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A necklace model for vesicles simulations in 2D

Profile image of Mourad IsmailMourad IsmailProfile image of Aline Lefebvre-LepotAline Lefebvre-Lepot

2014, International Journal for Numerical Methods in Fluids

https://doi.org/10.1002/FLD.3960
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21 pages

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Abstract

The aim of this paper is to propose a new numerical model to simulate 2D vesicles interacting with a newtonian fluid. The inextensible membrane is modeled by a chain of circular rigid particles which are maintained in cohesion by using two different type of forces. First, a spring force is imposed between neighboring particles in the chain. Second, in order to model the bending of the membrane, each triplet of successive particles is submitted to an angular force.

Figures (16)
Figure 1: Necklace model of the vesicle. These two last domains are supposed to be filled with fluids governed by Stokes equations with respective viscosities jn and Upy;-
Figure 1: Necklace model of the vesicle. These two last domains are supposed to be filled with fluids governed by Stokes equations with respective viscosities jn and Upy;-
The forces derived from these energies will be plugged in the right hand side of fluids model. This gives the major advantage to decouple entirely fluid solver and membrane model. Thus, the unknowns of the problem are the velocity and pressure fields in Q \ B, respectively denoted by u = (wu, U2) and p, together with the translational and angular velocities of the rigid particles, denoted by V = (V;,..., Vy) € R27” and w = (wj,..., wy) € RY. The Constant Area is insured by imposing a constraint which will be described in section 3.2.
The forces derived from these energies will be plugged in the right hand side of fluids model. This gives the major advantage to decouple entirely fluid solver and membrane model. Thus, the unknowns of the problem are the velocity and pressure fields in Q \ B, respectively denoted by u = (wu, U2) and p, together with the translational and angular velocities of the rigid particles, denoted by V = (V;,..., Vy) € R27” and w = (wj,..., wy) € RY. The Constant Area is insured by imposing a constraint which will be described in section 3.2.
Figure 3: Area variation induced by the displacement dx; of particle i. A second projection step is added in order to deal with the constant area constraint. To do so, let’s assume that the area of the vesicle is approximated by the area of the polygon defined by the center of the particles. Then, we compute the area variation dA induced by the displacement dx; of particle i. We denote by n7 and n; the two outer unit normals to the polygon at point x; (see figure 3) and we define n; = 5(|xi+1 - xi|n* + |x;-1 — xiln;) (note that indices has to be adapted for i= 1 andi=N). Then, the considered area variation is given by Obviously, in the case where all particles move we obtain
Figure 3: Area variation induced by the displacement dx; of particle i. A second projection step is added in order to deal with the constant area constraint. To do so, let’s assume that the area of the vesicle is approximated by the area of the polygon defined by the center of the particles. Then, we compute the area variation dA induced by the displacement dx; of particle i. We denote by n7 and n; the two outer unit normals to the polygon at point x; (see figure 3) and we define n; = 5(|xi+1 - xi|n* + |x;-1 — xiln;) (note that indices has to be adapted for i= 1 andi=N). Then, the considered area variation is given by Obviously, in the case where all particles move we obtain
Figure 4 shows the evolution of a vesicle suspended in a fluid at rest by plotting the position of each rigid particle center. One can see that the vesicle reaches its equilibrium shape which is biconcave in the case of small reduced area. Note that this form is typical of red blood cells Figure 4: Equlibrium shapes. Shapes versus ¢ for a = 0.42. In our simulations the initial distribution of the rigid particles (that form the vesicle mem- brane) is done by placing them in an ellipse. Then, the Stokes problem is solved in a rectangular domain containing the vesicle with homogeneous Dirichlet boundary conditions.
Figure 4 shows the evolution of a vesicle suspended in a fluid at rest by plotting the position of each rigid particle center. One can see that the vesicle reaches its equilibrium shape which is biconcave in the case of small reduced area. Note that this form is typical of red blood cells Figure 4: Equlibrium shapes. Shapes versus ¢ for a = 0.42. In our simulations the initial distribution of the rigid particles (that form the vesicle mem- brane) is done by placing them in an ellipse. Then, the Stokes problem is solved in a rectangular domain containing the vesicle with homogeneous Dirichlet boundary conditions.
