IJDE International Journal of Differential Equations 1687-9651 1687-9643 Hindawi Publishing Corporation 10.1155/2016/5801728 5801728 Research Article Boundary Layers and Shock Profiles for the Broadwell Model http://orcid.org/0000-0003-1232-3272 Bernhoff Niclas Galdi Giovanni P. Department of Mathematics Karlstad University 65188 Karlstad Sweden kau.se 2016 2072016 2016 26 04 2016 19 06 2016 2016 Copyright © 2016 Niclas Bernhoff. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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) . 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  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  for a simplified model, the Broadwell model , 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  for the DBE and  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 .

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) . In  existence of shock profiles for the DBE in the case of weak shocks was proved. We exemplify the theory in  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  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 b . In  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  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  for a mixture model, where one of the two species was modelled by the Broadwell model.

We consider the classical Broadwell model  in space (with velocities ξ 1 = ( 1,0 , 0 ) , ξ 2 = ( - 1,0 , 0 ) , ξ 3 = ( 0,1 , 0 ) , ξ 4 = ( 0 , - 1,0 ) , ξ 5 = ( 0,0 , 1 ) , and ξ 6 = ( 0,0 , - 1 ) ) (1) f ~ 1 t + f ~ 1 x = 2 σ 3 f ~ 3 f ~ 4 + f ~ 5 f ~ 6 - 2 f ~ 1 f ~ 2 , f ~ 2 t - f ~ 2 x = 2 σ 3 f ~ 3 f ~ 4 + f ~ 5 f ~ 6 - 2 f ~ 1 f ~ 2 , f ~ 3 t = - 2 σ 3 f ~ 3 f ~ 4 - f ~ 1 f ~ 2 , f ~ 4 t = - 2 σ 3 f ~ 3 f ~ 4 - f ~ 1 f ~ 2 , f ~ 5 t = - 2 σ 3 f ~ 5 f ~ 6 - f ~ 1 f ~ 2 , f ~ 6 t = - 2 σ 3 f ~ 5 f ~ 6 - f ~ 1 f ~ 2 , where σ is the mutual collision cross section. For a flow axially symmetric around the x -axis we can reduce system (1) to (with f 1 = f ~ 1 , f 2 = f ~ 3 = f ~ 4 = f ~ 5 = f ~ 6 , and f 3 = f ~ 2 )  (2) f 1 t + f 1 x = 4 σ 3 f 2 2 - f 1 f 3 , 4 f 2 t = - 8 σ 3 f 2 2 - f 1 f 3 , f 3 t - f 3 x = 4 σ 3 f 2 2 - f 1 f 3 .

The collision invariants are (3) ϕ = α 1,1 , 1 + β 1,0 , - 1 , α , β R , and the Maxwellians (equilibrium distributions) are (4) M = s 4 a 4 , a 2 , 1 , with   s = e α - β / 4 > 0 ,    a = e β / 4 > 0 ,    α , β R . The density, momentum, and internal energy can be obtained by (5) ρ = f 1 + 4 f 2 + f 3 , ρ u = f 1 - f 3 , 2 ρ e = f 1 + f 3 .

Let b , the speed of the wall, be a real number such that (6) b - 1,0 , 1 . We define the projections R + : R 3 R n + and R - : R 3 R n - , n - = 3 - n + , by (7) R + h = h + = h 1 , , h n + , R - h = h - = h n + + 1 , , h 3 , where (8) n + = 0 if   b > 1 1 if   0 < b < 1 2 if   - 1 < b < 0 3 if   b < - 1 , h = h 1 , h 2 , h 3 .

Consider the problem (9) D f t + B 0 f x = Q f , f , x > b t ,    t > 0 , f + b t , t = C ~ f - b t , t + φ ~ 0 , f x , 0 = f 0 x , f 0 x M as   x , where f = f 1 , f 2 , f 3 , D = ( 1,4 , 1 ) , B 0 = ( 1,0 , - 1 ) , M = s 4 ( a 4 , a 2 , 1 ) , C ~ is a given n + × n - matrix, φ ~ 0 R n + , and Q f , f is defined by the bilinear expression: (10) Q f , g = 2 σ 3 2 f 2 g 2 - f 1 g 3 - g 1 f 3 1 , - 2,1 .

