Stream function solutions for some contact line boundary conditions: Navier slip, super slip and the generalized Navier boundary condition
Yash Kulkarni, Tomas Fullana, Stéphane Zaleski
Abstract
The stream function solution for the inner region Stokes flow, for a locally plane moving fluid interface near the triple point, is derived considering three different boundary conditions: the Navier slip boundary condition (NBC), the super-slip boundary condition and the generalized Navier boundary condition (GNBC). The NBC, incorporating a slip length parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> , is a well-known method for regularization in the context of the three-phase dynamic contact line problem. It is demonstrated that the velocity field solution under this boundary condition maintains a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msup> </mml:math> continuity at the contact line, resulting in a logarithmic divergence of the pressure at the contact line. By contrast, the super-slip boundary condition establishes a proportional relationship between the wall velocity and the normal derivative of the shear stress, leading to a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> velocity field. Furthermore, the GNBC, which introduces an uncompensated Young stress to drive the contact line, yields a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> velocity field. The dominant terms are explicitly derived, and the analytical approach presented here can be extended to other bi-harmonic problems as well.