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A numerical method for the simulation of viscoelastic fluid surfaces. (English) Zbl 07512371

Summary: Viscoelastic surface rheology plays an important role in multiphase systems. A typical example is the actin cortex which surrounds most animal cells. It shows elastic properties for short time scales and behaves viscous for longer time scales. Hence, realistic simulations of cell shape dynamics require a model capturing the entire elastic to viscous spectrum. However, currently there are no numerical methods to simulate deforming viscoelastic surfaces. Thus models for the cell cortex, or other viscoelastic surfaces, are usually based on assumptions or simplifications which limit their applicability. In this paper we develop a first numerical approach for simulation of deforming viscoelastic surfaces. To this end, we derive the surface equivalent of the upper convected Maxwell model using the GENERIC formulation of nonequilibrium thermodynamics. The model distinguishes between shear dynamics and dilatational surface dynamics. The viscoelastic surface is embedded in a viscous fluid modelled by the Navier-Stokes equation. Both systems are solved using Finite Elements. The fluid and surface are combined using an Arbitrary Lagrange-Eulerian (ALE) Method that conserves the surface grid spacing during rotations and translations of the surface. We verify this numerical implementation against analytic solutions and find good agreement. To demonstrate its potential we simulate the experimentally observed tumbling and tank-treading of vesicles in shear flow. We also supply a phase-diagram to demonstrate the influence of the viscoelastic parameters on the behaviour of a vesicle in shear flow. Finally, we explore cytokinesis as a future application of the numerical method by simulating the start of cytokinesis using a spatially dependent function for the surface tension.

MSC:

76Dxx Incompressible viscous fluids
76Mxx Basic methods in fluid mechanics
65Mxx Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems

Software:

AMDiS
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Full Text: DOI arXiv

References:

