Abstract
The current paper is focused on analyzing the flow of viscoplastic fluid in a cavity that is driven by the two walls. The Lattice Boltzmann method (LBM) is used to solve the discrete Boltzmann equation. To represent the stress-strain rate relationship of viscoplastic fluids, the Bingham Papanastasiou constitutive model is considered. Cavity flow filled with Bingham fluids is considered for validating the present LBM code. After successful validation of the code, the analysis is extended for three dissimilar wall motions-simultaneous and opposed movement of the parallel facing walls, and the simultaneous motion of non-facing walls. The flow dynamics of Bingham fluid is influenced by Reynolds and Bingham numbers which can be studied using velocity and streamline plots. Subsequently, the yielded and un-yielded zones in a cavity have been effectively tracked using the limiting condition of yield stress. Further, the effect of wall motion on the variation of those zones inside a cavity has been studied. Finally, the drag coefficient for considered wall motions is presented.
Similar content being viewed by others
References
Aharonov, E. and D.H. Rothman, 1993, Non-Newtonian flow (through porous media): a lattice Boltzmann method, Geophys. Res. Lett. 20, 679–682.
Albensoeder, S., H.C. Kuhlmann, and H.J. Rath, 2001, Multiplicity of steady two - dimensional flows in two-sided lid driven cavities, Theor. Comput. Fluid Dyn. 14, 223–241.
Alleborn, N., H. Raszillier, and F. Durst, 1999, Lid-driven cavity with heat and mass transport, Int. J. Heat Mass Transfer 42, 833–853.
Artoli, A., 2003, Mesoscopic computational haemodynamic, Ph.D. Thesis, University of Amsterdam.
Balmforth, N.J., I.A. Frigaard, and G. Ovarlez, 2014, Yielding to stress: Recent developments in viscoplastic fluid mechanics, Annu. Rev. of Fluid Mech. 46, 121–146.
Benzi, R., S. Succi, and M. Vergassola, 1992, The lattice Boltzmann equation: theory and applications, Phys. Rep. 222, 145–197.
Bhatnagar, P.L., E.P. Gross, and M. Krook, 1954, A model for collision process in gasses, Phys. Rev. 94, 511–525.
Bird, R.B., G.C. Dai, and B.J. Yarusso, 1983, The rheology and flow of viscoplastic materials, Rev. Chem. Eng. 1, 1–70.
Blohm, C.H. and H.C. Kuhlmann, 2002, The two-sided lid-driven cavity: experiments on stationary and time-dependent flows, J. Fluid Mech. 450, 67–95.
Buick, J.M., 2009, Lattice Boltzmann simulation of power-law fluid flow in the mixing section of a single-screw extruder, Chem. Eng. Sci. 64, 52–58.
Cao, Z. and M.N. Esmail, 1995, Numerical study on hydrodynamics of short-dwell paper coaters, AIChE J. 41, 1833–1842.
Chai, Z., B. Shi, Z. Guo, and F. Rong, 2011, Multiple-relaxation-time lattice Boltzmann model for generalized Newtonian fluid flows, J. Non-Newtonian Fluid Mech. 166, 332–342.
Chen, S. and G.D. Doolen, 1998, Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech. 30, 329–364.
Cheng, M. and K.C. Hung, 2006, Vortex structure of steady flow in a rectangular cavity, Comput. Fluids 35, 1046–1062.
Chhabra, R.P., 2007, Bubbles, Drops, and Particles in non-Newtonian fluids, CRC Press, Boca Raton, FL.
Frey, S., F.S. Silveira, and F. Zinani, 2010, Stabilized mixed approximations for inertial viscoplastic fluid flows, Mech. Res. Commun. 37, 145–152.
Frigaard, I.A. and C. Nouar, 2005, On the usage of viscosity regularisation methods for visco-plastic fluid flow computation, J. Non-Newtonian Fluid Mech. 127, 1–26.
Gabbanelli, S., G. Drazer, and J. Koplik, 2005, Lattice Boltzmann method for non-Newtonian (power-Law) fluids, Phys. Rev. E: Stat. Nonlinear Soft Matter Phys. 72, 6312.
Gaskell, P.H., J.L.H.M. Summers, and M.D. Thompson, 1996, Savage Creeping flow analyses of free surface cavity flows, Theor. Comput. Fluid Dyn. 8, 415–433.
Ginzburg, I. and K. Steiner, 2002, free-surface lattice Boltzmann method for modelling the filling of expanding cavities by Bingham fluids, Philos. Trans. R. Soc. London, Ser. A 360, 453–466.
Hellebrand, H., 1996, Tape Casting, in: R. J. Brook (ed.), Processing of Ceramics, Part1, VCH Verlagsgesellschaft mbH, Weinheim 17A, 189–265.
He, X., L.S. Luo, and M. Dembo, 1997, Some progress in the lattice Boltzmann method: Reynolds number enhancement in simulations, Physica A 239, 276–285.
Hedayat, M.M., M.H. Borghei, A. Fakhari, and K. Sadeghy, 2010, On the use of Lattice-Boltzmann model for simulating lid-driven cavity flows of strain-hardening fluids, J. Soc. of Rheol. Jpn. 38, 201–207.
Hou, S., Q. Zou, S. Chen, G. Doolen, and A.C. Cogley, 1995, Simulation of cavity flow by the lattice Boltzmann method, J. Comput. Phys. 118, 329–347.
Kuhlmann, H.C., M. Wanschura, and H.J. Rath, 1997, Flow in two - sided lid-driven cavities: non-uniqueness, instabilities, and cellular structures. J. Fluid Mech. 336, 267–299.
