Bulk equations and Knudsen layers for the regularized 13 moment equations

What does an ideal wall look like?

“…These depend on the details of the kernel K (c ′ k → c k ); explicit calculations for Maxwell boundary conditions can be found in [3][4] [11]. Note that for rarefied gases, the temperature jump and the velocity slip are followed by marked boundary layers, known as Knudsen layers, which are of the width of the mean free path l [2][4] [12]. In the hydrodynamic regime, where Kn is small, the Knudsen layers are narrow, and are often incorporated into jump and slip by means of the Knudsen layer correction [2].…”

Boundary conditions for Grad's 13 moment equations

“…Therefore the differential equations are linearly independent, and their solution requires seven boundary conditions: four for (16) plus three for the conservation laws (12). We shall consider jump and slip boundary conditions for temperature and velocity, and additional boundary conditions for normal stress σ 22 . The constant P 0 follows again from the constant mass between the walls as, see (12) …”

“…When the Grad 13 moment equations for Couette flow are considered only within second order terms in the Knudsen number, they can be reduced to the so-called bulk equations [17,22,4],…”

“…These are the essentially NSF equations with two additional equations for normal stress σ 22 and parallel heat flux q 1 . The solution of the bulk equations requires jump and slip boundary conditions for temperature and velocity, but no additional boundary conditions for normal stress.…”

“…The results differed considerably from exact solutions of the Boltzmann equation as the Knudsen number became larger, which indicates problems with the boundary conditions. An expansion of the Grad equations in Couette geometry yields the second order bulk equations that require only jump and slip boundary conditions [17,22,4].…”