Numerical simulations of nearly incompressible viscoelastic membranes. (English) Zbl 1410.76152
Summary: This work presents a novel numerical investigation of the dynamics of free-boundary flows of viscoelastic liquid membranes. The governing equation describes the balance of linear momentum, in which the stresses include the viscoelastic response to deformations of Maxwell type. A penalty method is utilized to enforce near incompressibility of the viscoelastic media, in which the penalty constant is proportional to the viscosity of the fluid. A finite element method is used, in which the slender geometry representing the liquid membrane, is discretized by linear three-node triangular elements under plane stress conditions. Two applications of interest are considered for the numerical framework provided: shear flow, and extensional flow in drawing processes.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 76A10 | Viscoelastic fluids |
Software:
FEAPpvReferences:
| [1] | Levine, A. J.; MacKintosh, F. C., Dynamics of viscoelastic membranes, Phys Rev E, 66, 061606, (2002) |
| [2] | Evans, E. A.; Hochmuth, M., Membrane viscoleasticity, Biophys J, 16, 1-11, (1976) |
| [3] | Lubarda, V. A.; Marzani, A., Viscoelastic response of thin membranes with application to red blood cells, Acta Mech, 202, 1-16, (2009) · Zbl 1301.74034 |
| [4] | Callister, W. D.J., Materials science and engineering, (2007), John Wiley & Sons, Inc. New York |
| [5] | Green, A. E.; Zerna, W., Theoretical elasticity, (1968), Dover Publications Inc. New York · Zbl 0155.51801 |
| [6] | Taylor, R. L.; Oñate, E.; Ubach, P.-A., Finite element analysis of membrane structures, 47-68, (2005), Springer Netherlands Dordrecht · Zbl 1185.74094 |
| [7] | Ribe, N. M., A general theory for the dynamics of thin viscous sheets, J Fluid Mech, 457, 255-283, (2002) · Zbl 1060.76014 |
| [8] | Howell, P. D., Models for thin viscous sheets, Eur J Appl Math, 7, 321-343, (1996) · Zbl 0863.76020 |
| [9] | Love, A. E.H., A treatise on the mathematical theory of elasticity, (1944), Dover Publications New York · Zbl 0063.03651 |
| [10] | Landau, L. D.; Lifshitz, E. M., Theory of elasticity, (1970), Pergamon Press Oxford |
| [11] | Ventsel, E.; Krauthammer, T., Thin plates and shells, (2001), Marcel Dekker New York |
| [12] | Bonet, J.; Wood, R. D.; Mahaney, J.; Heywood, P., Finite element analysis of air supported membrane structures, Comput Methods Appl Mech Eng, 190, 579-595, (2000) · Zbl 1007.74071 |
| [13] | Maxwell, J. C., On the dynamical theory of gases, Phil Trans R Soc Lond, 157, 49-88, (1867) |
| [14] | Chester, S. A., A constitutive model for coupled fluid permeation and large viscoelastic deformation in polymeric gels, Soft Matter, 8, 8223-8233, (2012) |
| [15] | Yue, P.; Feng, J. J.; Liu, C.; Shen, J., A diffuse-interface method for simulating two-phase flows of complex fluids, J Fluid Mech, 515, 293-317, (2004) · Zbl 1130.76437 |
| [16] | Chua, W. K.; Oyen, M. L., Viscoelastic properties of membranes measured by spherical indentation, Cell Mol Bioeng, 2, 49-56, (2009) |
| [17] | Harland, C. W.; Bradley, M. J.; Parthasarathy, R., Phospholipid bilayers are viscoelastic, P Natl Acad Sci USA, 107, 19146-19150, (2010) |
| [18] | Mainardi, F.; Spada, G., Creep, relaxation and viscosity properties for basic fractional models in rheology, Eur Phys J Spec Top, 193, 133-160, (2011) |
| [19] | Crawford, G. E.; Earnshaw, J. C., Viscoelastic relaxation of bilayer lipid membranes. frequency-dependent tension and membrane viscosity, Biophys J, 52, 87-94, (1987) |
| [20] | Bird, R.; Armstrong, R.; Hassager, O., Dynamics of polymeric liquids (volume 1 fluid mechanics), (1987), Wiley-Interscience Toronto |
| [21] | Jeffreys, H., The Earth: its origin, history, and physical constitution, (1952), Cambridge University Press Cambridge · JFM 55.1190.04 |