After the change of variables y = x - b t and the transformation (11) f = M + M 1 / 2 h , with   M = s 4 a 4 , a 2 , 1 , we obtain the new system (12) h t + B h y + L h = S h , h , y > 0 ,    t > 0 , h + 0 , t = C h - 0 , t + φ 0 , h y , 0 = h 0 y , h 0 y 0 as   y , where h = h 1 , h 2 , h 3 , B = B 0 - b D , C is an n + × n - matrix, φ 0 R n + , and (13) L h = - 2 M - 1 / 2 Q M , M 1 / 2 h = 4 s 4 σ 3 h 1 - 2 a h 2 + a 2 h 3 1 , - 2 a , a 2 , S h , g = M - 1 / 2 Q M 1 / 2 h , M 1 / 2 g = 2 s 2 σ 3 2 h 2 g 2 - h 1 g 3 - g 1 h 3 1 , - 2 a , a 2 .

Similar initial boundary value problems have been studied in a series of papers using Green’s functions (with s 4 = 1 / 6 , a = 1 ) for 1 / 3 < b < 1 in  (with C = ( c 1 c 2 ) ) and , for - 1 < b < - 1 / 3 in , for 0 < b < 1 / 3 , with C = 0 in [12, 13, 15], and for diffuse boundary conditions in .

Here we consider the stationary nonlinear system (14) B d h d y + L h = S h , h , h = h y ,    y > 0 , h + 0 = C h - 0 + φ 0 , h y 0 as   y . The linearized collision operator (15) L = 4 s 4 σ 3 1 - 2 a a 2 - 2 a 4 a 2 - 2 a 3 a 2 - 2 a 3 a 4 is symmetric and semipositive and have the null-space (16) N L = span a 2 , a , 1 , a 2 , 0 , - 1 = span e 1 , e 2 , for some e 1 and e 2 , such that (17) e 1 , e 2 B = γ i δ i j , i , j = 1,2 . Note also that (18) S h , h , e = 0 , e N L . Here and below, · , · denotes the Euclidean scalar product and we denote · , · B = · , B · . If b a 2 / ( 1 + a 2 ) we can choose (19) e 1 = a , 1 2 , 0 , e 2 = a 2 b , a 3 2 1 - b , a 2 1 - b - b , and then (20) γ 1 = a 2 - b 1 + a 2 , γ 2 = a 2 - b 1 + a 2 - a 2 + 1 - a 4 b + 1 + a 2 + a 4 b 2 .

We let m ± denote the number of the positive and negative eigenvalues of the matrix B - 1 L . The numbers n ± , with n + + n - = 3 , defined above, denote the numbers of the positive and negative eigenvalues of the matrix B . Moreover, we let k + , k - , and l denote the number of positive, negative, and zero eigenvalues of the 2 × 2 -matrix (21) K = y 1 , y 1 B y 1 , y 2 B y 2 , y 1 B y 2 , y 2 B , where y 1 = a 2 , a , 1 and y 2 = a 2 , 0 , - 1 . Then m ± = n ± - k ± - l [22, 23]. The eigenvalues of K are (22) λ ± = η ± η 2 - 4 a 2 κ , where (23) η = 1 + a 2 a 2 - 1 - b 1 + a 2 , κ = - a 2 + 1 - a 4 b + 1 + a 2 + a 4 b 2 . We find that sgn λ ± = sgn η if κ > 0 , sgn λ ± = ± 1 if κ < 0 , λ + = 2 η , and λ - = 0 if κ = 0 , but κ < 0 if η = 0 , sgn ( η ) = - sgn ( b ) if κ > 0 , and (24) κ = 0 b = b ± = a 4 - 1 ± 1 + 4 a 2 + 2 a 4 + 4 a 6 + a 8 2 1 + a 2 + a 4 , κ > 0 if   b > b + ,   or   b < b - , κ < 0 if   b - < b < b + . Hence, we obtain the following number of positive and negative eigenvalues for different values of b (25) b - 1 b - 0 b + 1 n + 3 2 2 2 1 1 1 0 n - 0 1 1 1 2 2 2 3 k + 2 2 1 1 1 0 0 0 k - 0 0 0 1 1 1 2 2 l 0 0 1 0 0 1 0 0 m + 1 0 0 1 0 0 1 0 m - 0 1 0 0 1 0 0 1 for (26) b ± = a 4 - 1 ± 1 + 4 a 2 + 2 a 4 + 4 a 6 + a 8 2 1 + a 2 + a 4 . Particularly, if a = 1 then b ± = ± 1 / 3 .

