Theory for the Wall Boundary Condition

The Slip condition assumes that there are no viscous effects at the slip wall and hence, no boundary layer develops. From a modeling point of view, this is a reasonable approximation if the important effect of the wall is to prevent fluid from leaving the domain. Mathematically, the constraint can be formulated as:

The no-penetration term takes precedence over the Neumann part of the condition and the above expression is therefore equivalent to

expressing that there is no flow across the boundary and no viscous stress in the tangential direction.

For a moving wall with translational velocity utr, u in the above equations is replaced by the relative velocity urel = u−utr.

For turbulent flow, turbulence variables are in general subject to homogeneous Neumann conditions. For example

The sliding wall option is appropriate if the wall behaves like a conveyor belt; that is, the surface is sliding in its tangential direction. The wall does not have to actually move in the coordinate system.

where t = (ny , −nx) for 2D and t = (nz, −nr) for axial symmetry.

The normalization makes u have the same magnitude as uw even if uw is not exactly parallel to the wall.

In the microscale range, the flow at a boundary is seldom strictly no slip or slip. Instead, the boundary condition is something in between, and there is a slip velocity at the boundary. Two phenomena account for this velocity: violation of the continuum hypothesis for the viscosity and flow induced by a thermal gradient along the boundary.

The following equation relates the viscosity-induced jump in tangential velocity to the tangential shear stress along the boundary:

For gaseous fluids, the coefficient β is given by

where μ is the fluid’s dynamic viscosity (SI unit: Pa·s), αv represents the tangential momentum accommodation coefficient (TMAC) (dimensionless), and λ is the molecules’ mean free path (SI unit: m). The tangential accommodation coefficients are typically in the range of 0.85 to 1.0 and can be found in Ref. 15.

A simpler expression for β is

where Ls, the slip length (SI unit: m), is a straight channel measure of the distance from the boundary to the virtual point outside the flow domain where the flow profile extrapolates to zero. This equation holds for both liquids and gases.

Thermal creep results from a temperature gradient along the boundary. The following equation relates the thermally-induced jump in tangential velocity to the tangential gradient of the natural logarithm of the temperature along the boundary:

where σT is the thermal slip coefficient (dimensionless) and ρ is the density of the fluid. The thermal slip coefficients range between 0.3 and 1.0 and can be found in Ref. 15.

Combining the previous relationships results in the following equation:

Relate the tangential shear stress to the viscous boundary force by

where the components of K are the Lagrange multipliers that are used to implement the boundary condition. Similarly, the tangential temperature gradient results from the difference of the gradient and its normal projection: