Modelling of hydrophobic surfaces by the Stokes problem with the stick-slip boundary conditions

Modelling of hydrophobic surfaces by the Stokes problem with the stick-slip boundary conditions S. Fialová, J. Haslinger 2,3, R. Ku era 2, F. Pochylý, and V. átek 2 Brno University of Technology, Technická 2896/2, 66 69 Brno, CZ 2 IT4Innovations, V B-TU Ostrava, 7. listopadu 5/272, 78 33 Ostrava-Poruba, CZ 3 IG CAS, Studentská 768, 78 Ostrava-Poruba, CZ January 3, 26 Abstract Unlike the Navier boundary condition, the present paper deals with the case when the slip may occur only when the shear stress attains certain bound which is given à-priori. The discrete velocity-pressure model is derived using P-bubble/P elements. To release the impermeability condition and to regularize the non-smooth term characterizing the stick-slip behavior in the algebraic formulation, two additional vectors of Lagrange multipliers were introduced. The resulting minimization problem in terms of the dual variables (the pressure, the normal and shear stress) is solved by the interior point type method. Introduction The biggest part of hydraulic losses arises from uid ow near the surface and depends on the uid velocity gradient in the normal direction to the surface. The velocity gradient is subject to liquid adhesion. If the surface is perfectly wetted by the liquid, it is possible to characterize the sticking of the liquid by the Dirichlet boundary condition prescribing zero velocity vector along the surface. This condition is fullled by the majority of known compounds, e.g. metals-water, water-glass, metals-oil, most plastics-water, etc. Reduced adhesion can be achieved by reducing the gradient of the liquid velocity,

thereby reducing hydraulic losses. This phenomena is usually modelled by the following boundary condition, which expresses the slip of the liquid at each point of the surface by σ t = κu t, () where u t, σ t denote the tangential components of the velocity and the shear stress along the surface, respectively. Further, κ is a positive adhesive function which generally depends on the spatial variable and its knowledge is assumed [5, 6]. If κ = const. > on the surface then () is the classical Navier condition with the adhesive coecient κ. In this paper we use another type of the slip condition, namely the stickslip boundary conditions. We assume that a nonnegative slip bound function g is prescribed along the surface and we require fulllment of the following relations at each point of the surface: (i) σ t g; (ii) if σ t < g, then u t = ; (iii) if σ t = g, then the slip may occur and its sign is opposite to the sign of σ t. It is worth noticing that the partition of the surface between the stick and slip part is one of unknowns of the problem. The condition (iii) yields the existence of κ s such that σ t = κ s u t. In contrast to (), its value is known only after solving the problem and depends on the computed tangential velocity component: κ s = g u t, u t. (2) It is worth noticing that if g is chosen in such a way that the slip occurs on the whole surface then from (2) and (iii) it follows that σ t = κ s u t. Thus we obtain () with κ = κ s. The conditions analogous to our stick-slip boundary conditions are wellknown in contact problems of solid mechanics. They represent the Tresca friction law on common interfaces of solid bodies [7, ]. As the algebraic problems arising from nite element approximations are formally the same, the algorithms, which are highly ecient for contact problems could serve as natural tools also for solving this type of ow problems. Numerical experiments in [] proved that the algorithm based on an interior point method [] is suitable for this kind of problems, too. In this paper we test this model for physically realistic problems. The paper is organized as follows: Section 2 presents the classical and weak formulation of the Stokes problem with the stick-slip boundary conditions. The algebraic formulation of this problem and its dual form in terms of the pressure, shear and normal stress are derived in Section 3. In Section 4 the path-following interior point method is described. Three model examples, namely ow between parallel plates, ow initialized by rotating 2

