Unlike the Navier boundary condition, this paper deals with the case when the slip of a fluid along the wall may occur only when the shear stress attains certain bound which is given a priori and does not depend on the solution itself. The mathematical model of the velocity–pressure formulation with this type of threshold slip boundary condition is given by the so-called variational inequality of the second kind. For its discretization, we use P1-bubble/P1 mixed finite elements. The resulting algebraic problem leads to the minimization of a nondifferentiable energy function subject to linear equality constraints representing the discrete impermeability and incompressibility condition. To release the former one and to regularize the nonsmooth term characterizing the stick–slip behavior of the algebraic formulation, two additional vectors of Lagrange multipliers are introduced. Further, the velocity vector is eliminated, and the resulting minimization problem for a quadratic function depending on the dual variables (the discrete pressure and the normal and shear stress) is solved by the interior point type method which is briefly described. To justify the threshold model and to illustrate the efficiency of the proposed approach, three physically realistic problems are solved and the results are compared with the ones solving the Stokes problem with the Navier boundary condition.

References

References
1.
Fialová
,
S.
,
2016
, “
Identification of the Properties of Hydrophobic Layers and its Usage in Technical Practice
,” Habilitation, VUTIUM, Brno University of Technology, Brno, the Czech Republic.
2.
Fialová
,
S.
, and
Pochylý
,
F.
,
2014
, “
Identification and Experimental Verification of the Adhesive Coefficient of Hydrophobic Materials
,”
Wasserwirtsch.
,
105
(
1
), pp.
125
129
.
3.
Haslinger
,
J.
,
Hlaváček
,
I.
, and
Nečas
,
J.
,
1996
, “
Numerical Methods for Unilateral Problems in Solid Mechanics
,”
Handbook of Numerical Analysis
, Vol. IV,
North Holland
,
Amsterdam
, Part 2, pp.
313
485
.
4.
Anitescu
,
M.
, and
Potra
,
F. A.
,
1997
, “
Formulating Dynamic Multi-Rigid-Body Contact Problems With Friction as Solvable Linear Complementarity Problems
,”
Nonlinear Dyn.
,
14
(
3
), pp.
231
247
.
5.
Kučera
,
R.
,
Haslinger
,
J.
,
Šátek
,
V.
, and
Jarošová
,
M.
, 2016, “
Efficient Methods for Solving the Stokes Problem With Slip Boundary Conditions
,”
Math. Comput. Simul.
(in press).
6.
Kučera
,
R.
,
Machalová
,
J.
,
Netuka
,
H.
, and
Ženčák
,
P.
,
2013
, “
An Interior Point Algorithm for the Minimization Arising From 3D Contact Problems With Friction
,”
Optim. Method Software
,
28
(
6
), pp.
1195
1217
.
7.
Hammad
,
K. J.
,
2015
, “
The Flow Behavior of a Biofluid in a Separated and Reattached Flow Region
,”
ASME J. Fluids Eng.
,
137
(
6
), p.
061104
.
8.
Corredor
,
F. E. R.
,
Bizhani
,
M.
, and
Kuru
,
E.
,
2015
, “
Experimental Investigation of Drag Reducing Fluid Flow in Annular Geometry Using Particle Image Velocimetry Technique
,”
ASME J. Fluids Eng.
,
137
(
8
), p.
081103
.
9.
Ozogul
,
H.
,
Jay
,
P.
, and
Magnin
,
A.
,
2015
, “
Slipping of the Viscoplastic Fluid Flowing on a Circular Cylinder
,”
ASME J. Fluids Eng.
,
137
(7), p.
071201
.
10.
ANSYS,
2015
, “fluent,” ANSYS, Inc., Canonsburg, PA, accessed Nov. 15,
2015
, http://www.ansys.com/Products/Fluids/ansys+Fluent
11.
Brdička
,
M.
,
Samek
,
L.
, and
Bruno
,
S.
,
2011
,
Mechanika kontinua
,
Academia, Prague
,
Czech Republic
.
