×

zbMATH — the first resource for mathematics

Numerical simulation of pressure-driven displacement of a viscoplastic material by a Newtonian fluid using the lattice Boltzmann method. (English) Zbl 1408.76017
Summary: The pressure-driven displacement of a non-Newtonian fluid by a Newtonian fluid in a two-dimensional channel is investigated via a multiphase lattice Boltzmann method using a non-ideal gas equation of state well-suited for two incompressible fluids. The code has been validated by comparing the results obtained using different regularized models, proposed in the literature, to model the viscoplasticity of the displaced material. Then, the effects of the Bingham number, which characterizes the behaviour of the yield-stress of the fluid and the flow index, which reflects the shear-thinning/thickening tendency of the fluid, are studied. It is found that by increasing the Bingham number and the flow index, the size of the unyielded region of the fluid in the downstream portion of the channel increases as well as the thickness of the residual layer of the fluid resident initially in the channel; the latter is left behind on the channel walls by the propagating ‘finger’ of the displacing fluid. This, in turn, reduces the growth rate of interfacial instabilities and the speed of finger propagation.

MSC:
76A05 Non-Newtonian fluids
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
76M28 Particle methods and lattice-gas methods
76E05 Parallel shear flows in hydrodynamic stability
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Taghavi, S. M.; Alba, K.; Séon, T.; Wielage-Burchard, K.; Martinez, D. M.; Frigaard, I. A., Miscible displacement flows in near-horizontal ducts at low atwood number, J. Fluid Mech., 696, 175-214, (2012) · Zbl 1250.76051
[2] Joseph, D. D.; Bai, R.; Chen, K. P.; Renardy, Y. Y., Core-annular flows, Annu. Rev. Fluid Mech., 29, 65-90, (1997)
[3] Homsy, G. M., Viscous fingering in porous media, Annu. Rev. Fluid Mech., 19, 271-311, (1987)
[4] Govindarajan, R.; Sahu, K. C., Instabilities in viscosity stratified flow, Annu. Rev. Fluid Mech., 46, 331-353, (2014) · Zbl 1297.76067
[5] Chen, C.-Y.; Meiburg, E., Miscible displacement in capillary tubes. part 2. numerical simulations, J. Fluid Mech., 326, 57-90, (1996) · Zbl 0921.76178
[6] Rakotomalala, N.; Salin, D.; Watzky, P., Miscible displacement between two parallel plates: BGK lattice gas simulations, J. Fluid Mech., 338, 277-297, (1997) · Zbl 0889.76067
[7] Goyal, N.; Meiburg, E., Miscible displacements in Hele-Shaw cells: two-dimensional base states and their linear stability, J. Fluid Mech., 558, 329-355, (2006) · Zbl 1156.76382
[8] Petitjeans, P.; Maxworthy, P., Miscible displacements in capillary tubes. part 1. experiments, J. Fluid Mech., 326, 37-56, (1996)
[9] Sahu, K. C.; Ding, H.; Valluri, P.; Matar, O. K., Prssure-driven miscible two-fluid channel flow with density gradients, Phys. Fluids, 21, 043603, (2009) · Zbl 1183.76448
[10] Taghavi, S. M.; Séon, T.; Martinez, D. M.; Frigaard, I. A., Buoyancy-dominated displacement flows in near-horizontal channels: the viscous limit, J. Fluid Mech., 639, 1-35, (2009) · Zbl 1183.76617
[11] Taghavi, S. M.; Séon, T.; Martinez, D. M.; Frigaard, I. A., Stationary residual layers in buoyant Newtonian displacement flows, Phys. Fluids, 23, 044105, (2011)
[12] Mishra, M.; Wit, A. D.; Sahu, K. C., Double diffusive effects on pressure-driven miscible displacement flow in a channel, J. Fluid Mech., 712, 579-597, (2012) · Zbl 1275.76092
[13] Joseph, D. D.; Renardy, M.; Renardy, Y. Y., Instability of the flow of two immiscible liquids with different viscosities in a pipe, J. Fluid Mech., 141, 309-317, (1984) · Zbl 0562.76103