concentric and eccentric circles are computed. The results of rst two examples are compared with the ones obtained by ANSYS Fluent [6] and also with the analytic solutions [3]. 2 Classical and weak formulation of the problem Let Ω be a bounded domain in R 2 with a suciently smooth boundary Ω that is split into three disjoint parts: Ω = γ D γ N γ C. We consider the model of a viscous incompressible Newtonian uid modelled by the Stokes system with the Dirichlet and Neumann boundary conditions on γ D and γ N, respectively, and with the impermeability and the stick-slip boundary conditions prescribed on γ C : η u + p = f in Ω, u = in Ω, u = u D on γ D, σ = σ N on γ N, u n = on γ C, σ t g on γ C, σ t (x) < g(x) u t (x) = x γ C, σ t (x) = g(x) κ s := κ s (x) : σ t (x) = κ s u t (x) x γ C, (3) where σ = η du pn. Here, u = (u dn, u 2 ) is the ow velocity, p is the pressure, f = (f, f 2 ) represents volume forces acting on the uid, η > is the dynamic viscosity, and u D, σ N are given Dirichlet and Neumann boundary data, respectively. Further n = (n, n 2 ), t = ( n 2, n ) is the unit outer normal and tangential vector to Ω, respectively, while u n = u n, u t = u t is the normal and tangential component of u along γ C, respectively. Finally σ t = σ t is the shear stress and g is a given slip bound function on γ C. Note that κ s is not known à-priori. Its value can be computed by (2) only having the solution of (3) at our disposal. We will assume that γ D and γ C. Next we present the weak velocity-pressure formulation of (3). For the sake of simplicity we will suppose here that u D =. We introduce the following notation: and V (Ω) = {v ( H (Ω) ) 2 : v = on γd, v n = on γ C } a(v, w) = η v : w dx, Ω 3 b(v, q) = q v dx, Ω

l(v) = f v dx + Ω σ N v ds, γ N j(v) = g v t ds, γ C where v : w = v w + v 2 w 2, v = (v, v 2 ), w = (w, w 2 ) and g is a bounded, nonnegative function on γ C. Finally, H (Ω) stands for the Sobolev space of functions which are together with their rst derivatives square integrable in Ω (see [2]). The velocity-pressure formulation of (3) reads as follows: Find (u, p) V (Ω) L 2 (Ω) such that a(u, v u) + b(v u, p) + j(v) j(u) l(v u) v V (Ω), (4) b(u, q) = q L 2 (Ω). This problem has a unique solution provided that γ N is nonempty. In the opposite case the pressure is determined up to a constant. These results have been established in [2, 4]. To discretize (4) we use mixed nite elements satisfying the inf-sup stability condition [4]. In particular, we use the P-bubble/P elements [8], for which we observed a stable behavior of the Lagrange multipliers representing the shear stress []. 3 Algebraic formulation of the problem The nite element approximation of (4) leads to the following algebraic problem: Find (u, p) R nu R np such that u A(v u) + (v u) B p + g ( Tv Tu ) l (v u) v R nu, q Bu = q R np, Nu =, where u, p is the vector of the nodal values of the velocity u and the pressure p, respectively, A R nu nu is a symmetric and positive denite stiness matrix, B R np nu, T, N R nc nu are full row-rank matrices, l R nu, g R nc + are the load vector and the vector of the nodal values of g on γ C, respectively. Further x = ( x,..., x nc ) for x R nc ; n p is the total number of the nodes of a used triangulation contained in Ω, n c is the number of the nodes lying on γ C \γ D, and n u is the dimension of the solution component representing the velocity u. It is easy to show that (5) is equivalent to the following minimization problem: Find u V such that J (u) J (v) v V, (6) 4 (5)

