Abstract
Within the approximation of Stokes hydrodynamics, several problems of a steady-state flow over a two-dimensional cavity containing a gas bubble are solved using the method of boundary integral equations. In contrast to previous publications, the method developed makes it possible to study the situation in which the cavity is only partially filled with gas, and the edges of a curved phase interface do not coincide with the cavity corners. Using periodic boundary conditions for the velocity, the flows with pure-shear and parabolic velocity profiles, and also the flow over a group of cavities were considered. The aim of the study was to calculate the effective (average) slip velocity over a microcavity, as applied to flows near textured superhydrophobic surfaces. A parametric numerical study of the effective velocity slip as a function of the radius of curvature of the interface and the position of the interface relative to the cavity boundaries was performed. The accuracy of the method is validated by the calculations of a number of limiting flows over a cavity, for which a quantitative agreement with the results known in the literature is demonstrated.
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References
A.B.D. Cassie and S. Baxter, “Wettability of Porous Surfaces,” Trans. Faraday Soc. 40, 546–551 (1944).
A. Lafuma and D. Quere, “Superhydrophobic States,”, 2, 457–463 (2003).
J.P. Rothstein, “Slip on Superhydrophobic Surfaces,” Annu. Rev. Fluid Mech. 42, 89–102 (2010).
M.Z. Bazant and O.I. Vinogradova, “Tensorial Hydrophobic Slip,” J. Fluid Mech. 613, 125–134 (2008).
J.R. Philip, “Flows Satisfying Mixed No-Slip and No-Shear Conditions,” J. Appl. Math. Phys. (ZAMP) 23, 353–372 (1972).
E. Lauga and H.A. Stone, “Effective Slip in Pressure-Driven Stokes Flow,” J. Fluid Mech. 489, 55–77 (2003).
C.J. Teo and B.C. Khoo, “Analysis of Stokes Flow in Microchannels with Superhydrophobic Surfaces Containing a Periodic Array of Micro-Grooves,” Microfluid and Nanofluid 7, 353–382 (2009).
O.I. Vinogradova and A.V. Belyaev, “Wetting, Roughness, and Flow Boundary Conditions,” J. Phys.: Condens. Matter 23, 184104 (2011).
E.S. Asmolov and O.I. Vinogradova, “Effective Slip Boundary Condition for Arbitrary One-Dimensional Surfaces,” J. Fluid Mech. 706, 108–117 (2012).
E.S. Asmolov, J. Zhou, F. Schmidt, and O.I. Vinogradova, “Effective Slip-Length Tensor for a Flow Over aWeakly Slipping Strips,” Phys. Rev. E 88, 023004 (2013).
C. Schonecker and S. Hardt, “Longitudinal and Transverse Flow Over a Cavity Containing a Second Immiscible Fluid,” J. Fluid Mech. 717, 376–394 (2013).
A.I. Ageev and A.N. Osiptsov, “Self-Similar Regimes of Liquid-Layer Spreading along a Superhydrophobic Surface,” Fluid Dynamics 49 (3), 330–342 (2014).
A.I. Ageev and A.N. Osiptsov, “Viscous-Fluid Streamlet Flow Down an Inclined Superhydrophobic Surface”, Doklady Physics 59 (10), 476–479 (2014).
C.O. Ng and C.Y. Wang, “Stokes Shear Flow Over a Grating: Implications for Superhydrophobic Slip,” Phys. Fluids 21, 013602 (2009).
A.M.J. Davis and E. Lauga, “Hydrodynamic Friction of Fakir-Like Superhydrophobic Surface,” J. Fluid Mech. 661, 402–411 (2010).
A. Steinberger, C. Cottin-Bizonne, P. Kleimann, and E. Charlaix, “High Friction on a Bubble Mattress,” Nature Materials 6, 665–668 (2007).
G. Bolognesi, C. Cottin-Bizonne, and C. Pirat, “Experimental Evidence of Slippage Breakdown for a Superhydrophobic Surface in a Microfluidic Device,” Phys. Fluids 26, 082004 (2014).
A.M.J. Davis and E. Lauga, “Geometric Transition in Friction for Flow over a Bubble Mattress,” Phys. Fluids 21, 011701 (2009).
D. Crowdy, “Slip Length for Longitudinal Shear Flow over a Periodic Mattress of Protruding Bubbles,” Phys. Fluids 22, 121703 (2010).
O.A. Ladyzhenskaya, Mathematical Problems of Dynamics of Viscous Incompressible Fluid [in Russian] (Nauka, Moscow, 1970).
C. Pozrikdis, Boundary Integral and Singularity Methods for Linearized Viscous Flow (Cambridge University Press, New York, 1992).
J.J.L. Higdon, “Stokes Flow in Arbitrary Two-Dimensional Domains: Shear Flow over Ridges and Cavities,” J. Fluid Mech. 159, 195–226 (1985).
V.A. Yakutenok, “Numerical Modeling of Slow Liquid Flows with a Free Surface Using the Boundary Element Method,” Metem. Model. 4(10), 62–70.
M.A. Ponomareva, G.R. Shrager, and V.A. Yakutenok, “Stability of a Plane Jet of a Highly Viscous Fluid Impinging on a Horizontal Solid Wall,” Fluid Dynamics 46(1), 44–50 (2011).
O.A. Abramova, Y.A. Itkulova, N.A. Gumerov, and I.Sh. Akhatov, “An Efficient Method for Simulation of the Dynamics of a Large Number of Deformable Droplets in the Stokes Regime,” Doklady Physics 59(5), 236–240 (2014).
H.K. Moffat, “Viscous and Resistive Eddies Near a Sharp Corner,” J. Fluid Mech. 18 (1), 1–18 (1964).
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Original Russian Text © A.I. Ageev, N.A. Osiptsov, 2015, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2015, Vol. 50, No. 6, pp. 35–49.
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Ageev, A.I., Osiptsov, N.A. Stokes flow over a cavity on a superhydrophobic surface containing a gas bubble. Fluid Dyn 50, 748–758 (2015). https://doi.org/10.1134/S0015462815060046
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DOI: https://doi.org/10.1134/S0015462815060046