Figure 5: Equilibrium shapes. Area and perimeter versus t for a = 0.42. It is important to check that the area and the perimeter are well conserved. During this simulation, the variation of the area ranges from —0.3% to 0.2% and the variation of the perimeter from —0.05% to 0.05% (see figure 5).
Figure 5: Equilibrium shapes. Area and perimeter versus t for a = 0.42. It is important to check that the area and the perimeter are well conserved. During this simulation, the variation of the area ranges from —0.3% to 0.2% and the variation of the perimeter from —0.05% to 0.05% (see figure 5).
Figure 6: Equilibrium shapes for different values of a. Figure 6 shows the computed equilibrium shapes of vesicles with different reduced area. As it is well known, the equilibre shape goes from circular one when @ = | to biconcave one when a is small enough (about 0.6).
Figure 6: Equilibrium shapes for different values of a. Figure 6 shows the computed equilibrium shapes of vesicles with different reduced area. As it is well known, the equilibre shape goes from circular one when @ = | to biconcave one when a is small enough (about 0.6).
Figure 7: Equilibrium shapes. Comparison with results from Kaoui et al. [28]
Figure 7: Equilibrium shapes. Comparison with results from Kaoui et al. [28]
Figure 8: Tank treading angles. Comparison between our results (in the case of small confinement t = 0.2 ) and those from [9] (in unbounded domain). We are interested in this section to the dynamic of a vesicle under linear shear flow. It is wel known that if A, the ratio between the inner and the outer fluid viscosities, is smaller than a certain critical value (around 5 in unbounded domain [9]), the vesicle takes a steady angle with respect to the horizontal axis and its membrane starts to rotate like a chain of a tank (this motion is called Tank-Treading). To validate our model in this regime, we refer to the results given in [9]. So, we consider vesicles of different reduced area subject to linear shear flow in a rectangular domain and the confinement parameter t is chosen to be small (around 0.2) to minimize the effect of walls. In figure 8 we plot the steady angle for different reduced area and we compare our results with those from [9].
Figure 8: Tank treading angles. Comparison between our results (in the case of small confinement t = 0.2 ) and those from [9] (in unbounded domain). We are interested in this section to the dynamic of a vesicle under linear shear flow. It is wel known that if A, the ratio between the inner and the outer fluid viscosities, is smaller than a certain critical value (around 5 in unbounded domain [9]), the vesicle takes a steady angle with respect to the horizontal axis and its membrane starts to rotate like a chain of a tank (this motion is called Tank-Treading). To validate our model in this regime, we refer to the results given in [9]. So, we consider vesicles of different reduced area subject to linear shear flow in a rectangular domain and the confinement parameter t is chosen to be small (around 0.2) to minimize the effect of walls. In figure 8 we plot the steady angle for different reduced area and we compare our results with those from [9].
Figure 9: Streamlines of the velocity field and vesicle position in Tank-Treading regime at different time. a = 0.85, Tt = 0.32, N = 38 and = 1.
Figure 9: Streamlines of the velocity field and vesicle position in Tank-Treading regime at different time. a = 0.85, Tt = 0.32, N = 38 and = 1.
Figure 10: Streamlines of the velocity field and vesicle position in Tumbling regime at different time. a = 0.85, t = 0.32, N = 38 and A = 20.
Figure 10: Streamlines of the velocity field and vesicle position in Tumbling regime at different time. a = 0.85, t = 0.32, N = 38 and A = 20.
Figure 11: Angles versus time in tumbling regime. a = 0.85 and t = 0.2.
Figure 11: Angles versus time in tumbling regime. a = 0.85 and t = 0.2.
We assume that the vesicle has a circular shape with radius R. For large N, namely for small ON = = am see figure 2 for notations, we have From this, together with the fact that the continuous bending energy is given by
We assume that the vesicle has a circular shape with radius R. For large N, namely for small ON = = am see figure 2 for notations, we have From this, together with the fact that the continuous bending energy is given by
Figure 12: Streamlines of the velocity field and vesicle position in Vacillating-Breathing regime at different time. a = 0.85, t = 0.32, N = 38 andA=7.5.
Figure 12: Streamlines of the velocity field and vesicle position in Vacillating-Breathing regime at different time. a = 0.85, t = 0.32, N = 38 andA=7.5.
Figure 13: Angles versus time for different viscosity ratios. a = 0.85, t = 0.2.
Figure 13: Angles versus time for different viscosity ratios. a = 0.85, t = 0.2.
Figure 14: TT Angles versus viscosity ration.
Figure 14: TT Angles versus viscosity ration.
Figure 15: Vacillating-Breathing. Area and perimeter versus t for a = 0.85
Figure 15: Vacillating-Breathing. Area and perimeter versus t for a = 0.85

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