| [22] | Vilmin, T.; Raphaël, E., Dewetting of thin polymer films, Eur Phys J E, 21, 161-174, (2006) |
| [23] | Tomar, G.; Shankar, V.; Shukla, S.; Sharma, A.; Biswas, G., Instability and dynamics of thin viscoelastic liquid films, Eur Phys J E, 20, 185-199, (2006) |
| [24] | Barra, V.; Afkhami, S.; Kondic, L., Interfacial dynamics of thin viscoelastic films and drops, J Non-Newt Fluid Mech, 237, 26-38, (2016) |
| [25] | O’Kiely, D.; Breward, C. J.W.; Griffiths, I. M.; Howell, P. D.; Lange, U., Edge behaviour in the Glass sheet redraw process, J Fluid Mech, 785, 248-269, (2015) · Zbl 1381.76027 |
| [26] | Taroni, M.; Breward, C. J.W.; Cummings, L. J.; Griffiths, I. M., Asymptotic solutions of Glass temperature profiles during steady optical fibre drawing, J Eng Math, 80, 1-20, (2013) · Zbl 1367.76019 |
| [27] | Zienkiewicz, O. C.; Taylor, R. L.; Zhu, J. Z., The finite element method: its basis and fundamentals, (2013), Elsevier Oxford · Zbl 1307.74005 |
| [28] | Palaniappan, D., On some general solutions of transient Stokes and Brinkman equations, J Theor Appl Mech, 52, 405-415, (2014) |
| [29] | Batchelor, G. K., An introduction to fluid dynamics, (1967), Cambridge University Press · Zbl 0152.44402 |
| [30] | Hughes, T. J.R., The finite element method: linear static and dynamic finite element analysis, (2000), Dover Publications Inc. New York · Zbl 1191.74002 |
| [31] | der Zanden, J. V.; Kuiken, G. D.C.; Segal, A.; Lindhout, W. J.; Hulsen, M. A., Numerical experiments and theoretical analysis of the flow of an elastic liquid of the upper-convected Maxwell type in the presence of geometrical discontinuities, Appl Sci Res, 42, 303-318, (1985) · Zbl 0585.76007 |
| [32] | Courant, R., Variational methods for the solution of problems of equilibrium and vibrations, Bull Amer Math Soc, 49, 1-23, (1943) · Zbl 0810.65100 |
| [33] | Shen, J., On error estimates of the penalty method for unsteady Navier-Stokes equations, SIAM J Numer Anal, 32, 386-403, (1995) · Zbl 0822.35008 |
| [34] | Zienkiewicz, O. C.; Taylor, R. L., The finite element method, 2, (2000), Butterworth-heinemann Oxford · Zbl 0991.74002 |
| [35] | Hansbo, P.; Larson, M. G.; Larsson, F., Tangential differential calculus and the finite element modeling of a large deformation elastic membrane problem, Comp Mech, 56, 87-95, (2015) · Zbl 1329.74162 |
| [36] | Siginer, D. A., Stability of non-linear constitutive formulations for viscoelastic fluids, (2014), Springer New York · Zbl 1284.76004 |
| [37] | Hilber, H.; Hughes, T.; Taylor, R., Improved numerical dissipation for time integration algorithms in structural dynamics, Earthq Eng Struct D, 5, 283-292, (1977) |
| [38] | Brink, U.; Stein, E., On some mixed finite element methods for incompressible and nearly incompressible finite elasticity, Comp Mech, 19, 105-119, (1996) · Zbl 0889.73066 |
| [39] | Simo, J. C.; Armero, F., Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes, Int J Numer Methods Eng, 33, 1413-1449, (1992) · Zbl 0768.73082 |
| [40] | Lu, X.; Lin, P.; Liu, J.-G., Analysis of a sequential regularization method for the unsteady Navier-Stokes equations, Math Comput, 77, 1467-1494, (2008) · Zbl 1285.76018 |
| [41] | Srinivasan, S.; Wei, Z.; Mahadevan, L., Wrinkling instability of an inhomogeneously stretched viscous sheet, Phys Rev Fluids, 2, 074103, (2017) |
| [42] | Pfingstag, G.; Audoly, B.; Boudaoud, A., Thin viscous sheets with inhomogeneous viscosity, Phys Fluids, 23, 063103, (2011) · Zbl 1241.76235 |
| [43] | Cerda, E.; Ravi-Chandar, K.; Mahadevan, L., Thin films: wrinkling of an elastic sheet under tension, Nature, 419, 579-580, (2002) |
| [44] | Cerda, E.; Mahadevan, L., Geometry and physics of wrinkling, Phys Rev Lett, 90, 074302, (2003) |
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.
