Capsule-substrate Adhesion in the Presence of Osmosis by the Immersed Interface Method
A two-dimensional thin-walled capsule of a flexible
semi-permeable membrane is adhered onto a rigid planar substrate
under adhesive forces (derived from a potential function) in the
presence of osmosis across the membrane. The capsule is immersed
in a hypotonic and diluted binary solution of a non-electrolyte
solute. The Stokes flow problem is solved by the immersed interface
method (IIM) with equal viscosities for the enclosed and
surrounding fluid of the capsule. The numerical results obtained are
verified against two simplified theoretical solutions and the
agreements are good. The osmotic inflation of the adhered capsule is
studied as a function of the solute concentration field, hydraulic
conductivity, and the initial capsule shape. Our findings indicate that
the contact length shrinks in dimension as capsule inflates in the
hypotonic medium, and the equilibrium contact length does not
depend on the hydraulic conductivity of the membrane and the
initial shape of the capsule.
[1] T. A. Springer, "Traffic signals on endothelium for lymphocyte
recirculation and leukocyte emigration," Annu. Rev. Physiol., vol. 57,
pp. 827-872, 1995.
[2] D. B. Khismatullin and G. A. Truskey, "A 3D numerical study of the
effect of channel height on leukocyte deformation and adhesion in
parallel plate flow chambers," Microvasc. Res., vol. 68, pp. 188-202,
2004.
[3] C. Dong and X. X. Lei, "Biomechanics of cell rolling: shear flow, cellsurface
adhesion, and cell deformability," J. Biomech., vol. 33, pp. 35-
43, 2000.
[4] V. Pappu, S. K. Doddi and P. Bagchi, "A computational study of
leukocyte adhesion and its effect on flow pattern in microvessels," J.
Theor. Biol., vol. 254, pp. 483-498, 2008.
[5] V. Pappu and P. Bagchi, "3D computational modeling and simulation
of leukocyte rolling adhesion and deformation," Comput. Biol. Med.,
vol. 38, pp. 738-753, 2008.
[6] K. T. Wan and K. K. Liu, "Contact mechanics of a thin-walled capsule
adhered onto a rigid planar substrate," Med. Biol. Eng. Comput., vol.
39, pp. 605-608, 2001.
[7] R. M. Servuss and W. Helfrich, "Mutual adhesion of lecithin
membranes at ultralow tensions," J. Phys., vol. 50, pp. 809-827, 1989.
[8] K. D. Tachev, J. K. Angarska, K. D. Danov, and P. A. Kralchevsky,
"Erythrocyte attachment to substrates: determination of membrane
tension and adhesion energy," Colloid. Surface B., vol. 19, pp. 61-80,
2000.
[9] J. J. Foo, V. Chan, and K. K. Liu, "Contact deformation of liposome in
the presence of osmosis," Ann. Biomed. Eng., vol. 31, pp. 1279-1286,
2003.
[10] J. J. Foo, V. Chan, and K. K. Liu, "Coupling bending and shear effects
on liposome deformation," J. Biomech., vol. 39, pp. 2338-2343, 2006.
[11] K. K. Liu and K. T. Wan, "New model to characterize cell-substrate
adhesion in the presence of osmosis," Med. Biol. Eng. Comput., vol.
38, pp. 690-691, 2000.
[12] K. K. Liu, V. Chan, and Z. Zhang, "Capsule-sunstrate contact
deformation: determination of adhesion energy," Med. Biol. Eng.
Comput., vol. 40, pp. 491-495, 2002a.
[13] K. K. Liu, H. G. Wang, K. T. Wan, T. Liu, and T. Zhang,
"Characterizing capsule-substrate adhesion in presence of osmosis,"
Colloid. Surface B., vol. 25, pp. 293-298, 2002b.
[14] C. S. Peskin, "Numerical analysis of blood flow in the heart," J.
Comput. Phys., vol. 25, pp. 220-252, 1977.
[15] N. A. N-Dri, W. Shyy, and R. Tran-Son-Tay, "Computational modeling
of cell adhesion and movement using a continuum-Kinetics Approach,"
Biophys. J., vol. 85, pp. 2273-2286, 2003.
[16] S. Jadhav, K.Y. Chan, K. Konstantinos, and C. D. Eggleton, "Shear
modulation of intercellular contact area between two deformable cells
colliding under flow," J. Biomech., vol. 40, pp. 2891-2897, 2007.
