Boundary Layers and Shock Profiles for the Broadwell Model

Abstract

We consider the existence of nonlinear boundary layers and the typically nonlinear problem of existence of shock profiles for the Broadwell model, which is a simplified discrete velocity model for the Boltzmann equation. We find explicit expressions for the nonlinear boundary layers and the shock profiles. In spite of the few velocities used for the Broadwell model, the solutions are (at least partly) in qualitatively good agreement with the results for the discrete Boltzmann equation, that is the general discrete velocity model, and the full Boltzmann equation.

1. Introduction

The Boltzmann equation (BE) is a fundamental equation in kinetic theory. Half-space problems for the BE are of great importance in the study of the asymptotic behavior of the solutions of boundary value problems of the BE for small Knudsen numbers [1, 2] and have been extensively studied both for the full BE [3, 4] and for the discrete Boltzmann equation (DBE) [5–8]. The half-space problems provide the boundary conditions for the fluid-dynamic-type equations and Knudsen-layer corrections to the solution of the fluid-dynamic-type equations in a neighborhood of the boundary. In [8] nonlinear boundary layers for the DBE, the general discrete velocity model (DVM) was considered. Existence of weakly nonlinear boundary layers was proved. Here we exemplify the theory in [8] for a simplified model, the Broadwell model [9], where the whole machinery is actually not really needed, even if it helps out. For the nonlinear Broadwell model, we obtain explicit expressions for boundary layers near a wall moving with a constant speed. The number of conditions, on the assigned data for the outgoing particles at the boundary, needed for the existence of a unique (in a neighborhood of the assigned Maxwellian at infinity) solution of the problem is in complete agreement with the results in [8] for the DBE and [3] for the full BE. Here we also want to mention a series of papers studying initial boundary value problems for the Broadwell model using Green’s functions [10–16].

We also consider the question of existence of shock profiles [17, 18] for the same model [9, 19]. The shock profiles can then be seen as heteroclinic orbits connecting two singular points (Maxwellians) [20]. In [20] existence of shock profiles for the DBE in the case of weak shocks was proved. We exemplify the theory in [20] for the Broadwell model, where again the whole machinery is not really needed, even if it helps out. In this way we, in a new way, obtain similar explicit solutions, not only for weak shocks, as the ones obtained in [19] for the same problem.

The paper is organized as follows. In Section 2 we introduce the Broadwell model and find explicit expressions for the nonlinear boundary layers near a wall moving with a constant speed, and in Section 3 we find explicit expressions for the shock profiles for the Broadwell model.

2. Nonlinear Boundary Layers for the Broadwell Model Near a Moving Wall

In this section we study boundary layers for the nonlinear Broadwell model near a wall moving with a constant speed . In [8] the nonlinear boundary layers for the DBE, the general discrete velocity model (DVM) was considered. Existence of nonlinear boundary layers was proved. Here we exemplify the theory in [8] in the case of a simplified model, where the whole machinery is actually not really needed, even if it helps out. The same problem was considered in [21] for a mixture model, where one of the two species was modelled by the Broadwell model.

We consider the classical Broadwell model [9] in space (with velocities , , , , , and )where is the mutual collision cross section. For a flow axially symmetric around the -axis we can reduce system (1) to (with , , and ) [9]

The collision invariants areand the Maxwellians (equilibrium distributions) areThe density, momentum, and internal energy can be obtained by

Let , the speed of the wall, be a real number such thatWe define the projections and , , bywhere

Consider the problemwhere , , , , is a given matrix, , and is defined by the bilinear expression:

After the change of variables and the transformationwe obtain the new system where , , is an matrix, , and

Similar initial boundary value problems have been studied in a series of papers using Green’s functions (with , ) for in [10] (with ) and [14], for in [11], for , with in [12, 13, 15], and for diffuse boundary conditions in [16].

