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On the Kinematics of Contact Line Motion

We investigate the moving contact line problem for two‐phase incompressible flows by a kinematic approach. The key idea is to derive an evolution equation for the contact angle assuming the transporting velocity field to be given. It turns out that the resulting equation expresses the time derivative of the contact angle in terms of the velocity gradient at the solid wall. Together with the additionally imposed boundary conditions for the velocity, it yields a more specific form of the contact angle evolution. In this paper we consider the Navier slip boundary condition, which is frequently used for the modeling of moving contact lines. Exploiting furthermore the interfacial transmission condition for the viscous stress, we derive an explicit form of the contact angle evolution for sufficiently regular solutions. In the absence of phase change it only involves the contact line velocity and the slip length from the Navier condition. From this equation we can read off the qualitative behavior of the contact angle evolution for sufficiently regular solutions to these models, which turns out to be unphysical.