[17] R. J. LeVeque and Z. Li, "The immersed interface method for elliptic
equations with discontinuous coefficients and singular sources," SIAM
J. Numer. Anal., vol. 31, pp. 1019-1044, 1994.
[18] Z. Li and M-C. Lai, "The immersed interface method for the Navier-
Stokes equations with singular forces," J. Comput. Phys., vol. 171, pp.
822-842, 2001.
[19] M-C. Lai and H. C. Tseng, "A simple implementation of the immersed
interface methods for Stokes flows with singular forces," Comput.
Fluids., vol. 37, pp. 99-106, 2008.
[20] D. V. Le, B. C. Khoo, and J. Peraire, "An immersed interface method
for viscous incompressible flows involving rigid and flexible
boundaries," J. Comput. Phys., vol. 220, pp. 109-138, 2006.
[21] L. Lee and R. J. LeVeque, "An immersed interface method for the
incompressible Navier-Stokes equations," SIAM J. Sci. Comput., vol.
25, pp. 832-856, 2003.
[22] R. J. LeVeque and Z. Li, "Immersed interface methods for Stokes flow
with elastic boundaries or surface tension," SIAM J. Sci. Comput., vol.
18, pp. 709-735, 1997.
[23] Z. Li, "An overview of the immersed interface method and its
applications," Taiwan. J. Math., vol. 7, pp. 1-49, 2003.
[24] M. N. Linnick and H. F. Fasel, "A high-order immersed interface
method for simulating unsteady incompressible flows on irregular
domains," J. Comput. Phys., vol. 204, pp. 157-192, 2005.
[25] Z. Tan, D. V. Le, Z. Li, K. M. Lim, and B. C. Khoo, "An immersed
interface method for solving incompressible viscous flows with
piecewise constant viscosity across a moving elastic membrane," J.
Comput. Phys., vol. 227, pp. 9955-9983, 2008.
[26] S. Xu and Z. J. Wang, "An immersed interface method for simulating
the interaction of a fluid with moving boundaries," J. Comput. Phys.,
vol. 216, pp. 454-493, 2006.
[27] A. T. Layton, "Modeling water transport across elastic boundaries
using an explicit jump method," SIAM J. Sci. Comput., vol. 28, pp.
2189-2207, 2006.
[28] O. Kedem and A. Katchalsky, "Thermodynamics Analysis of the
Permeability of Biological Membranes to Non-electrolytes," Biochim.
Biophys. Acta, vol. 27, pp. 229-246, 1958.
[29] Z. Li, X. Wan, K. Ito, and S. R. Lubkin, "An augmented approach for
the pressure boundary condition in a stokes flow," Commun. Comput.
Phys., vol. 5, pp. 874-885, 2006.
[30] Z. Li and K. Ito, "The Immersed Interface Method: Numerical solutions
of PDEs involving interfaces and irregular domains," USA, 2006.
[31] P. G. Jayathilake, Z.-J. Tan, B. C. Khoo, and N. E. Wijeysundera,
"Deformation and osmotic swelling of an elastic membrane capsule in
Stokes flows by the immersed interface method," Chem. Eng. Sci, to be
published, 2009.
[32] I. Contat and C. Misbah, "Dynamics and similarity laws for adhering
vesicles in haptotaxis," Phys. Rev. Lett., vol. 83, pp. 235-238, 1999.
[33] Y. Liu, L. Zhaing, X. Wang, and W. K. Liu, "Coupling of Navier-
Stokes equations with protein molecular dynamics and its application to
hemodynamics," Int. J. Numer. Meth. Fl., vol. 46, pp. 1237-1252,
2004.
[34] J. Zhang, P.C. Johnson, and A.S. Popel, "Red blood cell aggregation
and dissociation in shear flows simulated by lattice Boltzmann
method," J. Biomech., vol. 41, pp. 47-55, 2008.
[35] C. X. Wang, L. Wang, and C.R. Thomas, "Modelling the mechanical
properties of single suspension-cultured tomato cells," Ann. Bot-
London, vol. 93, pp. 443-453, 2004.
[36] D. Zinemanas, A. Nir, "Osmophoretic motion of deformable particles,"
Int. J. Multiphas. Flow, vol. 21, pp. 787-800, 1995.
[1] T. A. Springer, "Traffic signals on endothelium for lymphocyte
recirculation and leukocyte emigration," Annu. Rev. Physiol., vol. 57,
pp. 827-872, 1995.
