A multi-phase SPH method for macroscopic and mesoscopic flows. (English) Zbl 1136.76419

Summary: A multi-phase smoothed particle hydrodynamics (SPH) method for both macroscopic and mesoscopic flows is proposed. Since the particle-averaged spatial derivative approximations are derived from a particle smoothing function in which the neighboring particles only contribute to the specific volume, while maintaining mass conservation, the new method handles density discontinuities across phase interfaces naturally. Accordingly, several aspects of multi-phase interactions are addressed. First, the newly formulated viscous terms allow for a discontinuous viscosity and ensure continuity of velocity and shear stress across the phase interface. Based on this formulation thermal fluctuations are introduced in a straightforward way. Second, a new simple algorithm capable for three or more immiscible phases is developed. Mesocopic interface slippage is included based on the apparent slip assumption which ensures continuity at the phase interface. To show the validity of the present method numerical examples on capillary waves, three-phase interactions, drop deformation in a shear flow, and mesoscopic channel flows are considered.


76M28 Particle methods and lattice-gas methods
76D05 Navier-Stokes equations for incompressible viscous fluids


Full Text: DOI


[1] Brackbill, J. U.; Kothe, D. B.; Zemach, C., A continuum method for modeling surface tension, J. Comput. Phys, 100, 335 (1992) · Zbl 0775.76110
[2] Chen, S.; Doolen, G. D., Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech., 30, 329 (1998) · Zbl 1398.76180
[3] Cleary, P., Modelling confined multi-material heat and mass flows using SPH, Appl. Math. Modelling, 22, 981 (1998)
[4] Cleary, P.; Ha, J.; Alguine, V.; Nguyen, T., Flow modelling in casting processes, Appl. Math. Modelling, 26, 171 (2002) · Zbl 1205.76186
[5] Colagrossi, A.; Landrini, M., Numerical simulation of interfacial flows by smoothed particle hydrodynamics, J. Comput. Phys., 191, 448 (2003) · Zbl 1028.76039
[6] Cottet, G. H.; Koumoutsakos, P., Vortex Methods: Theory and Practice (2000), Cambridge University Press: Cambridge University Press Cambridge
[7] Cottin-Bizonne, C.; Jurine, S. J.; Baudry, J.; Crassous, J.; Restagno, F.; Charlaix, E., Nanorheology: an investigation of the boundary condition at hydrophobic and hydrophilic interfaces, Eur. Phys. J. E, 9, 47 (2002)
[8] Español, P.; Warren, P., Statistical mechanics of dissipative particle dynamics, Europhys. Lett., 30, 191 (1995)
[9] Español, P.; Revenga, M., Smoothed dissipative particle dynamics, Phys. Rev. E, 67, 026705 (2003)
[10] Flebbe, O.; Munzel, S.; Herold, H.; Ruder, H., Smoothed particle hydrodynamics: physical viscosity and the simulation of accretion disks, Astrophys. J., 431, 754 (1994)
[11] Flekkøy, E. G.; Coveney, P. V.; de Fabritiis, G., Foundations of dissipative particle dynamics, Phys. Rev., 62, 2140 (2000)
[12] Gingold, R. A.; Monaghan, J. J., Smoothed particle hydrodynamics - Theory and application to non-spherical stars, Mon. Not. R. Astron. Soc, 181, 375 (1977) · Zbl 0421.76032
[13] Granick, S.; Zhu, Y.; Lee, H., Slippery questions about complex fluids flowing past solids, Nat. Mater., 2, 221 (2003)
[14] Grmela, M.; Öttinger, H. C., Dynamics and thermodynamics of complex fluids, I. Development of a general formalism, Phys. Rev. E, 56, 6620 (1997)
[15] Öttinger, H. C.; Grmela, M., Dynamics and thermodynamics of complex fluids, II. Illustrations of a general formalism, Phys. Rev. E, 56, 6633 (1997)
[16] Groot, R. D.; Warren, P. B., Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation, J. Chem. Phys., 107, 4423 (1997)
[17] Hietel, D.; Steiner, K.; Struckmeier, J., A finite-volume particle method for compressible flows, Math. Models Meth. Appl. Sci., 10, 1363 (2000) · Zbl 0970.76074
[18] Hoogerbrugge, P. J.; Koelman, J., Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics, Europhys. Lett., 19, 155 (1992)