Kuhlmann, H.C., M. Wanschura, and H.J. Rath, 1998, Elliptic instability in two-sided lid-driven cavity flow, Eur. J. Mech., B. Fluids. 17, 561–569.
Lallemand, P. and L.S. Luo, 2000, Theory of the lattice Boltzmann method: Dispersion, Dissipation, Isotropy, Galilean invariance and stability, Phys. Rev. E: Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 61, 6546.
Leong, C.W. and J.M. Ottino, 1989, Experiments on mixing due to chaotic advection in a cavity, J.Fluid Mech. 209, 463–499.
Mendu, S.S. and P.K. Das, 2012, Flow of power-law fluids in a cavity driven by the motion of two facing lids-a simulation by lattice Boltzmann method, J. Non-Newtonian Fluid Mech. 175–176, 10–24.
Miller, W., 1995, Flow in the driven cavity calculated by the lattice Boltzmann method, Phys. Rev. E 51, 3659–3669.
Mitsoulis, E., 2007, Flows of viscoplastic materials: Models and computations, Rheol. Rev. 135–178.
Mitsoulis, E. and Th. Zisis, 2001, Flow of Bingham plastics in a lid-driven square cavity, J. Non-Newtonian Fluid Mech. 101, 173–180.
Neofytou, P., 2005, A 3rd order upwind finite volume method for generalized Newtonian fluid flows, Adv. Eng. Software, 36, 664–680.
Ohta, M., T. Nakamura, Y. Yoshida, and Y. Matsukuma, 2011, Lattice Boltzmann simulation of Viscoplastic fluid flows through complex flow channels, J. Non-Newtonian Fluid Mech. 166, 404–412.
Papanastasiou, T.C., 1987, Flow of materials with yield, J. Rheol. 31, 385–404.
Patil, D.V., K.N. Lakshmisha, and B. Rogg, 2006, Lattice Boltzmann simulation of lid-driven flow in deep cavities, Comput. Fluids 35, 1116–1125.
Perumal, D.A. and A.K. Das, 2008, Simulation of flow in two-sided lid-driven square cavities by the lattice Boltzmann method, WIT Trans. Eng. Sci. 59, 45–54.
Perumal, D.A. and A.K. Das, 2010, Simulation of incompressible flows in two-sided lid-driven square cavities, Part I-FDM, CFD Lett. 2, 13–24.
Prashant, and J.J. Derksen, 2011, Direct simulations of spherical particle motion in Bingham liquids, Comput. Chem. Eng., 35, 1200–1214.
Succi, S., 2001, The lattice Boltzmann equation for fluid dynamics and beyond, Oxford University Press, Oxford.
Sukop, M.C., and D.T. Thorne, 2006, Lattice Boltzmann Modeling: An introduction for Geoscientists and engineers, Springer, Heidelberg.
Sullivan, S.P., L.F. Gladden, and M.L. Johns, 2006, Simulation of power-law fluid flow through porous media using lattice Boltzmann techniques, J. Non-Newtonian Fluid Mech. 133, 91–98.
Syrakos, A., G.C. Georgiou, and A.N. Alexandrou, 2013, Solution of the square lid-driven cavity flow of a Bingham plastic using the finite volume method, J. Non-Newtonian Fluid Mech. 195, 19–31.
Tang, G.H., S.B. Wang, P.X. Ye, and W.Q. Tao, 2011, Bingham fluid simulation with the incompressible lattice Boltzmann model, J. Non-Newtonian Fluid Mech. 166, 145–151.
Triantafillopoulos, N.G. and C.K. Aidun, 1990, Relationship between flow instability in short-dwell ponds and cross directional coat weight non uniformities, TAPPI J, 73, 127–136.
Vikhansky, M., 2008, Lattice-Boltzmann method for yield-stress liquids, J. Non-Newtonian Fluid Mech. 155, 95–100.
Vola, D., L. Boscardin, and J.C. Latché, 2003, Laminar unsteady flows of Bingham fluids: a numerical strategy and some benchmark results, J. Comput. Phys. 187, 441–456.
Wahba, E.M., 2009, Multiplicity of states for two-sided and four-sided lid driven cavity flows, Comput. Fluids 38, 247–253.
Wang, C.H. and J.R. Ho, 2008, Lattice Boltzmann modelling of Bingham plastics, Physica A, 387, 4740–4748.
Wang, C. and J. Ho, 2011, A lattice Boltzmann approach for the non-Newtonian effect in the blood flow, Comput. Math. Appl. 62, 75–86.
Wu, J.S. and Y.L. Shao, 2004, Simulation of lid-driven cavity flows by parallel lattice Boltzmann method using multi-relaxation-time scheme, Int. J. Numer. Methods Fluids, 46, 921–937.
Xie, C., J. Zhang, V. Bertola, and M. Wang, 2016, Lattice Boltzmann modelling for multiphase viscoplastic fluid flow, J. Non-Newtonian Fluid Mech. 234, 118–128.
Zou, Q. and X. He, 1997, On pressure and velocity boundary conditions for the lattice Boltzmann BGK model. Phys. Fluids, 9, 1591–1596.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Mendu, S.S., Das, P.K. Simulations for the flow of viscoplastic fluids in a cavity driven by the movement of walls by Lattice Boltzmann Method. Korea-Aust. Rheol. J. 32, 213–231 (2020). https://doi.org/10.1007/s13367-020-0021-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13367-020-0021-6