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Numerical simulation of 3D viscoelastic flows with free surfaces. (English) Zbl 1173.76303
Summary: A numerical model is presented for the simulation of viscoelastic flows with complex free surfaces in three space dimensions. The mathematical formulation of the model is similar to that of the volume of fluid (VOF) method, but the numerical procedures are different.
A splitting method is used for the time discretization. The prediction step consists in solving three advection problems, one for the volume fraction of liquid (which allows the new liquid domain to be obtained), one for the velocity field, one for the extra-stress. The correction step corresponds to solving an Oldroyd-B fluid flow problem without advection in the new liquid domain.
Two different grids are used for the space discretization. The three advection problems are solved on a fixed, structured grid made out of small cubic cells, using a forward characteristics method. The Oldroyd-B problem without advection is solved using continuous, piecewise linear stabilized finite elements on a fixed, unstructured mesh of tetrahedrons. Efficient post-processing algorithms enhance the quality of the numerical solution. A hierarchical data structure reduces the memory requirements.
Convergence of the numerical method is checked for the pure extensional flow and the filling of a tube. Numerical results are presented for the stretching of a filament. Fingering instabilities are obtained when the aspect ratio is large. Also, results pertaining to jet buckling are reported.

76A10 Viscoelastic fluids
76M10 Finite element methods applied to problems in fluid mechanics
76M25 Other numerical methods (fluid mechanics) (MSC2010)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] Aulisa, E.; Manservisi, S.; Scardovelli, R., A mixed markers and volume-of-fluid method for the reconstruction and advection of interfaces in two-phase and free-boundary flows, J. comput. phys., 188, 2, 611-639, (2003) · Zbl 1127.76346
[2] Bach, A.; Rasmussen, H.K.; Longin, P.-Y.; Hassager, O., Growth of non-axisymmetric disturbances of the free surface in the filament stretching rheometer: experiments and simulation, J. non-Newtonian fluid mech., 108, 163-186, (2002) · Zbl 1018.76500
[3] Bonvin, J.; Picasso, M.; Stenberg, R., GLS and EVSS methods for a three-field Stokes problem arising from viscoelastic flows, Comput. methods appl. mech. engrg., 190, 29-30, 3893-3914, (2001) · Zbl 1014.76043
[4] Caboussat, A., Numerical simulation of two-phase free surface flows, Arch. comput. methods engrg., state of the art reviews, 12, 2, 165-224, (2005) · Zbl 1097.76047
[5] Caboussat, A.; Picasso, M.; Rappaz, J., Numerical simulation of free surface incompressible liquid flows surrounded by compressible gas, J. comput. phys., 203, 2, 626-649, (2005) · Zbl 1143.76530
[6] Chang, Y.C.; Hou, T.Y.; Merriman, B.; Osher, S., A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, J. comput. phys., 124, 2, 449-464, (1996) · Zbl 0847.76048
[7] Chorin, A.J., Flame advection and propagation algorithms, J. comput. phys., 35, 1-11, (1980) · Zbl 0425.76086
[8] Cormenzana, J.; Ledd, A.; Laso, M.; Debbaut, B., Calculation of free surface flows using CONNFFESSIT, J. rheol., 45, 1, 237-258, (2001)
[9] Derks, D.; Lindner, A.; Creton, C.; Bonn, D., Cohesive failure of thin layers of soft model adhesives under tension, J. appl. phys., 93, 3, 1557-1566, (2003)
[10] Enright, D.; Fedkiw, R.; Ferzigerc, J.; Mitchell, I., A hybrid particle level set method for improved interface capturing, J. comput. phys., 183, 1, 83-116, (2002) · Zbl 1021.76044
[11] Goktekin, T.; Bargteil, A.; O’Brien, J., A method for animating viscoelastic fluids, ACM trans. graphics (proc. of ACM SIGGRAPH 2004), 23, 3, 463-468, (2004)
[12] Gramberg, H.J.J.; van Vroonhoven, J.C.W.; van de Ven, A.A.F., Flow patterns behind the free flow front for a Newtonian fluid injected between two infinite parallel plates, Eur. J. mech. B fluids, 23, 4, 571-585, (2004) · Zbl 1154.76337
[13] Grande, E.; Laso, M.; Picasso, M., Calculation of variable-topology free surface flows using CONNFFESSIT, J. non-Newtonian fluid mech., 113, 127-145, (2003) · Zbl 1065.76559
[14] Gueyffier, D.; Li, J.; Nadim, A.; Scardovelli, R.; Zaleski, S., Volume-of-fluid interface tracking with smoothed surface stress methods for three-dimensional flows, J. comput. phys., 152, 2, 423-456, (1999) · Zbl 0954.76063
[15] Hirt, C.W.; Nichols, B.D., Volume of fluid (VOF) method for the dynamics of free boundaries, J. comput. phys., 39, 201-225, (1981) · Zbl 0462.76020
[16] Li, J.; Renardy, Y., Shear-induced rupturing of a viscous drop in a Bingham liquid, J. non-Newtonian fluid mech., 95, 2-3, 235-251, (2000) · Zbl 0994.76005
[17] Lorstad, D.; Fuchs, L., High-order surface tension vof-model for 3d bubble flows with high density ratio, J. comput. phys., 200, 1, 153-176, (2004) · Zbl 1288.76083
[18] V. Maronnier, Simulation numérique d’écoulements de fluides incompressibles avec surface libre, Ph.D. Thesis, Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, 2000.