Explicitly, the eigenvalues of the matrix B - 1 L are 0 (of multiplicity 2 ) and (27) λ = 4 s 4 σ κ 3 b - b 3 , with   κ = - a 2 + 1 - a 4 b + 1 + a 2 + a 4 b 2 . For b b ± an eigenvector corresponding to the nonzero eigenvalue λ is (28) v = b + b 2 , a 2 1 - b 2 , a 2 b 2 - b ; that is (29) B - 1 L v = λ v , v , e B = 0 , e N L . Furthermore, (30) B - 1 S v , v = k v , with   k = s 2 a 2 σ 3 b 1 + 3 b 2 .

Example 1.

If a = 1 , corresponding to a nondrifting Maxwellian M , then we get that (31) b ± = ± 1 3 , λ = 4 s 4 σ 3 b 2 - 1 3 b - b 3 , v = b + b 2 , 1 - b 2 2 , b 2 - b , k = s 2 σ 3 b 1 + 3 b 2 .

Note that if φ 0 = 0 , then we always have the trivial solution h = 0 , and if - 1 < b b - ( φ 0 = ( φ 01 , φ 02 ) , where both φ 01 and φ 02 must be zero, i.e., φ 01 = φ 02 = 0 ), 0 < b b + ( φ 0 R , where φ 0 must be zero, i.e., φ 0 = 0 ), or 1 < b (no boundary conditions at all at the wall), then we have no other solutions. Otherwise, we have solutions if and only if h 0 span ( v + - C v - ) .

Below we consider the remaining different cases.

If b + < b < 1 then C = ( c 1 c 2 ) and φ 0 R . Hence, if φ 0 0 we obtain the unique solution (32) h y = λ k + D e λ y b + b 2 , a 2 1 - b 2 , a 2 b 2 - b ,

with (33) D = 2 b + b 2 - c 1 a 1 - b 2 - 2 c 2 a 2 b 2 - b 2 h 0 λ - k .

If b - < b < 0 then C = c 1 c 2 and φ 0 = ( φ 01 , φ 02 ) R 2 . Hence, if φ 01 0 , c 1 2 b / a ( 1 - b ) , and (34) φ 02 = a 1 - b 2 - 2 c 2 a 2 b 2 - b 2 b + b 2 - 2 c 1 a 2 b 2 - b φ 01 ,

then we obtain the unique solution (32) with (35) D = b + b 2 - c 1 2 a b 2 - b φ 01 λ - k ,

and if c 1 = 2 b / a ( 1 - b ) , c 2 - ( 1 + b ) / 2 a b , φ 01 = 0 , and φ 02 0 , then we obtain the unique solution (32) with (36) D = a 1 - b 2 - 2 c 2 a 2 b 2 - b 2 φ 02 λ - k .

If b < - 1 then C = 0 and φ 0 = ( φ 01 , φ 02 , φ 03 ) R 3 . Hence, if φ 01 0 and (37) φ 02 = a 1 - b 2 b φ 01 , φ 03 = a 2 b - 1 1 + b φ 01 ,

then we obtain the unique solution (32), where (38) D = b + b 2 φ 01 λ - k .

We note that in each of the above cases k + conditions on the assigned data φ 0 are implied to have a unique solution. This is in good agreement with the results for the DBE in  and for the continuous BE in .

Remark 2.

Similar results can be obtained for the (reduced) plane Broadwell model (39) 1 - b d f 1 d y = σ f 2 2 - f 1 f 3 , - 2 b d f 2 d y = - 2 σ f 2 2 - f 1 f 3 , - 1 + b d f 3 d y = σ f 2 2 - f 1 f 3 . Particularly, with a = 1 we have (40) b ± = ± 1 2 , λ = 2 s 4 σ 2 b 2 - 1 3 b - b 3 , v = b + b 2 , 1 - b 2 , b 2 - b , k = s 2 σ b .

