, in relating the kinetic theory of moment equations to the usual theory of ordinary thermodynamics. Instead, since, from (4.2) and (4.3), the iterates depend on vx , θ and their derivatives only, we can evaluate their values from the solutions, vx (y; c1 , c2 ) and θ(y; c1 , c2 ), of the 13-moment equations, satisfying the essential boundary conditions. The dependence on the uncontrollable parameters c 1 and c2 is indicated explicitly.

4.2

Criterion of Minimum Iterative Difference

For simplicity, let u = (p, θ, vx , Thxyi , Thxxi , Thyyi , qx , qy ) and denote by (n)

u (y; c1 , c2 )

and

u(y; c1 , c2 ),

respectively, the n-th iterated approximation and the solution of the 13-moment equations, satisfying the essential boundary conditions for given values of c1 and c2 , as indicated. (n)

There may be considerable difference between u and u for arbitrary values of uncontrollable parameters c1 and c2 . However, following the idea of Maxwellian iteration, the iterates are meant to be approximations of the solution, so that, for certain values of these parameters, it must be “close” to u if the solution is physically sensible. As a measure of closeness, we introduce the following norm, ku(c1 , c2 )k =

nZ

L

σ(u(y; c1 , c2 )) dy 0

8

o1/2

,

(4.4)

where σ(p, θ, vx , Thiji , qi ) = a1 p2 + a2 θ2 + a3 vx2 + a4 Thiji Thiji + a5 qi qi .

(4.5)

This is equivalent to the L2 norm in IR8 , the space of (p, θ, vx , Thiji , qi ), with appropriate positive constant coefficients for different physical quantities. The coefficients will be adopted as a1 = a 4 =

1 , 2τ P θ0

a2 =

k τP 5m , 2 2L θ02

a3 =

τP , 2L2 θ0

a5 =

2 k 5m τ P θ02

(4.6)

.

so that all terms have the same physical dimension of the entropy production density (see [8]). Such a choice is for convenience only. Now, in order to find the sensible solution, we shall determine the uncontrollable parameters, by requiring that the iterates be as close as possible to the solution (see [1]). More specifically, we propose the following minimization problem: Criterion of Minimum Iterative Difference: Determine the uncontrollable parameters, cn1 and cn2 , by minimizing the difference of the n-th iterated approximation from the solution, i.e., fn (cn1 , cn2 ) = min fn (c1 , c2 ), c1 ,c2

where

(n)

fn (c1 , c2 ) = ku(c1 , c2 ) − u (c1 , c2 )k.

(4.7)

In practice, minimization can be carried out numerically, and if fn (cn1 , cn2 ) → 0 when n → ∞, then for m some m > 0 large enough, cm 1 and c2 can be regarded as good approximations of the uncontrollable parameters required for the uniqueness of the 13-moment solution. It is worthwhile to point out that the determination of uncontrollable parameters for each n, can be obtained independently from the results obtained for any other n. In other words, it is a straight determination for each n, and no iteration is needed, as long as the iterated approximations for each n have been calculated from the iterative scheme (4.1).

4.3

Convergence and Consistency

For each n, after the solution u(y; cn1 , cn2 ) has been obtained from minimization criterion, we can consider a complementary problem, with consistent essential boundary conditions, and denote its ˆ (y; cˆn1 , cˆn2 ). In general, the solution u and the complementary solution from minimization criterion by u ˆ may not be the same, nevertheless, we expect that in the limit they must be consistent, solution u i.e., ˆ (ˆ lim ku(cn1 , cn2 ) − u cn1 , cˆn2 )k = 0, (4.8) n→∞

if the solution obtained from our criterion is sensible. We have the following proposition: Proposition: Suppose that the minimization problem of the criterion has unique solution for c n1 and cn2 for every n, and lim fn (cn1 , cn2 ) = 0 n→∞

then the consistency requirement is satisfied in the limit as n → ∞. Proof: To be specific, we choose to consider the problem with the following essential boundary conditions: vx (0) = v0 , θ(0) = θ0 , (4.9) Thxyi (L) = S, qy (L) − Svx (L) = Q,