[2] D. B. Khismatullin and G. A. Truskey, "A 3D numerical study of the
effect of channel height on leukocyte deformation and adhesion in
parallel plate flow chambers," Microvasc. Res., vol. 68, pp. 188-202,
2004.
[3] C. Dong and X. X. Lei, "Biomechanics of cell rolling: shear flow, cellsurface
adhesion, and cell deformability," J. Biomech., vol. 33, pp. 35-
43, 2000.
[4] V. Pappu, S. K. Doddi and P. Bagchi, "A computational study of
leukocyte adhesion and its effect on flow pattern in microvessels," J.
Theor. Biol., vol. 254, pp. 483-498, 2008.
[5] V. Pappu and P. Bagchi, "3D computational modeling and simulation
of leukocyte rolling adhesion and deformation," Comput. Biol. Med.,
vol. 38, pp. 738-753, 2008.
[6] K. T. Wan and K. K. Liu, "Contact mechanics of a thin-walled capsule
adhered onto a rigid planar substrate," Med. Biol. Eng. Comput., vol.
39, pp. 605-608, 2001.
[7] R. M. Servuss and W. Helfrich, "Mutual adhesion of lecithin
membranes at ultralow tensions," J. Phys., vol. 50, pp. 809-827, 1989.
[8] K. D. Tachev, J. K. Angarska, K. D. Danov, and P. A. Kralchevsky,
"Erythrocyte attachment to substrates: determination of membrane
tension and adhesion energy," Colloid. Surface B., vol. 19, pp. 61-80,
2000.
[9] J. J. Foo, V. Chan, and K. K. Liu, "Contact deformation of liposome in
the presence of osmosis," Ann. Biomed. Eng., vol. 31, pp. 1279-1286,
2003.
[10] J. J. Foo, V. Chan, and K. K. Liu, "Coupling bending and shear effects
on liposome deformation," J. Biomech., vol. 39, pp. 2338-2343, 2006.
[11] K. K. Liu and K. T. Wan, "New model to characterize cell-substrate
adhesion in the presence of osmosis," Med. Biol. Eng. Comput., vol.
38, pp. 690-691, 2000.
[12] K. K. Liu, V. Chan, and Z. Zhang, "Capsule-sunstrate contact
deformation: determination of adhesion energy," Med. Biol. Eng.
Comput., vol. 40, pp. 491-495, 2002a.
[13] K. K. Liu, H. G. Wang, K. T. Wan, T. Liu, and T. Zhang,
"Characterizing capsule-substrate adhesion in presence of osmosis,"
Colloid. Surface B., vol. 25, pp. 293-298, 2002b.
[14] C. S. Peskin, "Numerical analysis of blood flow in the heart," J.
Comput. Phys., vol. 25, pp. 220-252, 1977.
[15] N. A. N-Dri, W. Shyy, and R. Tran-Son-Tay, "Computational modeling
of cell adhesion and movement using a continuum-Kinetics Approach,"
Biophys. J., vol. 85, pp. 2273-2286, 2003.
[16] S. Jadhav, K.Y. Chan, K. Konstantinos, and C. D. Eggleton, "Shear
modulation of intercellular contact area between two deformable cells
colliding under flow," J. Biomech., vol. 40, pp. 2891-2897, 2007.
[17] R. J. LeVeque and Z. Li, "The immersed interface method for elliptic
equations with discontinuous coefficients and singular sources," SIAM
J. Numer. Anal., vol. 31, pp. 1019-1044, 1994.
[18] Z. Li and M-C. Lai, "The immersed interface method for the Navier-
Stokes equations with singular forces," J. Comput. Phys., vol. 171, pp.
822-842, 2001.
[19] M-C. Lai and H. C. Tseng, "A simple implementation of the immersed
interface methods for Stokes flows with singular forces," Comput.
Fluids., vol. 37, pp. 99-106, 2008.
[20] D. V. Le, B. C. Khoo, and J. Peraire, "An immersed interface method
for viscous incompressible flows involving rigid and flexible
boundaries," J. Comput. Phys., vol. 220, pp. 109-138, 2006.
[21] L. Lee and R. J. LeVeque, "An immersed interface method for the
incompressible Navier-Stokes equations," SIAM J. Sci. Comput., vol.
25, pp. 832-856, 2003.
[22] R. J. LeVeque and Z. Li, "Immersed interface methods for Stokes flow
with elastic boundaries or surface tension," SIAM J. Sci. Comput., vol.