where J (v) = 2 v Av v l + g Tv and V = {v R nu : Nv =, Bv = }. To release the discrete impermeability condition Nv = and to regularize the last non-dierentiable slip term in J, we introduce two algebraic Lagrange multipliers λ n and λ t, respectively, and dene the Lagrangian L : R nu Λ R by L(v, λ) = 2 v Av v l + λ Cv, where Λ = {λ t R nc : λ t g} R nc+np, λ = (λ t, λ n, p ) Λ, and C = (T, N, B ). The minimization problem (6) is equivalent to the following saddle-point formulation: Find (u, λ) } R nu Λ such that L(u, λ) L(u, λ) L(v, λ) (v, λ) R nu Λ. From the second inequality in (7) we see that (7) u = A (l C λ). (8) Inserting (8) into the rst inequality in (7) we get the dual problem in terms of λ only: Find λ Λ such that S( λ) S(λ) λ Λ (9) with S(λ) = 2 λ Fλ λ d, where F = CA C is symmetric, positive definite and d = CA l. The dual formulation (9) is the minimization of the strictly quadratic function S subject to a small number (n c ) of constrained unknowns versus a large number (n c +n p = n c +O(n 2 c)) of the unconstrained ones. The optimization algorithm appropriate for problems with this structure is described in the next section. 4 Path-following interior point algorithm The Lagrangian associated with (9) is dened by L(λ, µ) = S(λ) + µ ( λ t g) + µ 2 (λ t g), where µ = (µ, µ 2 ) R 2nc is the Lagrange multiplier releasing two sided constraint appearing in the denition of Λ. Let z := µ L(λ, µ) be the new variable used in the function G : R 6nc+np R 6nc+np dened by G(w) := ( λ L(λ, µ), ( µ L(λ, µ) + z), e MZ), 5

where w = (λ, µ, z ) R 6nc+np, M = diag(µ), Z = diag(z), and e R 2nc is the vector whose all components are equal to. The solution λ to (9) is the rst component of the vector w = ( λ, µ, z ) which satises G(w) =, µ, z, () since () is equivalent to the respective Karush-Khun-Tucker conditions. To derive the path-following algorithm, we replace () by the following perturbed problem: G(w) = (,, τe ), µ >, z >, () where τ R +. Solutions w τ to () dene a curve C(τ) in R 6nc+np called the central path. This curve approaches w when τ tends to zero. We combine the damped Newton method for solving () with an appropriate change of τ which guarantees that the iterations belong to a neighbourhood N (c, c 2 ) of C(τ) dened by N (c, c 2 ) = {w = (λ, µ, z ) R 6nc+np : µ i z i c ϑ, i =,..., 2n c, µ, z, λ L(λ, µ) c 2 ϑ, µ L(λ, µ) + z c 2 ϑ}, where c (, ], c 2, and ϑ := ϑ(w) = µ z/(2n c ). In the k-th iteration, we modify τ := τ (k) by the product of ϑ (k) = ϑ(w (k) ) with the centering parameter c (k) chosen as in [3]. To get the monotonically decreasing sequence {ϑ (k) }, the algorithm uses also the Armijo-type condition (3). By J(w) in (2), we denote the Jacobi matrix of G at w. The bounds on the parameters mentioned in the initialization section follow from the convergence analysis presented in []. Algorithm PF: Given c (, ], c 2, < c min c max /2, ω (, ), and ε. Let w () N (c, c 2 ) and set k :=. (i). Choose c (k) [c min, c max ]; (ii). Solve J(w (k) ) w (k+) = G(w (k) ) + (,, c (k) ϑ (k) e ) ; (2) (iii). Set w (k+) = w (k) + α (k) w (k+) with the largest α (k) (, ] satisfying w (k+) N (c, c 2 ) and ϑ (k+) ( α (k) ω( c (k) ))ϑ (k) ; (3) (iv). Return w = w (k+), if err (k) := w (k+) w (k) / w (k+) ε, else set k := k + and go to step (i). 6

The computational eciency depends on a method used for solving the inner linear systems (2). The Jacobi matrix is non-symmetric and indenite with the following block structure: J(w (k) ) = F J 2 J 2 I Z M, J 2 = ( I I Eliminating the 2nd and 3rd unknown in w (k+), we get the reduced linear system for λ (k+) with the Schur complement: J SC = F + M Z + M 2 Z 2, where Z = diag(z, Z 2 ) and M = diag(m, M 2 ). As µ (k) >, z (k) >, the matrix J SC is symmetric, positive denite and the reduced linear system can be solved by the conjugate gradient method. In order to guarantee its convergence, we use the preconditioner: P SC = D + M Z + M 2 Z 2, where D = diag(f). All eigenvalues of the preconditioned matrix P SC J SC belong to an interval which does not depend on the iteration and the spectral condition number is bounded by (see []): cond(p SC J SC) cond(d)cond(f). In computations we approximate D so that A in F is replaced by diag(a). The conjugate gradient method in the k-th step of Algorithm PF is initialized and terminated adaptively. The initial iteration is taken as the computed result of the previous iteration and the (inner) iterations are terminated, if the relative residuum is less than the stopping tolerance given by tol (k) = min{r tol err (k ), c fact tol (k ) }, where < r tol <, < c fact <, err ( ) =, and tol ( ) = r tol /c fact. ). 5 Examples of uid slippage ow In this section we present numerical results of three physically realistic problems [6, 4, 5] governed by the Stokes system with the stick-slip boundary conditions. To justify this ow model we compare some characteristic quantities computed using the solutions of the rst two examples with the ones 7

