# Dynamics of Biomembranes: Effect of the Bulk Fluid

A. Bonito; R.H. Nochetto; M.S. Pauletti

Mathematical Modelling of Natural Phenomena (2011)

- Volume: 6, Issue: 5, page 25-43
- ISSN: 0973-5348

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topBonito, A., Nochetto, R.H., and Pauletti, M.S.. "Dynamics of Biomembranes: Effect of the Bulk Fluid." Mathematical Modelling of Natural Phenomena 6.5 (2011): 25-43. <http://eudml.org/doc/222241>.

@article{Bonito2011,

abstract = {We derive a biomembrane model consisting of a fluid enclosed by a lipid membrane. The
membrane is characterized by its Canham-Helfrich energy (Willmore energy with area
constraint) and acts as a boundary force on the Navier-Stokes system modeling an
incompressible fluid. We give a concise description of the model and of the associated
numerical scheme. We provide numerical simulations with emphasis on the comparisons
between different types of flow: the geometric model which does not take into account the
bulk fluid and the biomembrane model for two different regimes of parameters. },

author = {Bonito, A., Nochetto, R.H., Pauletti, M.S.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {biomembrane; vesicle; red blood cell; helfrich; willmore; bending energy; parametric finite element; Lagrange multipliers; Helfrich; Willmore; parametric finite elements},

language = {eng},

month = {8},

number = {5},

pages = {25-43},

publisher = {EDP Sciences},

title = {Dynamics of Biomembranes: Effect of the Bulk Fluid},

url = {http://eudml.org/doc/222241},

volume = {6},

year = {2011},

}

TY - JOUR

AU - Bonito, A.

AU - Nochetto, R.H.

AU - Pauletti, M.S.

TI - Dynamics of Biomembranes: Effect of the Bulk Fluid

JO - Mathematical Modelling of Natural Phenomena

DA - 2011/8//

PB - EDP Sciences

VL - 6

IS - 5

SP - 25

EP - 43

AB - We derive a biomembrane model consisting of a fluid enclosed by a lipid membrane. The
membrane is characterized by its Canham-Helfrich energy (Willmore energy with area
constraint) and acts as a boundary force on the Navier-Stokes system modeling an
incompressible fluid. We give a concise description of the model and of the associated
numerical scheme. We provide numerical simulations with emphasis on the comparisons
between different types of flow: the geometric model which does not take into account the
bulk fluid and the biomembrane model for two different regimes of parameters.

LA - eng

KW - biomembrane; vesicle; red blood cell; helfrich; willmore; bending energy; parametric finite element; Lagrange multipliers; Helfrich; Willmore; parametric finite elements