9

and therefore, we can take vL and θL as the two uncontrollable parameters. For each n, from the n n minimization criterion, we obtain the boundary values vx (L) = vL and θ(L) = θL satisfying n n fn (S, Q, vL ; θL ) = min fn (S, Q; vL , θL ), vL ,θL

(4.10)

where we have also included the constants S and Q as arguments of fn to emphasize the role of the boundary values given at y = L. In this manner, we can obtain a sequence of approximate solutions of the problem with boundary conditions (4.9). In addition to this sequence of solutions, we can also construct another sequence of approximate solutions for complementary problems with consistent boundary conditions, such that, for each n, vx (0) = v0 , vx (L) =

θ(0) = θ0 ,

n vL ,

n θ(L) = θL .

(4.11)

Note that the conditions at y = L are obtained from the minimization result (4.10) of the original problem for the same n. The complementary problem has as its uncontrollable parameters S and Q at y = L. Applying again the minimization criterion, we have n n n n , θL ), , θL ) = min fn (SL , QL ; vL fn (S n , Qn ; vL SL ,QL

(4.12)

n n are also indicated as arguments for direct comparison with for which, the boundary values vL and θL the relation (4.10).

It follows immediately from (4.10) and (4.12) that n n n n ; θL ), , θL ) ≤ fn (S, Q, vL fn (S n , Qn ; vL

and by the assumption that fn converges to 0, we have n n n n ; θL ) = 0. , θL ) = lim fn (S, Q, vL lim fn (S n , Qn ; vL n→∞

n→∞

Consequently, from the uniqueness assumption of the minimization problem, we obtain lim S n = S,

lim Qn = Q,

n→∞

n→∞

which implies that n n n n , θL )k = 0. lim ku(S, Q; vL , θL ) − u(S n , Qn ; vL

n→∞

This is the consistency condition (4.8) in the present case. Problems with other types of essential boundary conditions can be treated with similar arguments. 2 The validity of the assumptions, namely, the convergence of the n-th iterated approximation to the solution of moment equations as n increases, and the uniqueness of minima for every n, has not been proved. However, it will be verified numerically in the numerical examples presented in this paper. It is usually difficult to make an error estimate for the convergence of the sequence, without knowing the actual limit. On the other hand, in numerical calculations, it is easy to check the consistency from the following relative error between the original and its complementary problems: Err(n) =

ˆ (ˆ ku(cn1 , cn2 ) − u cn1 , cˆn2 )k . ku(cn1 , cn2 )k

(4.13)

In the numerical examples, the consistency error can be checked independently for any n, and the value of Err(n) will be regarded as a precision indicator for the approximate solution of the boundary value problem.

10

5

Numerical Results

We shall consider one-dimensional flow with heat conduction in an argon gas. The numerical data are given as follows: the molecular mass m = 6.64x10−26 Kg, the relaxation time τ = 10−5 s, the Boltzamnn constant k = 1.38x10−23 JK−1 , and the normal pressure P = 100 Pa. Since our purpose is to show the qualitative results of the problems, the convenient choice of the value of relaxation time may not confirm to the real property of the gas in each of the following examples.

Numerical procedures There are two types of boundary value problems, I and II as shown in Fig. 1. For type I problems, since S and Q are constants, they are regarded as prescribed as well, besides θ and v x , at y = 0. Therefore, the equations (2.6) can be solved directly by numerical integration from y = 0, for any given values of C1 and C2 . Second order Runge-Kutta scheme is used. For type II problems, since both θ and vx are prescribed at y = 0 and y = L, they are two-point boundary value problems. Shooting method with Newton-Raphson correction is used (see [11]). Every shooting is a type I problem which can be solved by numerical integration as before. Minimization is done with respect to the two parameters C1 and C2 of (2.6). Downhill simplex method is used (see [11]). The values of the uncontrollable parameters are then calculated from the solution after C1 and C2 for the minimum have been determined.