resulting from the Stokes problem with the Navier boundary condition () and also with known analytic solutions. The second and the third examples are motivated by a ow inside a concentric and eccentric bearing, respectively. All our codes are implemented in Matlab 23b [7]. The computations were performed by ANSELM supercomputer at IT4I V B-TU Ostrava. We use Algorithm PF with c =., c 2 = 9, c min = 2, c max =.5, ω =., ε = 4, r tol =.5, c fact =.9. It turns out that these values are optimal as follows from numerical tests in []. 5. Flow between parallel plates Let Ω = (.5,.5) (.75,.75) (in meters). The boundary Ω is split into the following parts: γ D = {.5} (.75,.75), γ N = {.5} (.75,.75), and γ C = γ C, γ C,2, γ C, = (.5,.5) {.75}, γ C,2 = (.5,.5) {.75}; see Figure. Further the liquid density ρ = 998.2 [kg m 3 ], the kinematical viscosity ν =.48 6 [m 2 s ], the dynamical viscosity η = ρν =.3 [Pa s], the volume forces f = [N m 3 ], the inow velocity u D = (., ) [m s ], and the outow stress σ N = [Pa]. The computations are carried out with n u = 2, n p = 25, n c = 8. Figure : Geometry of the channel Ω with diameter d =.5 [m] and length L = [m]. The outow velocity prole for the Stokes system with the Navier condition () and κ = const. > prescribed on the walls of a channel is given by (see [3]): u = p ( x2 2 x 2η r κ r2 2η ) on γ N, (4) 8

where u is the rst component of the velocity u. The value of κ [Pa s m ] can be computed from (4): κ = p r (x c ) x u (x c ), (5) where x c := γ C, γ N. To compare the model with the Navier and the stickslip conditions we introduce a new quantity κ num (x c ) which will be computed by (5) but using the solution to (3) this time. Now we solve the problem (3) for several suciently small constant thresholds g chosen in such a way that the slip occurs along the whole γ C. From the found solution we compute κ s at x c by (2) and compare with κ num at x c from (5). Both values practically coincide as seen from Figure 2(a). Note that if g is constant on the whole γ C then due to the symmetry of the setting also u t is constant there and consequently κ s is constant. Figure 2(b) shows that the computed u t (x c ) depends linearly on g for small values of g and hence on σ t which conrms (). The main benet of the stick-slip conditions is automatic switching between the stick/slip mode for g large enough as apparent from Figure 2(b)..2 κ s κ num..9.8 κ(x c ).8.6.4 u t (x c ).7.6.5.4.3.2.2..5..5.2.25.3 (a) g..2.3.4.5.6 (b) g Figure 2: (a) Comparison of κ num and κ s at x c ; (b) dependence of u t (x c ) on the slip bound g. The computed outow velocity proles on γ C of solutions to (3) for several values of g are depicted in Figure 3. Now we solve the Stokes system with the Navier condition () prescribed on γ C by ANSYS Fluent (based on the nite volume method) for dierent adhesive coecients κ. The computed values of σ t (x c ) and u t (x c ) are summarized in Table. In the next step we solve problem (3) with the constant 9

