6.2 Numerical discretization
6.3.6 Vesicles in Poiseuille flow
As mentioned before, vesicles have the similar mechanical properties that mimic RBCs in flows. There have been widely investigated in the rheology of red blood cells passing through capillaries in Poiseuille flows numerically such as [56, 52, 61] and the references therein. Therefore, it will be quite natural to study the vesicle dynamics in Poiseuille flows. Recently, Coupier et. al. [8] study the problem through experiments, numerical, and theoretical computations to characterize the shape diagram of vesicles in Poiseuille flow. In the following, we shall also investigate the problem and compare our axisymmetric results with the ones shown in [8].
In this subsection, we set up the Poiseuille flow as illustrated in Figure 6.6 by w = Wm
µ 1 − r2
L2
¶
, u = 0,
where L is the capillary radius, and Wmis mean flow velocity indicating the centerline velocity. Throughout this subsection, as in Figure 6.6, we draw the z-axis along the horizontal direction (different from those in previous subsections) since the flow is along the axial direction.
It is known that, in Poiseuille flow, a vesicle reaches its equilibrium shape and then moves with a constant velocity. As discussed in literature [8], the reduced volume ν
z−axis r−axis
L
Figure 6.6: The velocity profile of Poiseuille flow.
measured the deflation of the vesicle plays a significant role in vesicle dynamics which is defined as
ν = V0
4
3π(A0/4π)3/2,
where V0 and A0 represent the volume and the surface area, respectively. This di-mensionless number is nothing but the volume ratio of the vesicle to a sphere with the same surface area. Thus, for a sphere, the reduced volume equals to one. In ad-dition, we define a characteristic non-dimensional parameter ˆR to indicate the vesicle confinement in the flow by
R =ˆ R0 L ,
where the effect radius can be computed by R0 = (3V0/4π)1/3. Thus, a larger ˆR means a stronger confinement. In this subsection, we shall investigate three different effects by varying the reduced volume ν, the confinement ˆR, and the mean velocity Wmindividually. The computational r −z domain is chosen as Ω = [0, L]×[−5L, 5L].
Unless otherwise stated, we all use oblate vesicles as our initial shapes in the flows.
• Effect of the reduced volume. In this test, we choose an oblate vesicle with three different reduced volumes ν = 0.48, 0.75, 0.9 but the same volume in a Poiseuille flow with a weaker confinement ˆR = 0.3 and Wm = 1. We run the simulations until the equilibrium shapes are obtained. Figure 6.7(a) shows the snapshots toward to equilibrium shapes for those different reduced volumes. One can see the parachute-like shapes are observed in all three cases while the smaller reduced volume deforms significantly more since it is thinner. This behavior is
(a)
(b)
Figure 6.7: (a) Snapshots of the vesicle in Poiseuille flow with different reduced volume ν = 0.48 (top), ν = 0.75 (middle), and ν = 0.9 (bottom). The flow comes from left to right. (b) A vesicle with initial prolate shape with reduced volume ν = 0.9 results in the bullet-like shape eventually.
consistent with the results obtained in [8]. On the other hand, it is interesting to see that if we choose the prolate vesicle initially (with same surface area and volume as the oblate one with ν = 0.9), we can obtain the bullet-like shape as shown in Figure 6.7(b). Notice that, the parachute-like shape has a concave rear part while the bullet-like shape has a convex one instead.
• Effect of the confinement. To study the confinement effect, we keep the mean velocity Wm = 1 and reduced volume ν = 0.95 fixed but vary the confinement as R = 0.3, 0.375, 0.5, respectively. Figure 6.8 shows the cross-sectional view of theˆ equilibrium shapes with different confinements in which the confinement gets stronger from left to right. One can see that the equilibrium shape turns from a parachute-like shape into bullet-like one as the confinement gets stronger. This is a physically interesting result which can be explained as the confinement is sufficiently large, its effect will dominate the other flow effects. This is also in a good agreement with the result obtained in [8].
−0.5 0 0.5
r−axis
Figure 6.8: Cross-sectional view of the equilibrium shapes with different confinements Left: ˆR = 0.3; middle: ˆR = 0.375 and right: ˆR = 0.5. As the confinement increases from left to right, the shape turns from parachute-like to bullet-like.