[14] Kang, Q.; Zhang, D.; Chen, S., Immiscible displacement in a channel: simulations of fingering in two dimensions, Adv. Water Resour., 27, 13-22, (2004)
[15] Chin, J.; Boek, E. S.; Coveney, P. V., Lattice Boltzmann simulation of the flow of binary immiscible fluids with different viscosities using the — shan-Chen microscopic interaction model, Phil. Trans. Math. Phys. Eng. Sci., 360, 547-558, (2002) · Zbl 1001.76080
[16] Grosfils, P.; Boon, J. P.; Chin, J., Structural and dynamical characterization of Hele-Shaw viscous fingering, Phil. Trans. Math. Phys. Eng. Sci., 362, 1723-1734, (2004) · Zbl 1205.76089
[17] Dong, B.; Yan, Y. Y.; Li, W.; Song, Y., Lattice Boltzmann simulation of viscous fingering phenomenon of immiscible fluids displacement in a channel, Comput. & Fluids, 39, 768-779, (2010) · Zbl 1242.76267
[18] Redapangu, P. R.; Sahu, K. C.; Vanka, S. P., A study of pressure-driven displacement flow of two immiscible liquids using a multiphase lattice Boltzmann approach, Phys. Fluids, 24, 102110, (2012) · Zbl 1258.76125
[19] Sahu, K. C.; Ding, H.; Valluri, P.; Matar, O. K., Linear stability analysis and numerical simulation of miscible channel flows, Phys. Fluids, 21, 042104, (2009) · Zbl 1183.76447
[20] Yih, C. S., Instability due to viscous stratification, J. Fluid Mech., 27, 337-352, (1967) · Zbl 0144.47102
[21] Yiantsios, S. G.; Higgins, B. G., Numerical solution of eigenvalue problems using the compound matrix-method, J. Comput. Phys., 74, 25-40, (1988) · Zbl 0641.65065
[22] Sahu, K. C.; Matar, O. K., Three-dimensional linear instability in pressure-driven two-layer channel flow of a Newtonian and a Herschel-Bulkley fluid, Phys. Fluids, 22, 112103, (2010)
[23] Govindarajan, R., Effect of miscibility on the linear instability of two-fluid channel flow, Int. J. Multiph. Flow, 30, 1177-1192, (2004) · Zbl 1136.76516
[24] Selvam, B.; Merk, S.; Govindarajan, R.; Meiburg, E., Stability of miscible core-annular flows with viscosity stratification, J. Fluid Mech., 592, 23-49, (2007) · Zbl 1151.76457
[25] Malik, S. V.; Hooper, A. P., Linear stability and energy growth of viscosity stratified flows, Phys. Fluids, 17, 024101, (2005) · Zbl 1187.76330
[26] Sahu, K. C.; Govindarajan, R., Linear stability of double-diffusive two-fluid channel flow, J. Fluid Mech., 687, 529-539, (2011) · Zbl 1241.76179
[27] Lajeunesse, E.; Martin, J.; Rakotomalala, N.; Salin, D.; Yortsos, Y. C., Miscible displacement in a Hele-Shaw cell at high rates, J. Fluid Mech., 398, 299-319, (1999) · Zbl 0942.76508
[28] Dimakopoulos, Y.; Tsamopoulos, J., Transient displacement of a viscoplastic material by air in straight and suddenly constricted tubes, J. Non-Newton. Fluid Mech., 112, 1, 43-75, (2003) · Zbl 1038.76505
[29] Papaioannou, J.; Karapetsas, G.; Dimakopoulos, Y.; Tsamopoulos, J., Injection of a viscoplastic material inside a tube or between two parallel disks: conditions for wall detachment of the advancing front, J. Rheol., 53, 5, 1155-1191, (2009)
[30] Allouche, M.; Frigaard, I. A.; Sona, G., Static wall layers in the displacement of two visco-plastic fluids in a plane channel, J. Fluid Mech., 424, 243-277, (2000) · Zbl 1003.76004
[31] Wielage-Burchard, K.; Frigaard, I. A., Static wall layers in plane channel displacement flows, J. Non-Newton. Fluid Mech., 166, 245-261, (2011) · Zbl 1281.76012
[32] Frigaard, I. A.; Nouar, C., On the usage of viscosity regularisation methods for visco-plastic fluid flow computation, J. Non-Newton. Fluid Mech., 127, 1-26, (2005) · Zbl 1187.76716
[33] Bercovier, M.; Engleman, M., A finite-element method for incompressible non-Newtonian flows, J. Comput. Phys., 36, 313-326, (1980) · Zbl 0457.76005