.6.6.4.4.2.2...8 u.8.6.4.2 κ s = noslip / g= κ s = / g=.29 κ s =.5 / g=.225 κ s =. / g=.8 u.6.4.2 κ =.69 / g =.29 s κ s =.34 / g γ =.225 N κ s =.66 / g =.8 κ = / g = s 8 6 4 2 2 4 6 8 γ N x 3 (a).2 8 6 4 2 2 4 6 8 x 3 (b) Figure 3: Outow velocity proles. (a) the same value of g on both parts of γ C ; (b) dierent values of g := g on γ C, and xed value g = on γ C,2 lead to the sticking eect on γ C,2. The values κ s := κ s (x c ) are computed by (2), and (5). slip bounds g := σ t (x c ) with σ t (x c ) taken from Table. Using the respective numerical solutions of (3), the values of κ s (x c ), κ num (x c ) computed by (2), and (5), respectively and u t (x c ) are shown in the columns of Table 2. Comparing Table and 2 we see a very good agreement of the results. κ σ t (x c ) u t (x c ).29822.29822.8.27836.34794.5.239.4638..823.8228..979.9797..99.99788 Table : Stokes problem with the Navier condition () for dierent κ (by ANSYS Fluent). There is yet another verication of the computed results. If a couple (u, p) satises the Stokes system (3),2 in the channel Ω with the right hand side f = together with the impermeability condition (3) 5 on γ C then due to the special geometry of Ω, the following identity holds :

g κ s (x c ) by (2) κ num (x c ) by (5) u t (x c ).29822.964.9744.2738.27836.865347.869922.3266.239.53323.53626.43492.823.2527.372.86.979.3.82.9764.99..6.99752 Table 2: Values of κ s (x c ), κ num (x c ) and u t (x c ) for dierent g. p in p out := d γ ( p dx 2 p dx 2 ) = D γ N d ( σ t dx σ t dx ) (6) γ C, γ C,2 making use of the Green theorem. The left-hand side in (6) represents the hydraulic losses. In particular, (6) holds for solutions (u, p) to (3) whose right hand side f =. Let us observe that the velocity u D = (u D,, ) prescribed on γ D does not appear explicitly but only implicitly through σ t and p in (6). Suppose that we solve (3) with u D := u D xed and g := g, where g is chosen in such a way that the slip occurs along the whole γ C. Then necessarily σ t = g on γ C, and σ t = g on γ C,2 as follows from the stick-slip boundary conditions. Consequently, the right hand side of (6) depends solely on g. Therefore the hydraulic losses are the same for any velocity u D used in (3) 3 which guarantees that the respective shear stress σ t still satises σ t = g on γ C, and σ t = g on γ C,2. In Table 3 we check this property for dierent u D, and xed g = 9.7965 4 on γ C. One can observe the satisfaction of (6) under the level of the terminating tolerance ε = 4. u D, κ s (x c ) by (2) κ num (x c ) by (5) (p in p out ) 2g L/d..29467.2983 5 6.3.3552.3567 5 6.5.2588.2646 6 6.7.4492.4532 5 6.9.82.23 5 6 Table 3: Verication of (6) for dierent u D,. It is evident that the slip bound g (and the adhesive coecient κ) significantly inuences the hydraulic losses.one can expect that wall features will fundamentally aect more sophisticated turbulence models.

Figure 4 shows the dependence of the hydraulic losses on the adhesive coecient κ num which is equal to κ s dened by (2) as mentioned above. From the practical point of view, surfaces with κ num < 2 are desired. 5 4.5 4 hydraulic losses 3.5 3 2.5 2.5.5 2 κ num 3 4 Figure 4: Dependence of the hydraulic losses on the adhesive coecient κ num. 5.2 Flow initialized by a rotating concentric inner cylinder Let Ω = {(x, x 2 ) R 2 : R 2 < x 2 + x 2 2 < R 2 2}, where R =.2 [m] and R 2 =.23 [m]. The decomposition of the boundary Ω is as follows: γ D = {(x, x 2 ) R 2 : x 2 +x 2 2 = R 2 }, γ C = {(x, x 2 ) R 2 : x 2 +x 2 2 = R 2 2}, and γ N =. The problem (3) is solved for f =, η =.3, and u D = ω R t, where t is the unit vector tangential to γ D and ω = 2πf q = 62.839 [rad s ] is the angular velocity of rotating cylinder γ D with the frequency f q = [Hz]; see Figure 5(a). Assuming γ D as perfectly wetted, the nonzero velocity u D simulates the initialization of owing by a rotation of the concentric inner cylinder. We assume the outer cylinder γ C partially wettable that is modelled by dierent g (and κ). Our computations are performed with n u = 55826, n p = 323, n c = 337. A MATLAB nite element mesh generator was used for the triangulation of Ω, see [9]. Due to the setting of the problem, the velocity eld u solving the Stokes system with the Navier condition on γ C is rotationally symmetric, i.e. it can be expressed in the polar coordinates with the radial function u := u(r) 2