• Effect of the mean velocity. In the final test, we keep the reduced volume ν = 0.95 and the weaker confinement ˆR = 0.3 fixed but vary the mean velocity as Wm = 1, 10, 100. Notice that, varying the mean velocity alone (but keep-ing other parameters fixed) means to vary the capillary number Ca = µRc40Wm
bL2 . Thus, the effect can be regarded as the effect of capillary number. Figure 6.9 shows the cross-sectional view of the equilibrium shapes with different mean velocity. By increasing the mean flow velocity, the equilibrium vesicle will turn from parachute-like to an unexpected bullet-like shape. This interesting result has been obtained from the experimental observations [8] (the authors used ν between 0.95 and 0.97) and our result in Figure 6.9 is qualitatively consistent with theirs.
−0.5 0 0.5
r−axis
Figure 6.9: Cross-sectional view of the equilibrium shapes with different mean ve-locity. Left: Wm = 1; middle: Wm = 10 and right: Wm = 100. As the mean flow velocity increases, the shape turns from parachute-like to bullet-like.
Chapter 7
Conclusions and future works
In this thesis, we developed three kinds of numerical methods to simulate vesicle dynamics. The governing equations are formulated based on immersed boundary framework where a mixture of Eulerian fluid variables and Lagrangian interfacial variables is used, with the linkage of these two kinds of variable is by smoothed Dirac delta function. Firstly, We state the model of an inextensible interface (vesicle with-out bending effect) suspended in Stokes flow. By taking advantage of the property of skew-adjointness between spreading operator and surface divergence operator, the re-sultant linear system can be solved efficiently by iterative conjugate gradient method.
Secondly, we turn our direction to employ the nearly inextensible approach to mimic vesicle dynamics. We proposed an unconditionally energy stable scheme so that the time restriction can be released substantially. This scheme involves solving a positive definite linear system which can be done efficiently by multigrid method. Lastly, we consider the realistic case of three-dimensional vesicle. The vesicle model is approx-imated by penalized nearly inextensible approach. Moreover, differ to our previous work, the derivative of interfacial variables are evaluated by high accuracy spectral method which outperforms finite difference method. The fluid equation is discretized by projection method and then fast Poisson solver can be used. We investigated vesicle dynamics numerically, such as relaxation to equilibrium state, tank-treading motion under shear flow, with presence of gravity field and shape rheology under Poiseuille flow.
As a next step, we will generalized the present model to with consideration of viscosity contrast which is close to the realistic world. In particular, we plan to extend our work to real three-dimensional space, while the presentation of vesicle shall be treated carefully when it deforms.
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Appendix A
Geometrical operators and quantities on a surface
In this appendix, we provide computation process of mean curvature H and Gaussian curvature K which can be obtained by first and second fundamental forms [51] and geometrical differential operators on surface.
A three dimensional surface can be presented by parametric form as X(α, β) = (X(α, β), Y (α, β), Z(α, β)), 0 ≤ α ≤ Lα and 0 ≤ β ≤ Lβ, where α and β are two Lagrangian parameters. From first fundamental form, we have
E = Xα· Xα, F = Xα· Xβ, G = Xβ· Xβ, W =√
EG − F2; by second fundamental form, we have
L = Xαα· n, M = Xαβ · n, N = Xββ· n,
where unit normal vector can be obtained by n = Xα×Xβ/W . With these elementary geometrical quantities, the mean curvature H and Gaussian curvature K is computed by
H = −1 2
EN − 2F M + GL
W2 , K = LN − M2 W2 .
Suppose φ is a scalar function and f is a vector function, then we have surface gradient
∇s, surface divergence ∇s· and surface Laplace operator ∆s (or Laplace-Beltrami operator) as follows.
∇sφ = GXα− F Xβ
W2 φα+EXβ− F Xα
W2 φβ,
∇s· f = Gfα− F fβ
W2 · Xα+Efβ− F fα W2 · Xβ,
∆sφ = 1 W
"µ
Gφα− F φβ W
¶
α
+
µEφβ− F φα W
¶
β
# .
Appendix B
Discrete skew-adjoint operators
In this appendix, we give a direct derivation to the matrix obtained from the discrete spreading operator Sh of σh and the matrix obtained from discrete surface divergence operator ∇sh of fU are transpose with each other. First, let us rewrite the operator Sh(σh) by
Sh(σh) =
M −1X
k=0
Ds(στ )k δh(x − Xk) ∆s
=
M −1X
k=0
σk+1/2τk+1/2− σk−1/2τk−1/2
∆s δh(x − Xk)∆s
= XM
k=1
σk−1/2τk−1/2δh(x − Xk−1) − XM
k=1
σk−1/2τk−1/2δh(x − Xk)
− σ−1/2τ−1/2δh(x − X0) + σM −1/2τM −1/2δh(x − XM)
= XM
k=1
(δh(x − Xk−1) − δh(x − Xk)) τk−1/2σk−1/2.