3. Shock Profiles

In this section we are concerned with the existence of shock profiles [17, 18] (41) F = F x 1 , ξ , t = f x 1 - b t , ξ for the Boltzmann equation (42) F t + ξ · x F = Q F , F . Here x = ( x 1 , , x d ) R d , ξ = ( ξ 1 , , ξ d ) R d , and t R + denote position, velocity, and time, respectively. Furthermore, b denotes the speed of the wave. The profiles are assumed to approach two given Maxwellians (43) M ± = ρ ± 2 π T ± d / 2 e - ξ - u ± 2 / 2 T ± ( ρ , u , and T denote density, bulk velocity, and temperature, resp.) as x ± , which are related through the Rankine-Hugoniot conditions.

The (shock wave) problem is to find a solution f = f ( y , ξ ) ( y = x 1 - b t ) of the equation (44) ξ 1 - b f y = Q f , f , such that (45) f M ± as   y ± .

In  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  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  for the same problem.

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

The shock wave problem for the Broadwell model reads (46) B d f d y = Q f , f , where   f M ±   as   y ± , where B = B ( b ) = diag ( 1 - b , - 4 b , - 1 + b ) , f = f 1 , f 2 , f 3 , and Q f , f is defined by the bilinear expression (10).

The density ρ , momentum ρ u , and internal energy 2 ρ E can be obtained by (5). The Maxwellians M - = s - 4 ( a - 4 , a - 2 , 1 ) and M + = s + 4 ( a + 4 , a + 2 , 1 ) must fulfill the Rankine-Hugoniot conditions (47) ρ + u + - b = ρ - u - - b , ρ + 2 E + - b u + = ρ - 2 E - - b u - , with (48) ρ ± = s ± 4 1 + 4 a ± 2 + a ± 4 , ρ ± u ± = s ± 4 a ± 4 - 1 , 2 ρ ± E ± = s ± 4 1 + a ± 4 . After some manipulations we obtain that (49) 2 E ± = 1 3 2 1 + 3 u ± 2 - 1 .

We consider (50) B d f d y = Q f , f , where   f M +   as   y and denote (51) F = M + M 1 / 2 h , with   M = M + = s + 4 a + 4 , a + 2 , 1 = s 4 a 4 , a 2 , 1 . Then we obtain (52) B d h d y + L h = S h , h , where   h 0   as   y , with the linearized operator L and the quadratic part S ( h , h ) given by (13). The linearized collision operator is given by (15) and then fulfills properties (16)–(20).

We assume that B is nonsingular; that is b - 1,0 , 1 . Then by (52) we obtain the system (53) d h d y + B - 1 L h = B - 1 S h , h .

In (25) we obtain that (54) b ± = a 4 - 1 ± 1 + 4 a 2 + 2 a 4 + 4 a 6 + a 8 2 1 + a 2 + a 4 = 2 u + ± 2 2 E + 1 + 6 E + , and the eigenvalues of the matrix B - 1 L are 0 (of multiplicity 2 ) and (55) λ = 4 s 4 σ κ 3 b - b 3 = ρ + σ 3 b - b 3 2 E + - 1 - 4 u + b + 1 + 6 E + b 2 , with   κ = - a 2 + 1 - a 4 b + 1 + a 2 + a 4 b 2 .

Let (56) h = ϑ e 1 + χ e 2 + μ v , where e 1 and e 2 are eigenvectors (19) corresponding to the zero eigenvalue and v is eigenvector (28) corresponding to the nonzero eigenvalue λ . Then (57) d ϑ d y = d χ d y = 0 , which implies that (58) ϑ = χ = 0 , since lim y ϑ = lim y χ = 0 . Therefore (59) d μ d y + λ μ = k μ 2 , where k is given in (30). We obtain that (60) μ = λ k + D e λ y . Assume that D 0 and let (61) b + < b < 1 , or   b - < b - 1 1 + 2 a 2 . Then (62) l i m y μ = 0 , l i m y - μ = λ k , and therefore (63) h y = λ k + D e λ y v , D 0 . We conclude that the solution of system (50) is of the form (64) f y = M + + λ k + D e λ y M + 1 / 2 v , where   λ = 4 s 4 σ 3 b - b 3 - a 2 + 1 - a 4 b + 1 + a 2 + a 4 b 2 ,    k = s 2 a 2 σ 3 b 1 + 3 b 2 ,    v = b + b 2 , a 2 1 - b 2 , a 2 b 2 - b ,    D 0 . It follows that (65) M - = M + + λ k M + 1 / 2 v = s 4 1 + 3 b 2 p 2 1 + b 1 - b , p q , q 2 1 - b 1 + b , with   p = 2 + a 2 b - a 2 ,    q = 1 + 1 + 2 a 2 b , which is a Maxwellian. Formally we can allow b < - 1 and - 1 / ( 1 + 2 a 2 ) < b < 0 . However, then, the equilibrium distribution (65) will not be nonnegative and, hence, not a Maxwellian.

