Navier Slip

The Navier-slip boundary condition says the velocity at the wall is proportional to the normal derivative of the velocity at the wall. The constant of proportionality is called the slip length. In some situations this condition may be a more accurate, or a necessary, replacement of the common no-slip condition. Factors that affect slip can include the scale of the flow (where the physics at the small scale make the no-slip assumption unrealistic but, as long as the Knudsen number is of the order of about 1/10, one can model the flow with the Navier-Stokes equations and the Navier-Maxwell slip condition [1]), surface roughness or coating, or interfacial wetability, for example. At high Reynolds numbers, surfaces with slip have the potential to reduce drag and vorticity generation.

Vorticity plots of a dipole colliding at and angle of 45 degrees with a wall with slippage at Re=7500 and different times (a-d). From Mohammed et al (2020)


Luckily for lattice Boltzmann, the shear stress is a locally attainable quantity, making the Navier-slip condition reasonable straightforward to implement without sacrificing the efficiency of the algorithm. My former PhD student, Seemaa Mohammed, imposed the Navier-slip condition using lattice Boltzmann and studied the dynamics of dipoles colliding with slip walls at high Reynolds number. She considered the influence of slip length, Reynolds number, and angle of collision of the flow. Her work suggests that an increase in wall slippage causes a reduction in the number of higher-order dipoles created by the wall-dipole collision. This leads to a decrease in the magnitude of the enstrophy peaks and reduces the dissipation of energy. She also found a theoretical relationship between entrophy and dissipation in the presence of slip.

Total enstropy as a function of time for different slip lengths for collision angles of 45 degrees at Re=2500. From Mohammed et al (2020)


Another doctoral student, Zainab Bu Sinnah, has shown that the approach can be used to study pulsatile flows with slip, too. She is also extending the method to three-dimensional flows with slip boundary conditions.

Simulated and exact solutions of the velocity in pulsatile flow with slip. The non-dimensional slip lengths (analogous to the Knudsen number) from left the right: 0.388, 0.194, 0,0194. Taken from Sinnah, Graham, and Reis (2018).



[1] Hadjiconstantinou, Phys. Fluids 18 (2006)