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Research Papers: Fundamental Issues and Canonical Flows

Modeling of Hydrophobic Surfaces by the Stokes Problem With the Stick–Slip Boundary Conditions

[+] Author and Article Information
R. Kučera

IT4Innovations,
VŠB-TU Ostrava,
17 listopadu 15/2172,
Ostrava-Poruba 708 33, Czech Republic
e-mail: radek.kucera@vsb.cz

V. Šátek

IT4Innovations,
VŠB-TU Ostrava,
17 listopadu 15/2172,
Ostrava-Poruba 708 33, Czech Republic
e-mail: vaclav.satek@vsb.cz

J. Haslinger

IG CAS,
Studentská 1768,
Ostrava-Poruba 708 00, Czech Republic
e-mail: hasling@karlin.mff.cuni.cz

S. Fialová

Victor Kaplan Department of Fluid Engineering,
Brno University of Technology,
Technická 2896/2,
Brno 616 69, Czech Republic
e-mail: fialova@fme.vutbr.cz

F. Pochylý

Victor Kaplan Department of Fluid Engineering,
Brno University of Technology,
Technická 2896/2,
Brno 616 69, Czech Republic
e-mail: pochyly@fme.vutbr.cz

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received February 3, 2016; final manuscript received July 12, 2016; published online October 10, 2016. Assoc. Editor: Matevz Dular.

J. Fluids Eng 139(1), 011202 (Oct 10, 2016) (9 pages) Paper No: FE-16-1073; doi: 10.1115/1.4034199 History: Received February 03, 2016; Revised July 12, 2016

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.

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Figures

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Fig. 1

Geometry of the channel Ω with diameter d = 0.015 (m) and length L = 1 (m)

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Fig. 2

(a) Comparison of κnum and κs at xc and (b) dependence of |ut(xc)| on the slip bound g

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Fig. 3

Outflow velocity profiles: (a) the same value of g on both parts of γC and (b) different values of g:=g1 on γC,1 and fixed value g = 1 on γC,2 lead to the sticking effect on γC,2. The values κs:=κs(xc) are computed by Eqs. (2) and (23).

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Fig. 4

Dependence of the hydraulic losses on the adhesive coefficient κnum

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Fig. 5

(a) Rotating cylinder γD and the stick–slip boundary conditions on γC and (b) comparison of κs and κnum on γC

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Fig. 6

Dependence of ut on (a) the slip bound g and (b) the adhesive coefficient κnum

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Fig. 7

Example: (a) eccentric rotating cylinder and (b) zoom of the mesh

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Fig. 8

Distribution of (a) adhesive coefficient κs and (b) tangential velocity ut on γC for g = 0.18094

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Fig. 9

Distribution of (a) shear stress |σt| and (b) tangential velocity ut on γC for g = 1.60313

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Fig. 10

The distribution of |σt| on γC for g = 50

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Fig. 11

Zoomed velocity fields u

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Fig. 12

Distribution of the pressure p on γC

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