We note that (66) f y = Θ y M + + 1 - Θ y M - , with   Θ y = 1 1 + C e - λ y , where C = k / D 0 is an arbitrary nonzero constant. The structure coincides with the one for the Mott-Smith approximation  in . However, λ is obtained in different ways.

Remark 3.

We can instead of system (50) consider (67) B d f d y = Q f , f , where   f M -   as   y - , with (68) a 2 2 + a 2 < b < b + , or   - 1 < b < b - , and in a similar way as above, we obtain (69) f y = M - + λ k + D e λ y M - 1 / 2 v , M + = M - + λ k M - 1 / 2 v .

Example 4.

If a = 1 then we have (70) ρ + = 6 s + 4 , u + = 0 , E + = 1 6 , b ± = ± 1 3 , λ = 4 s 4 σ 3 b 2 - 1 3 b - b 3 , v = b + b 2 , 1 - b 2 2 , b 2 - b , k = s 2 σ 3 b 1 + 3 b 2 . Furthermore, (71) f y = M + + λ k + D e λ y M + 1 / 2 v = s 4 1,1 , 1 + 4 s 4 3 b 2 - 1 1 + 3 b 2 + D ~ e λ y b 1 - b , 1 2 , - b 1 + b = r 3 b - 1 2 1 + b 1 - b , 9 b 2 - 1 , 3 b + 1 2 1 - b 1 + b + D ~ e λ y 1,1 , 1 , where   r = s 4 1 + 3 b 2 + D ~ e λ y , and the other Maxwellian is (72) M - = s 4 1 + 3 b 2 3 b - 1 2 1 + b 1 - b , 3 b + 1 3 b - 1 , 3 b + 1 2 1 - b 1 + b .

Example 5.

Similar results can be obtained for the (reduced) plane Broadwell model (73) 1 - b d f 1 d y = σ f 2 2 - f 1 f 3 , - 2 b d f 2 d y = - 2 σ f 2 2 - f 1 f 3 , - 1 + b d f 3 d y = σ f 2 2 - f 1 f 3 . Particularly, with a = 1 we have (74) ρ + = 4 s + 4 , u + = 0 , E + = 1 4 , c ± = ± 1 2 , λ = 2 s 4 σ 2 b 2 - 1 3 b - b 3 , v = b + b 2 , 1 - b 2 , b 2 - b , k = s 2 σ b . The other Maxwellian is then (75) M - = s 4 2 b - 1 2 1 + b 1 - b , 2 b + 1 2 b - 1 , 2 b + 1 2 1 - b 1 + b .

The shock strength (cf. ) is given by the density ratio (76) σ ρ = ρ - - ρ + ρ + = 2 a 2 ρ + λ k = 8 s 2 ρ + b + b 2 + a 2 b 2 - 1 + a 4 b 2 - b 1 - b 2 1 + 3 b 2 , if b > 0 and if b < 0 by (77) σ ρ = ρ - - ρ + ρ - = 2 a 2 ρ - λ k = 8 s 2 ρ - b + b 2 + a 2 b 2 - 1 + a 4 b 2 - b 1 - b 2 1 + 3 b 2 . Then the shock strength σ ρ tends to infinity as b approaches 1 and to zero as b approaches b ± ; that is (78) σ ρ as   b 1 , σ ρ 0 as   b b ± .

The shock width (cf. ) is given by the density ratio (79) d ρ = ρ - - ρ + m a x y d ρ / d y = 4 λ = 3 b 1 - b 2 s 4 σ b + b 2 + a 2 b - 1 + a 4 b 2 - b or by the velocity ratio (80) d u = u - - u + max y d u / d y = 4 λ 1 - u - / b ρ + . We conclude that the shock widths d ρ and d u tend to zero as b approaches 1 and to infinity as b approaches b ± ; that is (81) d ρ 0 , d u 0 as   b 1 , d ρ , d u as   b b ± .

Competing Interests

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

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