9 8 κ s 7 κ num 6 5 κ 4 3 2 (a).5.5 2 2.5 3 g (b) Figure 5: (a) Rotating cylinder γ D and the stick-slip boundary conditions on γ C ; (b) comparison of κ s and κ num on γ C. which depends only on the radius r. The velocity prole of this ow in Ω is given by ( ω R 2 R 2 u(r) = 2 R2 2 + R(2η/(κR 2 2 ) ) r r + 2ηr ), r R, R 2. (7) κr 2 The tangential velocity u t (x), x γ C is equal to u(r 2 ), i.e. ( ) ω R 2 2η u(r 2 ) =. (8) R2 2 + R(2η/(κR 2 2 ) ) κ Let us mention that u t and σ t are constant on γ C. The constant value of κ on γ C can be expressed from (8) as κ = 2ηR 2 ω u(r 2 )/R 2 (R2 2 R)u(R 2 2 ). (9) The presentation of the results has the same structure as in the previous example. First, problem (3) is computed for small values of g = const. which guarantee that the slip occurs on γ C. Clearly, u t and σ t are again constant on γ C and so κ s is. In Figure 5(b) we compare κ s computed by (2) with κ num computed by (9) but using the solution to (3) for dierent values of g. We see that κ s and κ num are the same. The dependence of u t on g, κ num is depicted in Figure 6(a), and (b), respectively. We see that u t is a 3

linear function of g which conrms () while the graph of u t as a function of κ num coincides with the graph of the right hand side of (8) considered as a function of κ..3.3.2.2...9.9 u t.8 u t.8.7.7.6.6.5.5.4.4.5.5 2 2.5 3 g (a) 2 3 4 5 6 7 8 9 κ num (b) Figure 6: Dependence of u t on (a) the slip bound g, (b) the adhesive coecient κ num. Then we compute by ANSYS Fluent the Stokes system with the Navier condition () for dierent κ (Table 4-left). The rst row (no slip) of this table is obtained from the solution with the no slip boundary condition u t = on γ C. The resulting σ t denes the slip bound g which will be used now in (3). Table 4-right now collects the results: κ s computed by (2), the tangential velocity u t and the tangential velocity u t computed by (8) but using κ s in place of κ. 5.3 Flow initialized by a rotating eccentric inner cylinder The previous two examples were used to compare solutions of the Stokes system with the Navier boundary condition on one hand and the stick-slip conditions on the other hand. In the next example we chose data to obtain all three modes: complete stick or slip on γ C and simultaneous stick on one part and slip on another part of γ C. We modify the domain from the previous example by shifting the inner cylinder. Let Ω = {(x, x 2 ) R 2 : x 2 + x 2 2 < R 2 2} \ ω, where ω = {(x, x 2 ) R 2 : (x e) 2 + x 2 2 < R 2 } with e =.5 [m], γ D = ω and γ C = Ω \ γ D ; 4