[1] Brenner, H., Interfacial Transport Processes and Rheology (2013), Elsevier
[2] Slattery, J. C.; Sagis, L.; Oh, E.-S., Interfacial Transport Phenomena (2007), Springer Science & Business Media · Zbl 1116.76001
[3] Jaensson, N. O.; Anderson, P. D.; Vermant, J., Computational interfacial rheology, J. Non-Newton. Fluid Mech., Article 104507 pp. (2021)
[4] Rey, A. D., Polar fluid model of viscoelastic membranes and interfaces, J. Colloid Interface Sci., 304, 1, 226-238 (2006)
[5] Sagis, L. M., Modelling surface rheology of complex interfaces with extended irreversible thermodynamics, Phys. A, Stat. Mech. Appl., 389, 4, 673-684 (2010)
[6] Nitschke, I.; Voigt, A., Observer-invariant time derivatives on moving surfaces (2020)
[7] Salbreux, G.; Charras, G.; Paluch, E., Actin cortex mechanics and cellular morphogenesis, Trends Cell Biol., 22, 536 (2012)
[8] Fischer-Friedrich, E.; Toyoda, Y.; Cattin, C. J.; Müller, D. J.; Hyman, A. A.; Jülicher, F., Rheology of the active cell cortex in mitosis, Biophys. J., 111, 3, 589-600 (2016)
[9] Mietke, A.; Jemseena, V.; Kumar, K. V.; Sbalzarini, I. F.; Jülicher, F., Minimal model of cellular symmetry breaking, Phys. Rev. Lett., 123, 18, Article 188101 pp. (2019)
[10] Mokbel, M.; Hosseini, K.; Aland, S.; Fischer-Friedrich, E., The Poisson ratio of the cellular actin cortex is frequency-dependent, Biophys. J. (2020)
[11] Ong, K. C.; Lai, M.-C., An immersed boundary projection method for simulating the inextensible vesicle dynamics, J. Comput. Phys., 408, Article 109277 pp. (2020) · Zbl 07505613
[12] Veerapaneni, S. K.; Gueyffier, D.; Zorin, D.; Biros, G., A boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2D, J. Comput. Phys., 228, 7, 2334-2353 (2009) · Zbl 1275.76175
[13] Aland, S.; Egerer, S.; Lowengrub, J.; Voigt, A., Diffuse interface models of locally inextensible vesicles in a viscous fluid, J. Comput. Phys., 277, 32-47 (2014) · Zbl 1349.76933
[14] Marth, W.; Aland, S.; Voigt, A., Margination of white blood cells - a computational approach by a hydrodynamic phase field model, J. Fluid Mech., 790, 389-406 (2016) · Zbl 1382.76304
[15] Lowengrub, J.; Allard, J.; Aland, S., Numerical simulation of endocytosis: viscous flow driven by membranes with non-uniformly distributed curvature-inducing molecules, J. Comput. Phys., 309, 112-128 (2016) · Zbl 1351.76067
[16] Barrett, J. W.; Garcke, H.; Nürnberg, R., A stable numerical method for the dynamics of fluidic membranes, Numer. Math., 134, 4, 783-822 (2016) · Zbl 1391.76298
[17] Pozrikidis, C., Computational Hydrodynamics of Capsules and Biological Cells (2010), CRC Press · Zbl 1446.92003
[18] Le, D.-V.; White, J.; Peraire, J.; Lim, K.; Khoo, B., An implicit immersed boundary method for three-dimensional fluid-membrane interactions, J. Comput. Phys., 228, 22, 8427-8445 (2009) · Zbl 1400.76024
[19] Mokbel, M.; Aland, S., An ALE method for simulations of axisymmetric elastic surfaces in flow, Int. J. Numer. Methods Fluids, 92, 1604-1625 (2020)
[20] Mokbel, M.; Mokbel, D.; Mietke, A.; Traber, N.; Girardo, S.; Otto, O.; Guck, J.; Aland, S., Numerical simulation of real-time deformability cytometry to extract cell mechanical properties, ACS Biomater. Sci. Eng., 3, 11, 2962-2973 (2017)
[21] Reuther, S.; Voigt, A., Solving the incompressible surface Navier-Stokes equation by surface finite elements, Phys. Fluids, 30, 1, Article 012107 pp. (2018)
[22] Reuther, S.; Voigt, A., Incompressible two-phase flows with an inextensible Newtonian fluid interface, J. Comput. Phys., 322, 850-858 (2016) · Zbl 1351.76326
[23] Jankuhn, T.; Reusken, A., Higher order trace finite element methods for the surface Stokes equation (2019), arXiv preprint
[24] Reusken, A., Stream function formulation of surface Stokes equations, IMA J. Numer. Anal., 40, 1, 109-139 (2020) · Zbl 1466.65201
[25] Arroyo, M.; DeSimone, A., Relaxation dynamics of fluid membranes, Phys. Rev. E, 79, Article 031915 pp. (2009)
[26] Barrett, J.; Garcke, H.; Nürnberg, R., Stable numerical approximation of two-phase flow with a Boussinesq-Scriven surface fluid, Commun. Math. Sci., 13, 7, 1829-1874 (2015) · Zbl 1329.35241
[27] Phan-Thien, N., Understanding Viscoelasticity - An Introduction to Rheology (2013), Springer Science & Business Media