[34] Papanastasiou, T. C., Flows of materials with yield, J. Rheol., 31, 5, 385-404, (1987) · Zbl 0666.76022
[35] Chen, S.; Doolen, G. D., Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech., 30, 329-364, (1998) · Zbl 1398.76180
[36] Gunstensen, A. K.; Rothman, D. H.; Zaleski, S.; Zanetti, G., Lattice Boltzmann model for immiscible fluids, Phys. Rev. A, 43, 4320-4327, (1991)
[37] Shan, X.; Chen, H., Lattice Boltzmann model for simulating flows with multiple phases and components, Phys. Rev. E, 47, 3, 1815-1819, (1993)
[38] Swift, M. R.; Osborn, W. R.; Yeomans, J. M., Lattice-Boltzmann simulation of nonideal fluids, Phys. Rev. Lett., 75, 830-833, (1995)
[39] Zhang, R.; He, X.; Chen, S., Interface and surface tension in incompressible lattice Boltzmann multiphase model, Comput. Phys. Commun., 129, 121-130, (2000) · Zbl 0990.76073
[40] He, X.; Zhang, R.; Chen, S.; Doolen, G. D., On the three-dimensional Rayleigh-Taylor instability, Phys. Fluids, 11, 5, 1143-1152, (1999) · Zbl 1147.76410
[41] He, X.; Chen, S.; Zhang, R., A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of Rayleigh-Taylor instability, J. Comput. Phys., 152, 642-663, (1999) · Zbl 0954.76076
[42] A. Vikhansky, Lattice-Boltzmann method for yield-stress liquids 155 (2008) 95-100. · Zbl 1274.76285
[43] Vikhansky, A., Construction of lattice-Boltzmann schemes for non-Newtonian and two-phase flows, Can. J. Chem. Eng., 90, 1081-1091, (2012)
[44] J.J. Derksen, Simulations of mobilization of bingham layers in a turbulently agitated tank 191 (2013) 25-34.
[45] Sahu, K. C.; Vanka, S. P., A multiphase lattice Boltzmann study of buoyancy-induced mixing in a tilted channel, Comput. & Fluids, 50, 199-215, (2011) · Zbl 1271.76272
[46] S.P. Vanka, A.F. Shinn, K.C. Sahu, Computational fluid dynamics using graphics processing units: challenges and opportunities, in: Proceedings of the ASME 2011 International Mechanical Engineering Congress and Exposition, Denver, Colorado, USA, 2011.
[47] Bhatnagar, P. L.; Gross, E. P.; Krook, M., A model for collision process in gases. I. small amplitude processes in charged and neutral one-component system, Phys. Rev., 94, 511-525, (1954) · Zbl 0055.23609
[48] Carnahan, N. F.; Starling, K. E., Equation of state for non-attracting rigid spheres, J. Chem. Phys., 51, 635-636, (1969)
[49] Premnath, K. N.; Abraham, J., Lattice Boltzmann model for axisymmetric multiphase flows, Phys. Rev. E, 71, 056706, (2005)
[50] Chang, Q.; Alexander, J. I.D., Application of the lattice Boltzmann method to two-phase Rayleigh-benard convection with a deformable interface, J. Comput. Phys., 212, 473-489, (2006) · Zbl 1084.76061
[51] Fakhari, A.; Rahimian, M. H., Simulation of falling droplet by the lattice Boltzmann method, Commun. Nonlinear Sci. Numer. Simul., 14, 3046-3055, (2009) · Zbl 1221.76163
[52] Fakhari, A.; Rahimian, M. H., Investigation of deformation and breakup of a moving droplet by the method of lattice Boltzmann equations, Internat. J. Numer. Methods Fluids, 64, 827-849, (2010) · Zbl 1342.76099
[53] Evans, R., The nature of the liquid-vapor interface and other topics in the statistical mechanics of non-uniform classical fluids, Adv. Phys., 28, 143-200, (1979)
[54] Lee, T.; Lin, C.-L., A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio, J. Comput. Phys., 206, 16-47, (2005) · Zbl 1087.76089
[55] Redapangu, P. R.; Vanka, S. P.; Sahu, K. C., Multiphase lattice Boltzmann simulations of buoyancy-induced flow of two immiscible fluids with different viscosities, Eur. J. Mech. B Fluids, 34, 105-114, (2012) · Zbl 1258.76125
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.