Navier by Fluent Stick-slip by MATLAB κ σ t u(r 2 ) g κ s by (2) u t u t by (8) no slip 4.695 4.695 2.58e4.73e-4.62e-4 2.5784.7892.5784 2.22.7926.7882.9745.975.9745.2.9773.974.8.88.226.88.899.25.22.5.5523.46.5523.566.64.44..236.2362.236.387.2375.237..27.269.27..275.275...273...2749.275 Table 4: left - Stokes problem with the Navier condition for dierent κ (solved by ANSYS Fluent); right - the adhesive coecient κ s and u t on γ C for different g (except the rst row) solved by MATLAB. see Figure 7(a). The remaining data of the problem are the same as in Subsection 5.2. Our computations are performed with n u = 33, n p = 754, n c = 248. A MATLAB nite element mesh generator was used for the triangulation of Ω, see [9]. A part of the zoomed mesh is in Figure 7(b). -4 5 4 x 2 3 2 (a) -.24 -.23 -.22 -.2 -.2 -.99 -.98 -.97 x (b) Figure 7: Example: (a) eccentric rotating cylinder; (b) zoom of the mesh. First we solve problem (3) with the slip bound g =.894 when the slip occurs on the whole γ C. In Figure 8(a),(b) we see the distribution of κ s computed by (2) and u t, respectively along γ C. Next we solve problem (3) with the constant slip bound g =.633 on γ C (a part of the circle of radius g is in red in Figure 9(a)). The distribution 5

.4 2.2 κ s.8.6.4 u t.5.5.2 5 5 2 25 clockwise from A (a) γ 5 5 2 γ 25 C C clockwise from A Figure 8: Distribution of: (a) adhesive coecient κ s (b) tangential velocity u t on γ C for g =.894. (b) of σ t is in green and γ C itself in blue. One can see that there is a small part of γ C (on the left) where σ t is below g and accordingly to (3) 7 the tangential component u t should be equal to zero there. This is conrmed by Figure 9(b) which depicts the distribution of u t on γ C. γ C.5 g u t.5 σ t (a) 5 5 2 25 clockwise from A (b) γ C Figure 9: Distribution of (a) shear stress σ t (b) tangential velocity u t on γ C for g =.633. Finally, we solve (3) with g = 5. In this case σ t is strictly less than g on the whole γ C and so u t = on γ C, see Figure. Let us note that in this case the solution is the same as the one to the Stokes system with the no-slip boundary condition u = on γ C. One can observe the reverse ow in a part of the domain which is a consequence of the continuity equation for incompressible ows, see Fig- 6

ure (g = 5). This eect is caused by a relatively high velocity gradient in the wider part of the domain. g γ C σ t Figure : The distribution of σ t on γ C for g = 5. Velocity elds in the part of Ω highlighted in Figure 7(a) and the distribution of the pressure on γ C are shown in Figure and 2, respectively. 7 6-4 7 6-4 7 6-4 x 2 5 4 x 2 5 4 5 x 2 4 2 γ C γ D 2 γ C γ D 2 γ C γ D 3 3 3 -.24 -.23 -.22 -.2x -.2 -.99 -.98 -.97 -.24 -.23 -.22 -.2x -.2 -.99 -.98 -.97 -.24 -.23 -.22 -.2x -.2 -.99 -.98 -.97 g =.894 g =.633 g = 5 Figure : Zoomed velocity elds u. 7

2 8 25 8 p 6 4 2 7 6 5 p 4 3 2 2 5 p 5 γ clockwise from A C 5 5 2 25 5 5 2 25 clockwise from A γ C 5 5 2 25 clockwise from A g =.894 g =.633 g = 5 Figure 2: Distribution of the pressure p on γ C. γ C 6 Conclusions and future plans The paper is devoted to numerical solution of the Stokes system with stickslip boundary conditions prescribed on a part of the boundary. Unlike the classical Navier condition, this time the slip may occur only when the shear stress attains a threshold g given à-priori. The resulting numerical model leads to minimization of a strictly convex quadratic function subject to a small number of simple constraints. The minimization itself was carried out by the interior point method. In order to validate this ow model, the obtained numerical results of two model examples for small values of g are compared with the ones computed by ANSYS Fluent and also with the known analytic solutions. The results are practically identical. As the next step of our research we plan to extend the stick-slip boundary conditions to the case when the threshold g depends on the solution itself (this seems to be more practical for the real hydrophobic materials), more precisely g := g( u t ) and to the stick-slip conditions of the Coulomb type. Acknowledgement This work has been supported by the IT4IXS - IT4Innovations Excellence in Science project (LQ62) (JH,RK,VS). Grant Agency of the Czech Republic within the project and GA/5-662S is gratefully acknowledged for the support of this work (SF,FP). References [] M. Anitescu and F. A. Potra. Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity prob- 8

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