Maxwell was the first person to consider the boundary conditions required for the solution of the Boltzmann equation in the case of a slightly rarefied gas (Ref. 1). The boundary conditions he derived can be used as boundary conditions for the Navier Stokes equations in the slip flow regime, although Maxwell considered the particular case of a monatomic hard-sphere gas, so some generalization is required.
where uslip is the slip velocity,
n is the boundary normal,
τ is the viscous stress tensor,
μ is the viscosity of the gas,
ρ is its density, and
Tg is its temperature. The factor
G (which has dimensions of length) is given by:
where λ is the mean free path and
av is the tangential momentum accommodation coefficient (for a model in which the surface reflects some molecules diffusely and some specularly, this is equivalent to the fraction of molecules which are reflected diffusely).
In Equation 7-1 the left-hand term represents the phenomena of viscous slip, whilst the right-hand term produces thermal slip or transpiration. Maxwell was aware that the temperature of the gas was not necessarily equal to that of the wall but formulated the boundary condition using only the gas temperatures. When the wall temperature is included, the following equations are obtained (
Ref. 2 and
Ref. 3):
where Tw is the wall temperature,
σs is the viscous slip coefficient,
σT is the thermal slip coefficient, and
ζT is the temperature jump coefficient. Within a generalized Maxwell’s model the three coefficients
σs,
σT, and
ζT are given by:
where κ is the thermal conductivity of the gas. The mean free path can be computed from the gas properties using the following equation (
Ref. 3):
