Geodesic
Dynamics of Biomembranes: Effect of the Bulk Fluid
A. Bonito1 ; R.H. Nochetto2 ; M.S. Pauletti1
1 Department of Mathematics, Texas A&M University, College Station, Texas, USA
2 Department of Mathematics, University of Maryland, College Park, Maryland, USA
Mathematical modelling of natural phenomena, Tome 6 (2011) no. 5, pp. 25-43 Cet article a éte moissonné depuis la source EDP Sciences

Voir la notice de l'article

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.
DOI : 10.1051/mmnp/20116502

A. Bonito 1 ; R.H. Nochetto 2 ; M.S. Pauletti 1

1 Department of Mathematics, Texas A&M University, College Station, Texas, USA
2 Department of Mathematics, University of Maryland, College Park, Maryland, USA 
@article{10_1051_mmnp_20116502,
     author = {A. Bonito and R.H. Nochetto and M.S. Pauletti},
     title = {Dynamics of {Biomembranes:} {Effect} of the {Bulk} {Fluid}},
     journal = {Mathematical modelling of natural phenomena},
     pages = {25--43},
     year = {2011},
     volume = {6},
     number = {5},
     doi = {10.1051/mmnp/20116502},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20116502/}
}
TY  - JOUR
AU  - A. Bonito
AU  - R.H. Nochetto
AU  - M.S. Pauletti
TI  - Dynamics of Biomembranes: Effect of the Bulk Fluid
JO  - Mathematical modelling of natural phenomena
PY  - 2011
SP  - 25
EP  - 43
VL  - 6
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20116502/
DO  - 10.1051/mmnp/20116502
LA  - en
ID  - 10_1051_mmnp_20116502
ER  - 
%0 Journal Article
%A A. Bonito
%A R.H. Nochetto
%A M.S. Pauletti
%T Dynamics of Biomembranes: Effect of the Bulk Fluid
%J Mathematical modelling of natural phenomena
%D 2011
%P 25-43
%V 6
%N 5
%U http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20116502/
%R 10.1051/mmnp/20116502
%G en
%F 10_1051_mmnp_20116502
A. Bonito; R.H. Nochetto; M.S. Pauletti. Dynamics of Biomembranes: Effect of the Bulk Fluid. Mathematical modelling of natural phenomena, Tome 6 (2011) no. 5, pp. 25-43. doi: 10.1051/mmnp/20116502

[1] R.A. Adams, J.J.F. Fournier. Sobolev spaces. Pure and Applied Mathematics, second edition, Amsterdam, 2003.

[2] E. Bänsch Numer. Math. 2001 203 235

[3] E. Bänsch, P. Morin, R.H. Nochetto J. Comput. Phys. 2005 321 343

[4] J.W. Barrett, H. Garcke, R. Nürnberg SIAM J. Sci. Comput. 2008 225 253

[5] A. Bonito, R. H. Nochetto, M. S. Pauletti SIAM J. Numer. Anal. 2010 1877 1899

[6] A. Bonito, R.H. Nochetto, M.S. Pauletti J. Comput. Phys. 2010 3171 3188

[7] P.B. Canham Journal of Theoretical Biology 1970 61 81

[8] P.G. Ciarlet, J.-L. Lions, editors. Finite element methods. Part 1. Handbook of numerical analysis, 2, North-Holland, Amsterdam, 1991.

[9] U. Clarenz, U. Diewald, G. Dziuk, M. Rumpf, R. Rusu Comput. Aided Geom. Design 2004 427 445

[10] T.A. Davis ACM Trans. Math. Software 2004 196 199

[11] K. Deckelnick, G. Dziuk Interfaces Free Bound. 2006 21 46

[12] K. Deckelnick, G. Dziuk, C.M. Elliott Acta Numer. 2005 139 232

[13] M.C. Delfour, J.-P. Zolésio. Shapes and geometries. Advances in Design and Control, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001.

[14] A. Demlow SIAM J. Numer. Anal. 2009 805 827

[15] G. Doǧan, P. Morin, R.H. Nochetto, M. Verani Comput. Methods Appl. Mech. Engrg. 2007 3898 3914

[16] M. Droske, M. Rumpf Interfaces Free Bound. 2004 361 378

[17] Q. Du, C. Liu, X. Wang J. Comput. Phys. 2004 450 468

[18] Q. Du, C. Liu, X. Wang J. Comput. Phys. 2006 757 777

[19] Q. Du, Ch. Liu, R. Ryham, X. Wang Physica D 2009 923 930

[20] G. Dziuk Numer. Math. 1991 603 611

[21] G. Dziuk Numer. Math. 2008 55 80

[22] S. Esedoglu, S.J. Ruuth, R. Tsai Interfaces Free Bound. 2008 263 282

[23] E.A. Evans, R. Skalak CRC Critical reviews in bioengineering 1979 331 418

[24] M. Giaquinta, S. Hildebrandt. Calculus of variations. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 310, Springer-Verlag, Berlin, 1996.

[25] W. Helfrich Zeitschrift Fur Naturforschung C-A Journal Of Biosciences 1973 693

[26] D. Hu, P. Zhang, W. E. Continuum theory of a moving membrane. Phys. Rev. E (3), 75 (2007), No. 4, 11.

[27] J.T. Jenkins SIAM J. Appl. Math. 1977 755 764

[28] D. Köster, O. Kriessl, K.G. Siebert Numer. Math. Theor. Meth. Appl. 2008 245 274

[29] W. Losert. Personal communication, 2009.

[30] M.S. Pauletti. Parametric AFEM for geometric evolution equation and coupled fluid-membrane interaction. Thesis (Ph.D.)–University of Maryland, College Park, 2008.

[31] R.E. Rusu Interfaces Free Bound. 2005 229 239

[32] 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.

[33] U. Seifert Advances in Physics 1997 13 137

[34] D.J. Steigmann Arch. Ration. Mech. Anal. 1999 127 152

[35] D.J. Steigmann, E. Baesu, R.E. Rudd, J. Belak, M. Mcelfresh Interfaces Free Bound. 2003 357 366

[36] R. Temam. Navier-Stokes equations. Theory and numerical analysis. North-Holland Publishing Co., Amsterdam, 1977.

Cité par Sources :