[19] Maronnier, V.; Picasso, M.; Rappaz, J., Numerical simulation of free surface flows, J. comput. phys., 155, 2, 439-455, (1999) · Zbl 0952.76070
[20] Maronnier, V.; Picasso, M.; Rappaz, J., Numerical simulation of 3d free surface flows, Int. J. numer. methods fluids, 42, 7, 697-716, (2003) · Zbl 1143.76539
[21] McKinley, G.H.; Sridhar, T., Filament-stretching rheometry of complex fluids, Annu. rev. fluid mech., 34, 375-415, (2002) · Zbl 0994.76502
[22] Noh, W.F.; Woodward, P., SLIC (simple line interface calculation), Lectures notes in physics, vol. 59, (1976), Springer-Verlag, pp. 330-340 · Zbl 0382.76084
[23] Phillips, T.N.; Williams, A.J., Viscoelastic flow through a planar contraction using a semi-Lagrangian finite volume method, J. non-Newtonian fluid mech., 87, 2-3, 215-246, (1999) · Zbl 0945.76052
[24] Picasso, M.; Rappaz, J., Existence, a priori and a posteriori error estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows, M2AN math. model. numer. anal., 35, 5, 879-897, (2001) · Zbl 0997.76051
[25] Picheli, E.; Coupez, T., Finite element solution of the 3D mold filling problem for viscous incompressible fluid, Comput. methods appl. mech. engrg., 163, 1-4, 359-371, (1998) · Zbl 0963.76051
[26] Pironneau, O., Finite element methods for fluids, (1989), Wiley Chichester · Zbl 0665.73059
[27] Pironneau, O.; Liou, J.; Tezduyar, T., Characteristic-Galerkin and Galerkin/least-squares space-time formulations for the advection-diffusion equation with time-dependent domains, Comput. methods appl. mech. engrg., 100, 117-141, (1992) · Zbl 0761.76073
[28] Quarteroni, A.; Valli, A., Numerical approximation of partial differential equations, Springer series in computational mathematics, 23, (1991), Springer-Verlag · Zbl 0723.65098
[29] Rasmussen, H.K.; Hassager, O., Three-dimensional simulations of viscoelastic instability in polymeric filaments, J. non-Newtonian fluid mech., 82, 189-202, (1999) · Zbl 0947.76033
[30] Renardy, M., Existence of slow steady flows of viscoelastic fluids of integral type, Z. angew. math. mech., 68, 4, T40-T44, (1988) · Zbl 0669.76022
[31] Renardy, Y.; Renardy, M., PROST: a parabolic reconstruction of surface tension for the volume-of-fluid method, J. comput. phys., 183, 2, 400-421, (2002) · Zbl 1057.76569
[32] Rider, W.J.; Kothe, D.B., Reconstructing volume tracking, J. comput. phys., 141, 2, 112-152, (1998) · Zbl 0933.76069
[33] R. Scardovelli, S. Zaleski, Direct numerical simulation of free-surface and interfacial flow, in: Annual Review of Fluid Mechanics, vol. 31, Ann. Rev. Fluid Mech., pp. 567-603, Annual Reviews, Palo Alto, CA, 1999.
[34] Sethian, J.A., Level set methods, Cambridge monographs on applied and computational mathematics, (1996), Cambridge University Press Cambridge · Zbl 0859.76004
[35] J.A. Sethian, P. Smereka, Level set methods for fluid interfaces, in: Annual Review of Fluid Mechanics, vol. 35, Ann. Rev. Fluid Mech., pp. 341-372, Annual Reviews, Palo Alto, CA, 2003. · Zbl 1041.76057
[36] Shelley, M.J.; Tian, F.-R.; Wlodarski, K., Hele-Shaw flow and pattern formation in a time-dependent gap, Nonlinearity, 10, 6, 1471-1495, (1997) · Zbl 0911.76026
[37] Sizaire, R.; Legat, V., Finite element simulation of a filament stretching extensional rheometer, J. non-Newtonian fluid mech., 71, 1-2, 89-107, (1997)
[38] Spiegelberg, S.H.; McKinley, G.H., Stress relaxation and elastic decohesion of viscoelastic polymer solutions in extensional flow, J. non-Newtonian fluid mech., 77, 49-76, (1996)
[39] Thompson, E., Use of pseudo-concentrations to follow creeping viscous flows during transient analysis, Int. J. numer. methods fluids, 6, 749-761, (1986)
[40] Tomé, M.F.; Mangiavacchi, N.; Cuminato, J.A.; Castelo, A.; McKee, S., A finite difference technique for simulation unsteady viscoelastic free surface flows, J. non-Newtonian fluid mech., 106, 61-106, (2002) · Zbl 1015.76060
[41] Tomé, M.F.; McKee, S., Numerical simulation of viscous flow: buckling of planar jets, Int. J. numer. methods fluids, 29, 705-718, (1999) · Zbl 0940.76072
[42] van der Pijl, S.P.; Segal, A.; Vuik, C.; Wesseling, P., A mass-conserving level-set method for modelling of multi-phase flows, Int. J. numer. methods fluids, 47, 4, 339-361, (2005) · Zbl 1065.76160
[43] Yao, M.; McKinley, G.H., Numerical simulation of extensional deformations of viscoelastic liquid bridges in filament stretching devices, J. non-Newtonian fluid mech., 74, 1-3, 47-88, (1998) · Zbl 0957.76005
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