[28] Sagis, L. M., Dynamic properties of interfaces in soft matter: experiments and theory, Rev. Mod. Phys., 83, 4, 1367 (2011)
[29] Grmela, M.; Öttinger, H. C., Dynamics and thermodynamics of complex fluids. I. Development of a general formalism, Phys. Rev. E, 56, 6620 (1997)
[30] Öttinger, H. C.; Grmela, M., Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism, Phys. Rev. E, 56, 6633 (1997)
[31] Öttinger, H. C., Beyond Equilibrium Thermodynamics (2005), Wiley Interscience: Wiley Interscience Hoboken
[32] Sagis, L. M.C.; Öttinger, H. C., Dynamics of multiphase systems with complex microstructure. I. Development of the governing equations through nonequilibrium thermodynamics, Phys. Rev. E, 88, Article 022149 pp. (2013)
[33] Scriven, L. E., Dynamics of a fluid interface equation of motion for Newtonian surface fluids, Chem. Eng. Sci., 12, 2, 98-108 (1960)
[34] Bothe, D.; Prüss, J., On the two-phase Navier-Stokes equations with Boussinesq-scriven surface fluid, J. Math. Fluid Mech., 12, 1, 133-150 (2010) · Zbl 1261.35100
[35] Jankuhn, T.; Olshanskii, M. A.; Reusken, A., Incompressible fluid problems on embedded surfaces: modeling and variational formulations, Interfaces Free Bound., 20, 3, 353-377 (2018) · Zbl 1406.35224
[36] Barrett, J. W.; Garcke, H.; Nürnberg, R., Parametric Finite Element Approximations of Curvature- Driven Interface Evolutions, Handbook of Numerical Analysis, vol. 21, 275423 (2020), Elsevier · Zbl 1455.35185
[37] Landau, L.; Lifshitz, E., Theory of Elasticity (1986), Elsevier Ltd.
[38] Vey, S.; Voigt, A., AMDiS - adaptive multidimensional simulations, Comput. Vis. Sci., 10, 57-66 (2007)
[39] Witkowski, T.; Ling, S.; Praetorius, S.; Voigt, A., Software concepts and numerical algorithms for a scalable adaptive parallel finite element method, Adv. Comput. Math., 41, 6, 1145-1177 (2015) · Zbl 1334.65160
[40] Chen, L., Finite element methods for Stokes equations
[41] Ma, B.; Wang, Y.; Kikker, A., Analytical solutions of oscillating Couette-Poiseuille flows for the viscoelastic Oldroyd B fluid, Acta Mech., 230, 6, 2249-2266 (2019) · Zbl 1428.76022
[42] Abreu, D.; Levant, M.; Steinberg, V.; Seifert, U., Fluid vesicles in flow, Adv. Colloid Interface Sci. (2014)
[43] Bächer, C.; Gekle, S., Computational modeling of active deformable membranes embedded in three-dimensional flows, Phys. Rev. E, 99, 6, Article 062418 pp. (2019)
[44] Yazdani, A.; Bagchi, P., Influence of membrane viscosity on capsule dynamics in shear flow, J. Fluid Mech., 718, 569-595 (Mar. 2013) · Zbl 1284.76426
[45] Carrozza, M.; Hulsen, M.; Anderson, P., Benchmark solutions for flows with rheologically complex interfaces, J. Non-Newton. Fluid Mech., 286, Article 104436 pp. (2020)
[46] Kessler, S.; Finken, R.; Seifert, U., Swinging and tumbling of elastic capsules in shear flow, J. Fluid Mech., 605, 207-226 (2008) · Zbl 1154.76021
[47] Kantsler, V.; Steinberg, V., Transition to tumbling and two regimes of tumbling motion of a vesicle in shear flow, Phys. Rev. Lett., 96, 3, Article 036001 pp. (2006)
[48] Fischer-Friedrich, E.; Hyman, A. A.; Jülicher, F.; Müller, D. J.; Helenius, J., Quantification of surface tension and internal pressure generated by single mitotic cells, Sci. Rep., 4, 4-11 (2014)
[49] Pollard, T. D., Nine unanswered questions about cytokinesis, J. Cell Biol., 216, 10, 3007-3016 (Aug. 2017)
[50] Grill, S. W., Growing up is stressful: biophysical laws of morphogenesis, Curr. Opin. Genet. Dev., 21, 5, 647-652 (2011)
[51] Green, R. A.; Paluch, E.; Oegema, K., Cytokinesis in animal cells, Annu. Rev. Cell Dev. Biol., 28, 29-58 (2012)
[52] Salbreux, G.; Prost, J.; Joanny, J.-F., Hydrodynamics of cellular cortical flows and the formation of contractile rings, Phys. Rev. Lett., 1035, Article 058102 pp. (2009)
[53] Sedzinski, J.; Biro, M.; Oswald, A.; Tinevez, J.-Y.; Salbreux, G.; Paluch, E., Polar actomyosin contractility destabilizes the position of the cytokinetic furrow, Nature, 476, 7361, 462-466 (2011)
[54] Mietke, A.; Jülicher, F.; Sbalzarini, I. F., Self-organized shape dynamics of active surfaces, Proc. Natl. Acad. Sci., 116, 1, 29-34 (2018)
[55] Zhao, J.; Wang, Q., Modeling cytokinesis of eukaryotic cells driven by the actomyosin contractile ring, Int. J. Numer. Methods Biomed. Eng., 32, 12, Article e02774 pp. (2016)
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