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June  2011, 15(4): 1019-1044. doi: 10.3934/dcdsb.2011.15.1019

## Boundary integral equation approach for stokes slip flow in rotating mixers

 1 Grupo de Energía y Termodinámica, Escuela de Ingenierías, Universidad Pontificia Bolivariana, Medellín, Circular 1 No. 73-34, Colombia, Colombia 2 Division of Energy and Sustainability, The University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom

Received  March 2010 Revised  June 2010 Published  March 2011

In order to employ continuum models in the analysis of the flow behaviour of a viscous Newtonian fluid at micro scale devices, it is necessary to consider at the wall surfaces appropriate slip boundary conditions instead of the classical non-slip condition. To account for the slip condition at the nano-scale, we used the Navier's type boundary condition that relates the tangential fluid velocity at the boundaries to the tangential shear rate. In this work a boundary integral equation formulation for Stokes slip flow, based on the normal and tangential projection of the Green's integral representational formulae for the Stokes velocity field, which directly incorporates into the integral equations the local tangential shear rate at the wall surfaces, is presented. This formulation is used to numerically simulate concentric and eccentric rotating Couette mixers and a Single rotor mixer, including the effect of thermal creep in cases of rarefied gases. The numerical results obtained for the Couette mixers, concentric and eccentric, are validated again the corresponding analytical solutions, showing excellent agreements.
Citation: César Nieto, Mauricio Giraldo, Henry Power. Boundary integral equation approach for stokes slip flow in rotating mixers. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 1019-1044. doi: 10.3934/dcdsb.2011.15.1019
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##### References:
 [1] Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic & Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639 [2] Jaroslav Haslinger, Raino A. E. Mäkinen, Jan Stebel. Shape optimization for Stokes problem with threshold slip boundary conditions. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1281-1301. doi: 10.3934/dcdss.2017069 [3] W. G. Litvinov. Problem on stationary flow of electrorheological fluids at the generalized conditions of slip on the boundary. Communications on Pure & Applied Analysis, 2007, 6 (1) : 247-277. doi: 10.3934/cpaa.2007.6.247 [4] Chun Liu, Jie Shen. On liquid crystal flows with free-slip boundary conditions. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 307-318. doi: 10.3934/dcds.2001.7.307 [5] Boris Muha, Zvonimir Tutek. Note on evolutionary free piston problem for Stokes equations with slip boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1629-1639. doi: 10.3934/cpaa.2014.13.1629 [6] Xin Liu. Compressible viscous flows in a symmetric domain with complete slip boundary: The nonlinear stability of uniformly rotating states with small angular velocities. Communications on Pure & Applied Analysis, 2019, 18 (2) : 751-794. doi: 10.3934/cpaa.2019037 [7] Yosra Boukari, Houssem Haddar. The factorization method applied to cracks with impedance boundary conditions. Inverse Problems & Imaging, 2013, 7 (4) : 1123-1138. doi: 10.3934/ipi.2013.7.1123 [8] Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148 [9] Donatella Donatelli, Eduard Feireisl, Antonín Novotný. On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 783-798. doi: 10.3934/dcdsb.2010.13.783 [10] Hugo Beirão da Veiga. A challenging open problem: The inviscid limit under slip-type boundary conditions.. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 231-236. doi: 10.3934/dcdss.2010.3.231 [11] María Anguiano, Francisco Javier Suárez-Grau. Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions. Networks & Heterogeneous Media, 2019, 14 (2) : 289-316. doi: 10.3934/nhm.2019012 [12] Irena Lasiecka, To Fu Ma, Rodrigo Nunes Monteiro. Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1037-1072. doi: 10.3934/dcdsb.2018141 [13] Daniele Boffi, Lucia Gastaldi. Discrete models for fluid-structure interactions: The finite element Immersed Boundary Method. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 89-107. doi: 10.3934/dcdss.2016.9.89 [14] Mahamadi Warma. Semi linear parabolic equations with nonlinear general Wentzell boundary conditions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5493-5506. doi: 10.3934/dcds.2013.33.5493 [15] Frederic Rousset. The residual boundary conditions coming from the real vanishing viscosity method. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 605-625. doi: 10.3934/dcds.2002.8.606 [16] María J. Rivera, Juan A. López Molina, Macarena Trujillo, Enrique J. Berjano. Theoretical modeling of RF ablation with internally cooled electrodes: Comparative study of different thermal boundary conditions at the electrode-tissue interface. Mathematical Biosciences & Engineering, 2009, 6 (3) : 611-627. doi: 10.3934/mbe.2009.6.611 [17] Kersten Schmidt, Ralf Hiptmair. Asymptotic boundary element methods for thin conducting sheets. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 619-647. doi: 10.3934/dcdss.2015.8.619 [18] So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343 [19] Bruno Fornet, O. Guès. Penalization approach to semi-linear symmetric hyperbolic problems with dissipative boundary conditions. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 827-845. doi: 10.3934/dcds.2009.23.827 [20] Paul Sacks, Mahamadi Warma. Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 761-787. doi: 10.3934/dcds.2014.34.761

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