Geometric and constitutive dependence of Maxwell’s velocity slip boundary condition

The general form of Maxwell’s velocity slip boundary condition for rarefied gas flows depends on both the geometry of the surface and the constitutive relations used to relate the viscous stress to rate of strain. The dependence on geometry is often overlooked in current rarefied flow calculations, and the generality of the constitutive dependence means the condition can also be usefully applied in regions where the Navier-Stokes equations fail, e.g. rarefied flows close to surfaces. In this paper we give examples illustrating the importance of both these dependencies and show, therefore, that implementing the general Maxwell condition produces substantially different results to conventional implementations of the condition. Finally, we also investigate a common numerical instability associated with Maxwell’s boundary condition, and propose an implicit solution method to overcome the problem. MAXWELL’S VELOCITY SLIP BOUNDARY CONDITION In 1879, James Clerk Maxwell published a paper on the viscous stresses arising in rarefied gases [1]. At the time, a reviewer commented that it also might be useful if Maxwell could use his theoretical findings to derive a velocity boundary condition for rarefied gas flows at solid surfaces. Consequently, in an appendix to the paper, Maxwell proposed his now-famous velocity slip boundary condition. This boundary condition was successful in predicting two prior experimental observations: (a) that a rarefied gas could slide over a surface, and (b) that inequalities in temperature could give rise to a force tending to make the gas slide over a surface from colder to hotter regions (which had been discovered by Reynolds, and was known as “thermal transpiration” — now more commonly known as “thermal creep”). What has subsequently been overlooked by many current researchers is the general form of the slip expression derived by Maxwell, and this has some substantial consequences for modern simulations of, e.g., hypersonic aerodynamics and gas flows in microsystems. Maxwell related the tangential gas velocity slip, slip u r to the tangential shear stress, τ , and heat flux, q . Writing his expression in vector form so that it can be easily applied to flows over three-dimensional surfaces, we have: q p A uslip r r r γ γ τ λ σμ σ ) 1 Pr( 4 3 ) 2 ( 1 − − − − = (1) where ) ( ) ( n n n i i i r r r r − ⋅ ⋅ = 1 Π τ , ) ( n ni i Q q r r r r − ⋅ = 1 , an arrow denotes a vector quantity, σ is the momentum accommodation coefficient (equal to one for surfaces that reflect all incident molecules diffusely, and zero for purely specular reflection), μ is the gas viscosity at the wall, λ is the molecular mean free path at the wall, Pr is the Prandtl number, γ is the specific heat ratio, p is the gas pressure at the wall, n i r is a unit vector normal and away from the wall, Π is the stress tensor at the wall, 1 is the identity tensor and Q r is the heat flux vector at the wall. Maxwell proposed a value for the slip coefficient, A1=1, although a recent analytical solution of the linearized Boltzmann equation at a planar surface [2] estimates the “micro-slip” (the actual velocity slip at the wall) requires A1=√(2/π)≈0.8. In [2], a value of A1=1.146 is also proposed when the Navier-Stokes equations are used: this represents an additional “fictitious” velocity slip to accommodate for the Knudsen-layer structure that is not captured by linear constitutive relations [3].