5.1

Example 1: Plane Shearing Flow

In this case, the shear force and the energy flux are applied at the upper plate, while the lower plate is held fixed at constant temperature, shown as Prob. I in Fig. 1. The boundary conditions are given with the following boundary values: vx (0) = 0 ms−1 ,

θ(0) = 300 K,

S = 10 Pa,

Q = −3000 Wm−2 ,

and the distance between the parallel plates L = 0.02 m. For consistency test, for each n, a complementary problem with boundary conditions (3.3) of type II will also be considered, for vx (0) = 0 ms−1 ,

θ(0) = 300 K,

n , vx (L) = vL

n , θ(L) = θL

n n where vL and θL are the values of vx (y) and θ(y) at y = L, obtained by the minimization criterion for the original problem.

Shearing flow

Consistency test n

n

n vL

n θL

fn

S

1 2 3 4

202.2130 202.2708 202.2709 202.2709

376.1108 376.1002 376.1003 376.1003

2.81x10−1 5.01x10−4 1.53x10−5 8.74x10−6

9.99496 10.0001 10.0000 10.0000

Qn

Err(n)%

-2999.5342 -2999.9961 -3000.0000 -3000.0000

1.48x10−2 3.16x10−4 9.96x10−7 1.66x10−7

Table 1: Numerical results of Example 1. The numerical results of minimization criterion are given in Table 1. It seems that the minimum of the differences fn tends to zero as n increases, while for the complementary problem, we have S n → 10 and Qn → −3000, which are consistent with the values S and Q prescribed in the original problem. 11

Note that the consistency error Err(n) is less than 0.0004% for n ≥ 2. Indeed, the solution converges quickly, and for n > 2, they are almost indistinguishable graphically. n n From Table 1, the uncontrollable parameters are now being determined as limits of v L and θL , −1 namely, vx (L) ≈ 202.27 ms and θ(L) ≈ 376.10 K. 380

250 ET13 NSF 200

360 150

θ

vx

340

100 320 50 ET13 NSF 300

0 0

0.005

0.01

0.015

0.02

0

0.005

0.01

0.015

0.02

y

y -500

ET13 800 -1000

-1500

600

qy

qx

-2000

400 -2500 NSF

ET13 NSF -3000

qx = 0

200 0

0.005

0.01

0.015

0.02

0

y

0.005

0.01

0.015

0.02

y

Figure 2: Comparison of the results of 13-moment theory and NSF theory for Example 1 In Fig. 2, the results of the 13-moment theory and the NSF theory are plotted together for comparison. Note that for the data chosen, they are close, except of course, the heat flux parallel to the plate qx , which must vanishes in the NSF theory. The viscous stresses Thxxi and Thyyi , which must also vanish in the NSF theory, have not been plotted here, since they are almost constant, T hxxi ≈ −1.617, Thyyi ≈ 1.213, and small compared with the normal pressure P = 100.

5.2

Example 2: Plane Couette Flow

For a plane Couette flow shown as Prob. II in Fig. 1, both the velocity and the temperature are prescribed at the upper and the lower plates. The boundary conditions are given with the following values: vx (0) = −200 ms−1 , θ(0) = 300 K, vx (L) = 200 ms−1 , θ(L) = 300 K, and L = 0.02 m. For each n, a complementary problem with boundary conditions (3.2) of the type I will be considered: vx (0) = −200 ms−1 , θ(0) = 300 K, S = S n , Q = Qn , where S n and Qn are the (constant) values, of Thxyi and qy − Thxyi vx respectively (see (2.5)1,3 ), obtained by the minimization criterion for the original problem. The numerical results of minimization criterion are given in Table 2. Again, the minimum difference n n fn tends to zero as n increases, while for the complementary problem, we have v L → 200 and θL → 300, which are the values vx (L) and θ(L) prescribed in the original problem. Note that the solution 12