[19] Inutsuka, S., Reformulation of smoothed particle hydrodynamics with Riemann solver, J. Comput. Phys., 179, 238 (2002) · Zbl 1060.76094
[20] Koshizuka, S.; Nobe, A.; Oka, Y., Numerical analysis of breaking waves using the moving particle semi-implicit method, Int. J. Numer. Meth. Fluids, 26, 751 (1998) · Zbl 0928.76086
[21] Koplik, J.; Banavar, J. R., Continuum deductions from molecular hydrodynamics, Annu. Rev. Fluid Mech., 27, 257 (1995)
[22] Koumoutsakos, P., Multiscale flow simulations using particles, Annu. Rev. Fluid Mech., 37, 457 (2005) · Zbl 1117.76054
[23] Lafaurie, B.; Nardone, C.; Scardovelli, R.; Zaleski, S.; Zanetti, G., Modelling merging and fragmentation in multiphase flows with SURFER, J. Comput. Phys., 113, 134 (1994) · Zbl 0809.76064
[24] Lauga, E.; Stone, H. A., Effective slip in pressure-driven Stokes flow, J. Fluid Mech., 489, 55 (2003) · Zbl 1064.76028
[25] Lucy, L. B., A numerical approach to the testing of the fission hypothesis, Astron. J., 82, 1013 (1977)
[26] Monaghan, J. J., Smoothed particle hydrodynamics, Annu. Rev. Astronom. Astrophys., 30, 543 (1992)
[27] Monaghan, J. J., Simulating free surface flows with SPH, J. Comput. Phys., 110, 399 (1994) · Zbl 0794.76073
[28] Monaghan, J. J.; Kocharyan, A., SPH simulation of multi-phase flow, Comput. Phys. Commun., 87, 225 (1995) · Zbl 0923.76195
[29] Morris, J. P.; Fox, P. J.; Zhu, Y., Modeling low Reynolds number incompressible flows using SPH, J. Comput. Phys., 136, 214 (1997) · Zbl 0889.76066
[30] Morris, J. P., Simulating surface tension with smoothed particle hydrodynamics, Int. J. Numer. Meth. Fluids, 33, 333 (1999) · Zbl 0985.76072
[31] Nugent, S.; Posch, H. A., Liquid drops and surface tension with smoothed particle applied mechanics, Phys. Rev. E, 62, 4968 (2000)
[32] Pit, R.; Hervet, H.; Leger, L., Direct experimental evidence of slip in hexadecane: solid interfaces, Phys. Rev. Lett., 85, 980 (2000)
[33] Qian, T.; Wang, X. P.; Sheng, P., Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E, 68, 016306 (2003)
[34] Randles, P. W.; Libersky, L. D., Smoothed particle hydrodynamics: some recent improvements and applications, Meth. Appl. Mech. Eng., 139, 375 (1996) · Zbl 0896.73075
[35] Richie, B. W.; Thomas, P. A., Multiphase smoothed-particle hydrodynamics, Mon. Not. R. Astron. Soc, 323, 743 (2001)
[36] Scardovelli, R.; Zaleski, S., Direct numerical simulation of free-surface and interfacial flow calculations, Annu. Rev. Fluid Mech., 31, 567 (1999)
[37] Serrano, M.; Español, P., Thermodynamically consistent mesoscopic fluid particle model, Phys. Rev. E, 64, 046115 (2001)
[38] Serrano, M.; de Fabritiis, G.; Español, P.; Flekkoy, E.; Coveney, P. V., Mesoscopic dynamics of Voronoi fluid particles, J. Phys. A, 35, 1605 (2002) · Zbl 1019.76037
[39] Sethian, J. A.; Smereka, P., Level-set methods for fluid interfaces, Annu. Rev. Fluid Mech., 35, 341 (2003) · Zbl 1041.76057
[40] D. Shepherd, A two dimensional function for irregularly space data, in: ACM National Conference, 1968.; D. Shepherd, A two dimensional function for irregularly space data, in: ACM National Conference, 1968.
[41] Takeda, H.; Miyama, S. M.; Sekiya, M., Numerical simulation of viscous flow by smoothed particle hydrodynamics, Prog. Theor. Phys., 92, 939 (1994)
[42] Taylor, G. I., The formation of emulsions in de.nable.elds of flows, Proc. R. Soc. Lond. A, 146, 501 (1934)
[43] Thompson, P. A.; Robbins, M. O., Simulations of contact-line motion: slip and the dynamic contact angle, Phys. Rev. Lett., 63, 766 (1989)
[44] Wu, J.; Yu, S. T.; Jiang, B. N., Simulation of two-fluid flows by the least-square finite element method using a continuum surface tension model, Int. J. Numer. Meth. Eng., 42, 583 (1998) · Zbl 0912.76035
[45] Zhou, H.; Pozrikidis, C., The flow of suspensions in channels: single files of drops, Phys. Fluids A, 5, 311 (1993)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.