Here we consider the stationary nonlinear systemThe linearized collision operatoris symmetric and semipositive and have the null-spacefor some and , such thatNote also thatHere and below, denotes the Euclidean scalar product and we denote . If we can chooseand then

We let denote the number of the positive and negative eigenvalues of the matrix . The numbers , with , defined above, denote the numbers of the positive and negative eigenvalues of the matrix . Moreover, we let , , and denote the number of positive, negative, and zero eigenvalues of the -matrixwhere and . Then [22, 23]. The eigenvalues of arewhereWe find that if , if , , and if , but if , if , and Hence, we obtain the following number of positive and negative eigenvalues for different values of for Particularly, if then .

Explicitly, the eigenvalues of the matrix are (of multiplicity ) andFor an eigenvector corresponding to the nonzero eigenvalue isthat isFurthermore,

Example 1. If , corresponding to a nondrifting Maxwellian , then we get that

Note that if , then we always have the trivial solution , and if (, where both and must be zero, i.e., ), (, where must be zero, i.e., ), or (no boundary conditions at all at the wall), then we have no other solutions. Otherwise, we have solutions if and only if .

Below we consider the remaining different cases.(i) If then and . Hence, if we obtain the unique solution with(ii) If then and . Hence, if , , and then we obtain the unique solution (32) with and if , , , and , then we obtain the unique solution (32) with(iii) If then and . Hence, if and then we obtain the unique solution (32), where

We note that in each of the above cases conditions on the assigned data are implied to have a unique solution. This is in good agreement with the results for the DBE in [8] and for the continuous BE in [3].

Remark 2. Similar results can be obtained for the (reduced) plane Broadwell model Particularly, with we have

3. Shock Profiles

In this section we are concerned with the existence of shock profiles [17, 18] for the Boltzmann equation Here , , and denote position, velocity, and time, respectively. Furthermore, denotes the speed of the wave. The profiles are assumed to approach two given Maxwellians(, , and denote density, bulk velocity, and temperature, resp.) as , which are related through the Rankine-Hugoniot conditions.

The (shock wave) problem is to find a solution () of the equationsuch that

In [20] the shock wave problem (44), (45) for the DBE was considered. Existence of shock profiles in the case of weak shocks was proved. Here we exemplify the theory in [20] in the case of a simplified model, where the whole machinery is actually not really needed, even if it helps out. In this way we, in a different way, obtain similar results as is obtained in [19] for the same problem.

We study the reduced system (2) of the classical Broadwell model in (1) [9] in space. The collision invariants are given by (3) and the Maxwellians (equilibrium distributions) by (4).

The shock wave problem for the Broadwell model readswhere , , and is defined by the bilinear expression (10).

The density , momentum , and internal energy can be obtained by (5). The Maxwellians and must fulfill the Rankine-Hugoniot conditionswithAfter some manipulations we obtain that

We considerand denoteThen we obtainwith the linearized operator and the quadratic part given by (13). The linearized collision operator is given by (15) and then fulfills properties (16)–(20).

We assume that is nonsingular; that is . Then by (52) we obtain the system

In (25) we obtain that and the eigenvalues of the matrix are (of multiplicity ) and

Letwhere and are eigenvectors (19) corresponding to the zero eigenvalue and is eigenvector (28) corresponding to the nonzero eigenvalue . Then which implies that since . Therefore where is given in (30). We obtain that Assume that and let Thenand thereforeWe conclude that the solution of system (50) is of the form It follows thatwhich is a Maxwellian. Formally we can allow and . However, then, the equilibrium distribution (65) will not be nonnegative and, hence, not a Maxwellian.

We note thatwhere is an arbitrary nonzero constant. The structure coincides with the one for the Mott-Smith approximation [24] in [25]. However, is obtained in different ways.

Remark 3. We can instead of system (50) considerwithand in a similar way as above, we obtain

Example 4. If then we have Furthermore, and the other Maxwellian is

Example 5. Similar results can be obtained for the (reduced) plane Broadwell model Particularly, with we have The other Maxwellian is then

The shock strength (cf. [19]) is given by the density ratio if and if byThen the shock strength tends to infinity as approaches and to zero as approaches ; that is

The shock width (cf. [19]) is given by the density ratioor by the velocity ratioWe conclude that the shock widths and tend to zero as approaches and to infinity as approaches ; that is

Competing Interests

The author declares that there is no conflict of interests regarding the publication of this paper.