Couette flow n

n

S

Q

1 2 3 4

19.1454 19.1694 19.1695 19.1695

n

0.194261 0.002414 0.002075 0.004288

Consistency test fn

n vL

n θL

Err(n)%

2.60x10−0 2.63x10−2 2.13x10−3 7.51x10−4

198.2558 200.0003 200.0004 200.0004

300.6301 299.9998 299.9998 299.9998

2.70x10−1 1.90x10−4 1.86x10−4 1.86x10−4

Table 2: Numerical results of Example 2. 340 ET13 NSF

ET13 NSF

200

330 100

θ

vx

320

0 310 -100 300 -200 0

0.005

0.01

0.015

0

0.02

0.005

0.01

0.015

0.02

y

y 4000

4000

ET13 NSF 2000

2000

qy

qx

0

0

-2000

-2000 ET13 NSF

-4000

-4000 0

0.005

0.01

0.015

0

0.02

0.005

0.01

0.015

0.02

y

y

Figure 3: Comparison of the results of 13-moment theory and NSF theory for Example 2 converges quickly, and for n > 2 the consistency error Err(n) is less than 0.0002%. From the numerical results, the uncontrollable parameters are determined as limits of S n and Qn , namely, S ≈ 19.17 Pa and Q ≈ 0.00 Wm−2 . The comparison with NSF theory is shown in Fig. 3. The solutions are symmetric with respect to the midpoint between the two boundary plates, due to the given boundary conditions that the plates are moving in the opposite directions and are maintained at the same temperature. The difference between the two theories are more pronounced, but the general features are qualitatively the same as those mentioned in the previous example. The velocity profiles of the 13-moment theory and the NSF theory are almost indistinguishable in Fig. 3 and we also have Thxxi ≈ −6.133, Thyyi ≈ 4.600, and hence the equilibrium pressure p ≈ 104.600, they are almost constant. The existence of the non-vanishing parallel heat flux and normal viscous stresses distinguishes the theory of extended thermodynamics and the classical theory. Most notably, the parallel heat flux q x , whose value is not negligible compared to the value of qy , has been reported in the literature. Our results are qualitatively in agreement with those obtained in [3, 4, 5].

13

100 ET13 NSF

302

75

50

300

vx

θ

25

298

0

-25

296

ET13 NSF

-50 0

0.0005

0.001

0.0015

0

0.002

0.0005

0.001

0.0015

0.002

y

y 4000 4000

ET13 NSF

2000

0

2000

qx -2000

qy 0

-4000

-2000

-6000

ET13 NSF

-8000 0

0.0005

0.001

0.0015

0

0.002

0.0005

0.001

0.0015

0.002

y

y

Figure 4: Comparison of the results of 13-moment theory and NSF theory for Example 3

5.3

Example 3

We consider the following boundary conditions: vx (0) = −50 ms−1 ,

θ(0) = 300 K,

S = 50 Pa,

Q = 0 Wm−2 .

The distance between the parallel plates is taken to be L = 0.002 m. Therefore, the Knudsen number is ten times that of the previous two examples. For this example, from (2.8), the classical NSF theory gives the solution of a Couette flow, symmetric with respect to the midpoint between the two parallel plates, with the following corresponding boundary values: vx (0) = −50 ms−1 ,

θ(0) = 300 K,

vx (L) = 50 ms−1 ,

θ(L) = 300 K,

(5.1)

as plotted in Fig. 4, which shows considerable difference from the numerical results of the 13-moment theory. Indeed, we obtain from the minimization of difference with 4-th iterated approximation, the uncontrollable parameters (compared with (5.1)) vx (L) = 93.03 ms−1 ,

θ(L) = 295.86 K.

(5.2)

These boundary values, vx (L) and θ(L), together with the values of vx (0) and θ(0), can then be assigned for solving the complementary problem with the minimization criterion for n = 4, in order to check the consistency of the solution with the original one. If the complementary results are plotted, they are almost indistinguishable from the results shown in Fig 4. As an estimate of accuracy, the consistency error has also been calculated. The value is Err(4) = 0.0857%, which we shall regard as satisfactory. We point out that once the sequence of iterated approximations, such as (4.2), (4.3), has been calculated, the minimization can be done for any n-th iterated approximation independently from the 14

previous ones. Indeed, unlike the previous examples, in the present one, only the minimization for n = 4 has been performed and the corresponding consistency error checked for accuracy. If better accuracy is required, further minimization with higher iterates can be taken.