Note that, the last two terms are cancelled out due to the periodicity of the interface.
Now we can write down the discrete operator ∇sh as
∇sh· Uk = Uk− Uk−1
∆s · τk−1/2/ |DsX|k−1/2
= − h2
∆s |DsX|k−1/2 X
x
(δh(x − Xk−1) − δh(x − Xk)) τk−1/2· ui,j. Since the discrete surface divergence operator of the velocity is zero as described in Eq. (4.11), we can scale out the coefficient −∆s |Dh2
sX|k−1/2 so that the resultant matrices obtained from Sh and ∇sh· are transpose to each other.
Appendix C
Unconditionally stable IB method
In this appendix, we provide the numerical detail on how we compute the bending force in Eq. (5.36). To proceed, we first define the discrete force spreading operator Fhn, and the velocity interpolating operator Ihn by
Fhn[F](x) = XM
k=1
F(sk)δh(x − Xnk)∆s, Ihn[u](sk) = X
x
u(x)δh(x − Xnk) h2, (C.1)
respectively. It is well-known [50, 44] that the above both operators are adjoint with each other as
hFhn[F], uiΩh = X
x
à M X
k=1
F(sk)δh(x − Xnk)∆s
!
· u(x) h2
= XM
k=1
F(sk) ·
ÃX
x
u(x)δh(x − Xnk) h2
!
∆s = hF, Ihn[u]iΓh. The singular immersed boundary force arising from the bending is written as
fb(x) = −cb Z
Γ
∂4X
∂s4 δ(x − X(s, t)) ds. (C.2)
To simplify our notations, we define the discrete fourth-order centered difference oper-ator Ah as an approximation to the fourth-order derivative. Thus, the discretization for Eqs. (C.2) can be written as
fn+1b (x) = −cbFhn[AhXn+1](x).
By substituting Xn+1= Xn+ ∆tIhn[un+1] into the above equation, we have fn+1b (x) = −cbFhn[AhXn+ ∆tAhIhn[un+1]](x)
= −cbFhn[AhXn](x) − cb∆tFhnAhIhn[un+1](x). (C.3)
Since the discrete operators Fhnand Ihnare both adjoint to each other and the discrete fourth differential operator Ah is symmetric positive definite, we can conclude that the above composite operator FhnAhIhn is also symmetric positive definite. So in our modified BE scheme, we only need to add the additional singular force f∗b(x) as
f∗b(x) = −cbFhn[AhXn](x) − cb∆tFhnAhIhn[u∗](x). (C.4) One should note that, the symmetric positive definite matrix structure for u∗ in Eq. (5.24) does not change at all even we add this extra bending force term. The bending term for modified CN scheme can be similarly derived so we omit the detail here.
Appendix D
Geometrical differential operators in axisymmetric coordinate
In this Appendix, we derive the mean curvature H, Gaussian curvature K, and the surface Laplacian of mean curvature ∆sH in axisymmetric coordinates as shown in Eq. (6.10). The mean curvature H and Gaussian curvature K can be computed by the the first and second fundamental forms [51]. As before, under axisymmetric assumption, the interface can be parameterized as
X(s, θ) = (R(s) cos θ, R(s) sin θ, Z(s)).
Here, we omit the dependence of time t for simplicity. The coefficients E, F , G of the first fundamental form for the above surface, and the surface area dilating factor W can be obtained as
E = Xs· Xs= |Xs|2, F = Xs· Xθ = 0, G = Xθ· Xθ = R2, W =√
EG − F2 = R |Xs| . The unit outward normal vector is
n = Xθ× Xs
W = 1
|Xs|(Zscos θ, Zssin θ, −Rs).
The coefficients of the second fundamental form L, M and N can be obtained as L = Xss· n = 1
|Xs|(RssZs− RsZss), M = Xsθ· n = 0,
N = Xθθ· n = −RZs
|Xs| .
Therefore, the mean curvature H and Gaussian curvature K can be computed
It is interesting to note that there is another way to derive the above curvatures H and K. Using the formula of the surface divergence operator derived in Eq. (6.14) and substituting the normal vector into the definition of 2H = ∇s· n, we obtain the mean curvature H by One can immediately see that those two terms in above equation comprise two prin-cipal curvatures, thus their product leads to the Gaussian curvature as shown in (D.2).
Note that, the surface gradient operator in axisymmetric coordinates is defined as
∇s H = ∂H
where the tangent vector is defined in Eq. (6.12). By applying the surface divergence operator (as defined in Eq. (6.14)) to the above equation, one can obtain
∆sH = ∇s· (∇sH) = 1