Deformation of giant lipid bilayer vesicles in shear flow

Haas, Karl‐Heinz; Blom, C.; Ende, Dirk van den; Duits, Michael H.G.; Mellema, J.

doi:10.1103/physreve.56.7132

Cited by 144 publications

(145 citation statements)

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“…In contrast, the nonequilibrium dynamics of lipid bilayer membranes has been studied only to a limited extent. Experimental studies 7,8,9,10 of vesicle behavior in unbounded shear flow observe that in weak flows, and when the inner and outer fluids are the same, the vesicle deforms into a tanktreading stationary prolate ellipsoid with an inclination angle close to π/4 with respect to the flow direction; however, in striking contrast to drops, when the fluid inside is more viscous than outside the vesicle undergoes a tumbling motion. Numerical simulations 11,12,13 and analytical theories 14,15,16 of vesicle microhydrodynamics attempt to elucidate such experimental observations.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a viscous vesicle in linear flows

2007

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An analytical theory is developed to describe the dynamics of a closed lipid bilayer membrane (vesicle) freely suspended in a general linear flow. Considering a nearly spherical shape, the solution to the creeping-flow equations is obtained as a regular perturbation expansion in the excess area. The analysis takes into account the membrane fluidity, incompressibility and resistance to bending. The constraint for a fixed total area leads to a non-linear shape evolution equation at leading order. As a result two regimes of vesicle behavior, tank-treading and tumbling, are predicted depending on the viscosity contrast between interior and exterior fluid. Below a critical viscosity contrast, which depends on the excess area, the vesicle deforms into a tank-treading ellipsoid, whose orientation angle with respect to the flow direction is independent of the membrane bending rigidity. In the tumbling regime, the vesicle exhibits periodic shape deformations with a frequency that increases with the viscosity contrast. Non-Newtonian rheology such as normal stresses is predicted for a dilute suspension of vesicles. The theory is in good agreement with published experimental data for vesicle behavior in simple shear flow.

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Section: Introductionmentioning

confidence: 99%

Dynamics of a viscous vesicle in linear flows

2007

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“…This non-equilibrium problem has revealed a variety of new physical effects and became a subject of intense experimental and theoretical studies. Laboratory experiments [1,2,3,4,5] have shown that the vesicles immersed in a shear flow exhibit at least two qualitatively different types of behavior, either tank-treading or tumbling motion. In the tank-treading regime a vesicle shape is stationary, it is ellipsoid oriented at an angle with respect to the shear flow.…”

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confidence: 99%

“…whereâ is some dimensionless operator, reflecting all viscous mechanisms, the Z-axis of our reference frame is chosen to be directed opposite to the angular velocity ω, and F (3) is an expansion up to third order in u of the free energy (1). Note that the elongation and the rotation parts of the external flow are separated in Eq.…”

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confidence: 99%

Dynamics of Nearly Spherical Vesicles in an External Flow

2007

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We analytically derive an equation describing vesicle evolution in a fluid where some stationary flow is excited regarding that the vesicle shape is close to a sphere. A character of the evolution is governed by two dimensionless parameters, S and Λ, depending on the vesicle excess area, viscosity contrast, membrane viscosity, strength of the flow, bending module, and ratio of the elongation and rotation components of the flow. We establish the "phase diagram" of the system on the S − Λ plane: we find curves corresponding to the tanktreading to tumbling transition (described by the saddle-node bifurcation) and to the tank-treading to trembling transition (described by the Hopf bifurcation). PACS numbers: 87.16.Dg, 87.17.Jj, 05.45.a Vesicles are closed membranes which separate two regions occupied by possibly different fluids. The vesicles are attracting significant attention not only due to their resemblance with biological objects but also because of their importance in different industries such as pharmaceutics where they are used for drug transportation. A natural problem which arises in these applications is understanding of how a single vesicle behaves in an external flow. This non-equilibrium problem has revealed a variety of new physical effects and became a subject of intense experimental and theoretical studies. Laboratory experiments [1,2,3,4,5] have shown that the vesicles immersed in a shear flow exhibit at least two qualitatively different types of behavior, either tank-treading or tumbling motion. In the tank-treading regime a vesicle shape is stationary, it is ellipsoid oriented at an angle with respect to the shear flow. In the tumbling regime the vesicle experiences periodic flipping in the shear plane. A novel type of behavior: trembling, discovered in the work [6] is an intermediate regime between tank-treading and tumbling in which a vesicle trembles around the flow direction.Constructing a phase diagram for all these regimes depending on the external parameters is a challenging and an extremely difficult task because the problem in consideration is both strongly non-linear and non-equilibrium. As long as no analytic solution of this problem exists theoretical studies were based either on numerical simulations or on some approximations allowing analytical treatment. Numerical investigations of this problem involved several different computational schemes, including boundary element method [7], mesoscopic particle-based approximation [8,9,10,11,12], and an advected field approach [13]. These approaches have shown qualitative agreement with experiments however did not solve the problem of constructing the vesicle dynamics phase diagram completely. Analytical studies of the problem can be divided in two major classes. In the first one [9,14,15], phenomenological models of a vesicle dynamics based on the classical work of Keller and Skallak [16] were proposed and proved themselves to be rather efficient in explaining the experiments. In the second series of works [17,18,19,20] the studies foc...