5.4

Remark on the Equilibrium Pressure

In our examples, the numerical results seem to suggest that the viscous stress T hyyi converges to a constant, and hence so does the equilibrium pressure p. To take a more careful look at the limiting behavior from numerical results of Example 2, in Fig.5, we have plotted the difference of v x between the 13-moment theory and the NSF theory for n = 2, 3, 4, as well as Thyyi for n = 1, 2. It seems to confirm the constancy as far as it looks for increasing n. n=2 n=3 n=4

vx − vx (NSF)

0.001

n=1 n=2 4.7

Tyy

0

4.6

-0.001

4.5 0

0.005

0.01

0.015

0.02

0

y

0.005

0.01

0.015

0.02

y

Figure 5: Limiting behavior of vx − vx (NSF) and Thyyi In [4], the equilibrium pressure p is assumed to be constant in order to facilitate the solution. If this were true, one can readily determine one of the uncontrollable parameters. Indeed, by assuming constant p, the viscous stress Thyyi is also constant by (2.5)2 . Let k=

Thyyi , S

then integration of (2.6)1 for vx , and comparison of (2.6)2 with the derivative of (2.6)3 , lead to the following conditions: 5 y vx = vx (0) + ky = vx (0) + (vx (L) − vx (0)) , 6τ L (5.3) 25 2 4 2 5 2 3 5 2 S k − P Sk + P k − S k − 5P S = 0. 9 3 6 3 Therefore, the profile of vx is a straight line. For Example 1, with P = 100 and S = 10, we obtain from (5.3)2 that k = 0.1210. Hence Thyyi = 1.210 and from (5.3)1 , the uncontrollable parameter vx (L) = 201.6, which is about 0.3% difference from our numerical results. For Example 2, with vx (0) = −200 and vx (L) = 200, we obtain from (5.3)1 that k = 0.24 and from (5.3)2 the uncontrollable parameter S = 19.39, and hence Thyyi = 4.654. The difference from our numerical results is about 1%. Similarly, for Example 3, with P = 100, S = 50 and vx (0) = −50, we obtain k = 0.7625 from (5.3)2 and then from (5.3)1 , vx (L) = 77.08, which differs significantly from vx (L) = 93.03 of the numerical result (5.2)1 , a 17% error. Therefore, although for flows with small Knudsen number, the assumption for constant pressure seems to be a good approximation, in general, there is no reason to justify such an assumption. Indeed, for larger Knudsen numbers, this may result in considerable error, as we have seen from Example 3.

15

We also remark that the difference from a straight line of the velocity profile shown in Fig. 5 is not typical – its shape depends on the data. On the other hand, its magnitude seems to decrease as n increases, without substantially changing the sharp of the profile. This can be easily understood, because they are not the solutions of the equations of the iterated approximations; they are solutions of the 13-moment theory with the uncontrollable parameters determined in each n with fixed essential boundary conditions. 4

1x10−2 n = 4 n = 3 n = 2

3

fn (C1 , C2 )

1x10−3 2 1x10−4 1 n = 4 n = 3 n = 2

1x10−5

0 8.20x102

8.40x102

8.60x102

8.41x102

8.42x102

8.43x102

C1

C1 8x104

1x10−2 n = 4 n = 3 n = 2

fn (C1 , C2 )