UR - http://eudml.org/doc/222241

ER -

## References

top- R.A. Adams, J.J.F. Fournier. Sobolev spaces. Pure and Applied Mathematics, second edition, Amsterdam, 2003.
- E. Bänsch. Finite element discretization of the Navier-Stokes equations with a free capillary surface. Numer. Math., 88 (2001), No. 2, 203–235.
- E. Bänsch, P. Morin, R.H. Nochetto. A finite element method for surface diffusion: the parametric case. J. Comput. Phys., 203 (2005), No. 1, 321–343.
- J.W. Barrett, H. Garcke, R. Nürnberg. Parametric approximation of Willmore flow and related geometric evolution equations. SIAM J. Sci. Comput., 31 (2008), No. 1, 225–253.
- A. Bonito, R. H. Nochetto, M. S. Pauletti. Geometrically consistent mesh modification. SIAM J. Numer. Anal., 48 (2010), No. 5, 1877–1899.
- A. Bonito, R.H. Nochetto, M.S. Pauletti. Parametric fem for geometric biomembranes. J. Comput. Phys., 229 (2010), No. 9, 3171–3188.
- P.B. Canham. The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. Journal of Theoretical Biology, 26 (1970), No. 1, 61–81.
- P.G. Ciarlet, J.-L. Lions, editors. Finite element methods. Part 1. Handbook of numerical analysis, 2, North-Holland, Amsterdam, 1991.
- U. Clarenz, U. Diewald, G. Dziuk, M. Rumpf, R. Rusu. A finite element method for surface restoration with smooth boundary conditions. Comput. Aided Geom. Design, 21 (2004), No. 5, 427–445.
- T.A. Davis. Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software, 30 (2004), No. 2, 196–199.
- K. Deckelnick, G. Dziuk. Error analysis of a finite element method for the Willmore flow of graphs. Interfaces Free Bound., 8 (2006), No. 1, 21–46.
- K. Deckelnick, G. Dziuk, C.M. Elliott. Computation of geometric partial differential equations and mean curvature flow. Acta Numer., 14 (2005), 139–232.
- M.C. Delfour, J.-P. Zolésio. Shapes and geometries. Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001.
- A. Demlow. Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces. SIAM J. Numer. Anal., 47 (2009), No. 2, 805–827.
- G. Doǧan, P. Morin, R.H. Nochetto, M. Verani. Discrete gradient flows for shape optimization and applications. Comput. Methods Appl. Mech. Engrg., 196 (2007), No. 37-40, 3898–3914.
- M. Droske, M. Rumpf. A level set formulation for Willmore flow. Interfaces Free Bound., 6 (2004), No. 3, 361–378.
- Q. Du, C. Liu, X. Wang. A phase field approach in the numerical study of the elastic bending energy for vesicle membranes. J. Comput. Phys., 198 (2004), No. 2, 450–468.
- Q. Du, C. Liu, X. Wang. Simulating the deformation of vesicle membranes under elastic bending energy in three dimensions. J. Comput. Phys., 212 (2006), No. 2, 757–777.
- Q. Du, Ch. Liu, R. Ryham, X. Wang. Energetic variational approaches in modeling vesicle and fluid interactions. Physica D, 238 (2009), No. 9-10, 923–930.
- G. Dziuk. An algorithm for evolutionary surfaces. Numer. Math., 58 (1991), No. 6, 603–611.
- G. Dziuk. Computational parametric willmore flow. Numer. Math., 111 (2008), No. 1, 55–80.
- S. Esedoglu, S.J. Ruuth, R. Tsai. Threshold dynamics for high order geometric motions. Interfaces Free Bound., 10 (2008), No. 3, 263–282.
- E.A. Evans, R. Skalak. Mechanics and thermodynamics of biomembranes .2. CRC Critical reviews in bioengineering, 3 (1979), No. 4, 331–418.
- M. Giaquinta, S. Hildebrandt. Calculus of variations. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 310, Springer-Verlag, Berlin, 1996.
- W. Helfrich. Elastic properties of lipid bilayers - theory and possible experiments. Zeitschrift Fur Naturforschung C-A Journal Of Biosciences, 28 (1973), 693.
- D. Hu, P. Zhang, W. E. Continuum theory of a moving membrane. Phys. Rev. E (3), 75 (2007), No. 4, 11.
- J.T. Jenkins. The equations of mechanical equilibrium of a model membrane. SIAM J. Appl. Math., 32 (1977), No. 4, 755–764.
- D. Köster, O. Kriessl, K.G. Siebert. Design of finite element tools for coupled surface and volume meshes. Numer. Math. Theor. Meth. Appl., 1 (2008), No. 3, 245–274.
- W. Losert. Personal communication, 2009.
- M.S. Pauletti. Parametric AFEM for geometric evolution equation and coupled fluid-membrane interaction. Thesis (Ph.D.)–University of Maryland, College Park, 2008.
- R.E. Rusu. An algorithm for the elastic flow of surfaces. Interfaces Free Bound., 7 (2005), No. 3, 229–239.
- A. Schmidt, K.G. Siebert. Design of adaptive finite element software: The finite element toolbox ALBERTA, Lecture Notes in Computational Science and Engineering, 42, Springer-Verlag, Berlin, 2005.
- U. Seifert. Configurations of fluid membranes and vesicles. Advances in Physics, 46 (1997), No. 1, 13–137.
- D.J. Steigmann. Fluid films with curvature elasticity. Arch. Ration. Mech. Anal., 150 (1999), No. 2, 127–152.
- D.J. Steigmann, E. Baesu, R.E. Rudd, J. Belak, M. McElfresh. On the variational theory of cell-membrane equilibria. Interfaces Free Bound., 5 (2003), No. 4, 357–366.
- R. Temam. Navier-Stokes equations. Theory and numerical analysis. North-Holland Publishing Co., Amsterdam, 1977.

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