12.
Elman
,
H. C.
,
Silvester
,
D. J.
, and
Wathen
,
A. J.
,
2005
,
Finite Elements and Fast Iterative Solvers With Applications in Incompressible Fluid Dynamics
,
Oxford University Press
,
Oxford, UK
.
13.
Nečas
,
J.
,
1967
,
Les Méthodes Directes en Théorie des Equations Elliptiques
,
Masson
,
Paris, France
.
14.
Fujita
,
H.
,
1994
, “
A Mathematical Analysis of Motions of Viscous Incompressible Fluid Under Leak and Slip Boundary Conditions
,”
RIMS Kokyuroku
,
888
, pp.
199
216
.
15.
Bulíček
,
M.
, and
Málek
,
J.
,
2016
, “
On Unsteady Internal Flows of Bingham Fluids Subject to Threshold Slip on the Impermeable Boundary
,”
Recent Developments of Mathematical Fluid Mechanics
,
H.
Amann
,
Y.
Giga
,
H.
Kozono
,
H.
Okamoto
, and
M.
Yamazaki
, eds.,
Springer
,
Heidelberg, Germany
, pp.
135
156
.
16.
Ayadi
,
M.
,
Baffico
,
L.
,
Gdoura
,
M. K.
, and
Sassi
,
T.
,
2014
, “
Error Estimates for Stokes Problem With Tresca Friction Conditions
,”
ESAIM: Math. Modell. Numer. Anal.
,
48
(
5
), pp.
1413
1429
.
17.
Koko
,
J.
,
2012
, “
Vectorized matlab Codes for the Stokes Problem With P1-Bubble/P1 Finite Element
,” http://www.isima.fr/~jkoko/Codes/StokesP1BubbleP1.pdf, accessed July 28, 2016, ISIMA Public Engineering School in Computer Science, Auvergne, France.
18.
Jarošová
,
M.
,
Kučera
,
R.
, and
Šátek
,
V.
,
2015
. “
A New Variant of the Path-Following Algorithm for the Parallel Solving of the Stokes Problem With Friction
,”
Fourth International Conference on Parallel, Distributed, Grid and Cloud Computing for Engineering
, P. Iványi and B. H. V. Topping, eds.,
Civil-Comp Press
, Stirlingshire, Scotland, UK, Paper No. 11.
19.
Nocedal
,
J.
,
Wächter
,
A.
, and
Waltz
,
R. A.
,
2005
, “
Adaptive Barrier Strategies for Nonlinear Interior Methods
,”
Report No. TR RC 23563
, IBM T.J. Watson Research Center, Yorktown Heights, NY.
20.
Pochylý
,
F.
,
Fialová
,
S.
, and
Kozubková
,
M.
,
2011
, “
Journal Bearings With Hydrophobic Surface
,” Vibronadežnos i Germetičnos Centrobežnyh Mašin, Technical Study—Monography, pp.
314
320
.
21.
Pochylý
,
F.
,
Fialová
,
S.
, and
Malenovský
,
E.
,
2012
, “
Bearing With Magnetic Fluid and Hydrophobic Surface of the Lining
,”
IOP Conf. Ser.: Earth Environ. Sci.
,
15
(
2
), pp.
1
9
.
22.
MATLAB, 2016, “MATLAB: The Language of Technical Computing,” http://www.mathworks.com/products/matlab/, The Mathworks, Inc., Natick, MA.
23.
Hron
,
J.
,
Roux
,
C. L.
,
Málek
,
J.
, and
Rajagopal
,
K.
,
2015
, “
Flows of Incompressible Fluids Subject to Navier’s Slip on the Boundary
,”
Comput. Math. Appl.
,
56
(8), pp.
2128
2143
.
24.
Koko
,
J.
,
2015
, “
A matlab Mesh Generator for the Two-Dimensional Finite Element Method
,”
Appl. Math. Comput.
,
250
, pp.
650
664
.
You do not currently have access to this content.