Steady Stokes Flows With Threshold Slip Boundary Conditions

Abstract

We prove the existence, uniqueness and continuous dependence on the data of weak solutions to boundary-value problems that model steady flows of incompressible Newtonian fluids with wall slip in bounded domains. The flows satisfy the Stokes equations and a nonlinear slip boundary condition: for slip to occur, the magnitude of the tangential traction must exceed a prescribed threshold, which is independent of the normal stress, and where slip occurs the tangential traction is equal to a prescribed, possibly nonlinear, function of the slip velocity. In addition, a Dirichlet condition is imposed on a component of the boundary if the domain is rotationally symmetric. The method of proof is based on a variational inequality formulation of the problem and fixed point arguments which utilize wellposedness results for the Stokes problem with a slip condition of the "friction type".

Numerical methods for the Stokes and Navier–Stokes equations driven by threshold slip boundary conditions

“…Equations (2.1) and (2.2) are supplemented by nonlinear slip boundary of friction type, which is the main modeling assumption in this work. It should be pointed out that such boundary conditions have already been considered in [43,44]. Hence we will just state the mathematical equations governing this phenomenon as the physical merits of such model have been discussed elsewhere (see particularly [43]).…”

“…It should be pointed out that such boundary conditions have already been considered in [43,44]. Hence we will just state the mathematical equations governing this phenomenon as the physical merits of such model have been discussed elsewhere (see particularly [43]). So, we assume that the boundary of Ω, say, ∂Ω is made of two components S and Γ, and it is required that ∂Ω = S ∪ Γ, with S ∩ Γ = ∅.…”

“…Subsequently, many studies have focused on the properties of the solution of the resulting boundary value problem, for example, existence, uniqueness, regularity, and continuous dependence on data, for Stokes, Navier-Stokes and Brinkman-Forchheimer equations under such boundary conditions. Details can be found in [1,3,11,12,13,14,15,16,17,43,44,45,46] among others.…”

“…Recently, due to some advances in modeling it has been shown that some natural flows may have nonlinear frictional slip at the interface separating the fluid and the solid, giving rise to variational inequality (see [11,13,14,15,16,42,43,44,45]). The numerical investigation of these later problems have benefited a lot from the existing knowledge of general ways of treating variational inequalities and some works worth to be mentioned here include [2,7,18,30,39,40].…”

Discontinuous Galerkin finite element discretization for steady Stokes flows with threshold slip boundary condition

“…On S, we first assume the impermeability condition v N = v · n = 0 on S, (1.4) where n is the outward unit normal on the boundary ∂Ω, and v N is the normal component of the velocity, while v τ = v − v N n is its tangential component. In addition to (1.4) we also impose on S, a "friction type" boundary condition [5,6,7,8,16], which is the main ingredient of this work. The "friction type" boundary condition can be formulated with the knowledge of a positive function g : S −→ (0, ∞) called threshold slip or barrier function, and the use of sub-differential to link quantities of interest.…”