A Lagrangian finite element method for simulation of a suspension under planar extensional flow. (English) Zbl 1213.76117
Summary: A numerical simulation of a suspension of two-dimensional solid particles in a Newtonian fluid under planar extensional flow is presented. The method uses a finite element solution of the flow with a unit cell within the self-replicating lattice for planar extensional flow identified by A. M. Kraynik and D. A. Reinelt [Int. J. Multiphase Flow 18, No. 6, 1045–1059 (1992; Zbl 1144.76408)]. This is implemented using a quotient space representation that maps all points space onto points within the unit cell. This mapping is preserved by using fully Lagrangian grid movement, with grid quality preserved by a combination of Delaunay reconnection and grid adaptivity. The no-slip boundary conditions on the particles are enforced weakly via a traction force acting as a Lagrange multiplier. The method allows simulations of suspensions under planar extensional flow to be conducted to large strains in a truly periodic cell. The method is illustrated for both isotropic and anisotropic two-dimensional particles and can be easily extended to viscoelastic fluids and to non-rigid particles.
Citations:
Zbl 1144.76408Software:
TriangleReferences:
| [1] | Batchelor, G. K., The stress system in a suspension of force-free particles, J. Fluid Mech., 41, 545 (1970) · Zbl 0193.25702 |
| [2] | Bilby, B. A.; Kolbuszewski, M. L., The finite deformation of an inhomogeneity in two-dimensional slow viscous incompressible flow, Proc. Roy. Soc. Lond. A, 355, 335 (1977) · Zbl 0406.76086 |
| [3] | Boek, E. S.; Coveney, P. V.; Lekkerkerker, H. N.W.; van der Schoot, P., Simulating the rheology of dense colloidal suspensions using dissipative particle dynamics, Phys. Rev. E, 55, 33124 (1997) |
| [4] | Brady, J. F.; Bossis, G., Stokesian dynamics, Annu. Rev. Fluid Mech., 20, 111 (1988) |
| [5] | Brady, J. F., The Einstein viscosity correction in a \(n\) dimensions, Int. J. Multiphase Flow, 10, 113 (1984) · Zbl 0539.76111 |
| [6] | Chen, S.; Doolen, G. D., Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech., 30, 329 (1998) · Zbl 1398.76180 |
| [7] | D’Avino, G.; Maffettone, P. L.; Hulsen, M. A.; Peters, G. M.W., A numerical method for simulating concentrated rigid particle suspensions in an elongational flow using a fixed grid, J. Comput. Phys. (2007), doi:10.1016/j.jcp.2007.04.027 · Zbl 1310.76080 |
| [8] | Delaunay, B. N., Sur la sphere vide, Bull. Acad. Sci. USSR VII: Class. Sci. Math., 793 (1934) · Zbl 0010.41101 |
| [9] | Glowinski, R.; Pan, T.-W.; Hesla, T. I.; Joseph, D., A distributed Lagrange multiplier/fictitious domain method for particulate flows, Int. J. Multiphase Flow, 25, 755 (1999) · Zbl 1137.76592 |
| [10] | Harlen, O. G.; Rallison, J. M.; Szabo, P., A split Lagrangian-Eulerian method for simulating transient viscoelastic flows, J. Non-Newtonian Fluid Mech., 60, 81 (1995) |
| [11] | Heyes, D. M., Molecular dynamics simulations of extensional sheet and unidirectional flow, Chem. Phys., 98, 15 (1985) |
| [12] | Hwang, W. R.; Hulsen, M. A., Direct numerical simulations of hard particle suspensions in planar elongational flow, J. Non-Newtonian Fluid Mech., 136, 67 (2006) · Zbl 1195.76403 |
| [13] | Hwang, W. R.; Hulsen, M. A.; Meijer, H. E.H., Direct simulation of particle suspensions in sliding bi-periodic frames, J. Comput. Phys., 194, 742 (2004) · Zbl 1100.76539 |
| [14] | Hu, H. H., Direct simulation of flows of solid-liquid mixtures, Int. J. Multiphase Flow, 22, 335 (1996) · Zbl 1135.76442 |
| [15] | Jeffery, G. B., The motion of ellipsoidal particles immersed in a viscous fluid, Proc. Roy. Soc. Lond. A, 102, 61 (1922) · JFM 49.0748.02 |
| [16] | Kraynik, A. M.; Reinelt, D. A., Extensional motions of spatially periodic lattices, Int. J. Multiphase Flow, 18, 1045 (1992) · Zbl 1144.76408 |
| [17] | Lees, A. W.; Edwards, S. F., The computer study of transport processes under extreme conditions, J. Phys. C, 5, 1921 (1972) |
| [18] | N.F. Morrison, J.M. Rallison, Transient 3D flow of polymer solutions, in: Presented at XVth International Workshop on Numerical Methods for Non-Newtonian Flows, Rhodes, 2007.; N.F. Morrison, J.M. Rallison, Transient 3D flow of polymer solutions, in: Presented at XVth International Workshop on Numerical Methods for Non-Newtonian Flows, Rhodes, 2007. |
| [19] | Petrie, C. J.S., One hundred years of extensional flow, J. Non-Newtonian Fluid Mech., 137, 1 (2006) · Zbl 1195.01034 |
| [20] | Shaqfeh, E. S.G.; Koch, D. L., Orientational dispersion of fibers in extensional flows, Phys. Fluids A, 2, 1077 (1990) · Zbl 0703.76034 |
| [21] | Shewchuk, J. R., Triangle: engineering a 2D quality mesh generator and Delaunay triangulator, Lect. Notes Comput. Sci., 1148, 203 (1996) |
| [22] | Sibson, R., Locally equiangular triangulation, Comput. J., 21, 243 (1977) |
| [23] | Silvester, D.; Wathen, A., Fast iterative solution of stabilised Stokes systems. Part 2: using general block preconditioners, SIAM J. Numer. Anal., 30, 630 (1994) |
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.