18, pp. 709-735, 1997.
[23] Z. Li, "An overview of the immersed interface method and its
applications," Taiwan. J. Math., vol. 7, pp. 1-49, 2003.
[24] M. N. Linnick and H. F. Fasel, "A high-order immersed interface
method for simulating unsteady incompressible flows on irregular
domains," J. Comput. Phys., vol. 204, pp. 157-192, 2005.
[25] Z. Tan, D. V. Le, Z. Li, K. M. Lim, and B. C. Khoo, "An immersed
interface method for solving incompressible viscous flows with
piecewise constant viscosity across a moving elastic membrane," J.
Comput. Phys., vol. 227, pp. 9955-9983, 2008.
[26] S. Xu and Z. J. Wang, "An immersed interface method for simulating
the interaction of a fluid with moving boundaries," J. Comput. Phys.,
vol. 216, pp. 454-493, 2006.
[27] A. T. Layton, "Modeling water transport across elastic boundaries
using an explicit jump method," SIAM J. Sci. Comput., vol. 28, pp.
2189-2207, 2006.
[28] O. Kedem and A. Katchalsky, "Thermodynamics Analysis of the
Permeability of Biological Membranes to Non-electrolytes," Biochim.
Biophys. Acta, vol. 27, pp. 229-246, 1958.
[29] Z. Li, X. Wan, K. Ito, and S. R. Lubkin, "An augmented approach for
the pressure boundary condition in a stokes flow," Commun. Comput.
Phys., vol. 5, pp. 874-885, 2006.
[30] Z. Li and K. Ito, "The Immersed Interface Method: Numerical solutions
of PDEs involving interfaces and irregular domains," USA, 2006.
[31] P. G. Jayathilake, Z.-J. Tan, B. C. Khoo, and N. E. Wijeysundera,
"Deformation and osmotic swelling of an elastic membrane capsule in
Stokes flows by the immersed interface method," Chem. Eng. Sci, to be
published, 2009.
[32] I. Contat and C. Misbah, "Dynamics and similarity laws for adhering
vesicles in haptotaxis," Phys. Rev. Lett., vol. 83, pp. 235-238, 1999.
[33] Y. Liu, L. Zhaing, X. Wang, and W. K. Liu, "Coupling of Navier-
Stokes equations with protein molecular dynamics and its application to
hemodynamics," Int. J. Numer. Meth. Fl., vol. 46, pp. 1237-1252,
2004.
[34] J. Zhang, P.C. Johnson, and A.S. Popel, "Red blood cell aggregation
and dissociation in shear flows simulated by lattice Boltzmann
method," J. Biomech., vol. 41, pp. 47-55, 2008.
[35] C. X. Wang, L. Wang, and C.R. Thomas, "Modelling the mechanical
properties of single suspension-cultured tomato cells," Ann. Bot-
London, vol. 93, pp. 443-453, 2004.
[36] D. Zinemanas, A. Nir, "Osmophoretic motion of deformable particles,"
Int. J. Multiphas. Flow, vol. 21, pp. 787-800, 1995.
@article{"International Journal of Mechanical, Industrial and Aerospace Sciences:62933", author = "P.G. Jayathilake and B.C. Khoo and Zhijun Tan", title = "Capsule-substrate Adhesion in the Presence of Osmosis by the Immersed Interface Method", abstract = "A two-dimensional thin-walled capsule of a flexible
semi-permeable membrane is adhered onto a rigid planar substrate
under adhesive forces (derived from a potential function) in the
presence of osmosis across the membrane. The capsule is immersed
in a hypotonic and diluted binary solution of a non-electrolyte
solute. The Stokes flow problem is solved by the immersed interface
method (IIM) with equal viscosities for the enclosed and
surrounding fluid of the capsule. The numerical results obtained are
verified against two simplified theoretical solutions and the
agreements are good. The osmotic inflation of the adhered capsule is
studied as a function of the solute concentration field, hydraulic
conductivity, and the initial capsule shape. Our findings indicate that
the contact length shrinks in dimension as capsule inflates in the
hypotonic medium, and the equilibrium contact length does not
depend on the hydraulic conductivity of the membrane and the
initial shape of the capsule.", keywords = "Capsule-substrate adhesion, Fluid mechanics,Immersed interface method, Osmosis, Mass transfer.", volume = "3", number = "12", pages = "1612-10", }