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“…However, it was demonstrated that lipid vesicles without an attached polymer network may serve as a simple model which already captures the basic physics of the fluid-structure interaction problem. We note that the dynamics of RBC and vesicles in fluid flow has been studied both experimentally (see [1,2,19,20,26,37]) and theoretically (cf. [6,7,21,22,32,34]).…”

Section: % Iterationmentioning

confidence: 99%

“…at each time step the BE/BE FE-IB then amounts to the computation of z (k+1) , 0 ≤ k ≤ M − 1, as the solution of the nonlinear system 19) where the nonlinear mapping G(·; t k+1 ) :…”

Section: Fully Implicit Fe-ibmentioning

confidence: 99%

An adaptive Newton continuation strategy for the fully implicit finite element immersed boundary method

Hoppe

Linsenmann

2012

Journal of Computational Physics

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Abstract. The Immersed Boundary Method (IB) is known as a powerful technique for the numerical solution of fluid-structure interaction problems as, for instance, the motion and deformation of viscoelastic bodies immersed in an external flow. It is based on the treatment of the flow equations within an Eulerian framework and of the equations of motion of the immersed bodies with respect to a Lagrangian coordinate system including interaction equations providing the transfer between both frames. The classical IB uses finite differences, but the IBM can be set up within a finite element approach in the spatial variables as well (FE-IB). The discretization in time usually relies on the Backward Euler (BE) method for the semidiscretized flow equations and the Forward Euler (FE) method for the equations of motion of the immersed bodies. The BE/FE FE-IB is subject to a CFL-type condition, whereas the fully implicit BE/BE FE-IB is unconditionally stable. The latter one can be solved numerically by Newton-type methods whose convergence properties are dictated by an appropriate choice of the time step size, in particular, if one is faced with sudden changes in the total energy of the system. In this paper, taking advantage of the well developed affine covariant convergence theory for Newton-type methods, we study a predictor-corrector continuation strategy in time with an adaptive choice of the continuation steplength. The feasibility of the approach and its superiority to BE/FE FE-IB is illustrated by a representative numerical example.Key words. finite element immersed boundary method, fully implicit scheme, predictorcorrector continuation, red blood cells AMS subject classifications. 65H20, 65M60, 74L15, 76D05, 92C101. Introduction. A computationally attractive methodology for the numerical simulation of the motion and deformation of elastic and viscoelastic bodies in external flows is the Immersed Boundary Method (IB), which has been originally developed by Peskin [28] and further studied in [11,29,30,31,33]. The IBM uses an Eulerian coordinate system for the flow equations and Lagrangian coordinates for the boundary of the immersed bodies together with appropriate interaction equations to transform Eulerian to Lagrangian quantities and vice versa. The interaction equations feature multidimensional Dirac delta functions that have to be approximated appropriately within a finite difference approach. More recently, a variational formulation of the IBM has been provided in [8] and [9, 10] as a basis for a finite element realization referred to as the Finite Element Immersed Boundary Method (FE-IB). Both for the classical IB and the FE-IB, the most common approach with regard to discretization in time is to use the Backward Euler (BE) method for the flow equations and the Forward Euler (FE) method for the equation describing the motion and deformation of the immersed bodies which gives rise to the BE/FE IB and BE/FE FE-IB, respectively. However, these schemes typically require a CFL-type condition (cf., e.g., [9,16]). B...

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Deformation of giant lipid bilayer vesicles in shear flow

Cited by 144 publications

References 20 publications

Dynamics of a viscous vesicle in linear flows

Dynamics of a viscous vesicle in linear flows

Dynamics of Nearly Spherical Vesicles in an External Flow

An adaptive Newton continuation strategy for the fully implicit finite element immersed boundary method

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