6x104 1x10−3 4x104 1x10−4 2x104

n = 4 n = 3 n = 2

1x10−5

0 3.0x107

3.1x107

3.10021x107

3.2x107

C2

3.10023x107

3.10025x107

C2

Figure 6: Dependence of fn on the parameters C1 and C2 for Example 1

5.5

Remark on Iterative Approximations

It has been observed numerically, in [1] (Sec. 4.3), that the iterative approximations do not converge, to the solution of the extended theory, except for a particular value of the uncontrollable parameter – Although this has not been proved in a rigorous manner yet, it seems to explain why the proposed criterion is workable. This interesting feature of convergence can also be observed in the present examples. The values of fn , defined in (4.7) as the difference of the n-th iterated approximation from the 13-moment solution, can be represented as a surface over the plane of two uncontrollable parameters. In Fig. 6, vertical sections through the minimum of this surface will be presented. The data correspond to Example 1, where the uncontrollable parameters vx (L) = 202.27 and θ(L) = 376.10, for the minimum of fn at n = 4, are replaced by the corresponding values of C1 = 8.42012x102 and C2 = 3.10023x107 according to (2.6) – recall that C1 and C2 have been used as parameters in the actual numerical minimization. Therefore, the surface can be represented as fn (C1 , C2 ). Two profiles of the surface fn (C1 , C2 ) are presented in two sets of graphs on the left side of Fig. 6, the upper one with constant C2 = 3.10023x107 and the lower one with constant C1 = 8.42012x102 . We observe that curves are convex, so that unique minimum exists for every n = 2, 3, 4 (n = 1 has been omitted, because its value is nearly zero in the plotting scale). On the right side of Fig. 6, the corresponding curves are plotted again in the logscale so that the details near the minima can be seen. 16

It is clear from these profiles that except in the neighborhood of the point (C 1 , C2 ) = (8.42012x102 , 3.10023x107 ), the difference fn increases as n increases, and moreover, at this point fn tends to nearly zero. In other words, the iterative approximations converge only at this point to the solution of the 13-moment theory and diverge everywhere else. Therefore, the proposed criterion determines the point, which is the minimum of fn for large enough n to within a satisfactory accuracy. From examples in [1] and the present ones, the property of convergence at a particular point, seems to be a typical property of the differential iterations of boundary value problems with uncontrollable parameters. As a final remark, it seems worthwhile to mention the comment regarding Maxwellian iterations, by Truesdell in [10, 12], that “successive iterates by repeated differentiation is altogether untypical of methods for solving differential equations”, but nevertheless, “a differential iteration may converge to a special solution, the principal one”. Indeed, further studies of this “untypical” property of convergence of the differential iterations at a particular point will be of great interest for the proposed criterion. Acknowledgement: The author (ISL) gratefully acknowledges the support of CNPq-Brazil, through the Research Fellowship Proc. 300135/83-1.

References [1] Liu, I-Shih; Rincon, M. A.; M¨ uller, I.: Iterative approximation of stationary heat conduction in extended thermodynamics, Continuum Mech. Thermodyn. 14, 483-493 (2002). [2] Struchtrup, H.; Weiss, W.: Maximum of the local entropy production becomes minimal in stationary processes, Phy. Rev. Lett. 80, 5048-5051 (1998). [3] Risso, D.; Cordero, P.: Dilute gas Couette flow: theory and molecular dynamics simulation, Phys. Rev. E, 56, 489-496 (1998). [4] Marques Jr., W.; Kremer, G. M.: Couette flow from a thirteen field theory with slip and jump boundary conditions, Continuum Mech. Thermodyn. 13, 207-217 (2001). [5] Reitebuch, D.; Weiss, W.: Application of high moment theory to the plane Couette flow, Continuum Mech. Thermodyn. 11, 217-225 (1999). [6] Grad, H.: Principles of the kinetic theory of gases, Handbuch der Physik, Vol. XII, Ed. S. Fl¨ ugge, Springer (1958). [7] Liu, I-Shih; M¨ uller, I.: Extended thermodynamics of classical and degenerate gases, Arch. Rational Mech. Anal. 83, 285–332 (1983). [8] M¨ uller, I., Ruggeri, T.: Rational Extended Thermodynamics. 2nd edition, Tracts in Natural Philosophy 37, Springer, New York (1998). [9] Bhatnagar, P. L.; Gross, E. P.; Krook, M.: A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94. (1954). [10] Truesdell, C.: On the pressure and the flux of energy in a gas according to Maxwell’s Kinetic theory II, J. Rational Mech. Anal. 5, 55-128 (1956). [11] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P.: Numerical Recipes in C, The Art of Scientific Computing, 2nd Edition, Cambridge University Press, Cambridge New York (1992). [12] Truesdell, C., Muncaster, R. G.: Fundamentals of Maxwell’s Kinetic Theory of a Simple Monatomic Gas, Academic Press, New York (1980).

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