A SLIP CONDITION BASED ON MINIMAL ENERGY DISSIPATION

An intrinsic difficulty arises when solving Stokes equations in Ω ⊂ ℝ2 when the boundary conditions on the velocity are discontinuous: the solution is physically unacceptable since the force exerted by the fluid on the boundary is logarithmically singular. To illustrate this phenomena, we present an explicit solution in which the logarithmic singularity appears in a particularly simple form. A common method of avoiding the appearance of these singular forces is via an alteration of the boundary velocity profile in the vicinity of the discontinuity. However, there is no obvious physical criterion according to which the velocity profile along the boundary should be chosen. We consider a possible physically motivated criterion based on minimal energy dissipation. We prove the existence of a unique minimizing profile and demonstrate that the resultant velocity field does indeed exert a finite force along the boundary. Lastly, the minimizing profile is calculated numerically and the effect of free parameters is considered.