Biomech Model Mechanobiol DOI 10.1007/s10237-014-0641-1
ORIGINAL PAPER
The dynamics of inextensible capsules in shear flow under the effect of the natural state Xiting Niu · Tsorng-Whay Pan · Roland Glowinski
Received: 30 July 2014 / Accepted: 9 December 2014 © Springer-Verlag Berlin Heidelberg 2014
Abstract The effect of the natural state on the motion of an inextensible capsule in two-dimensional shear flow has been studied numerically. The energy barrier based on such natural state plays a role for having the transition between two well-known motions, tumbling and tank-treading (TT) with the long axis oscillating about a fixed inclination angle (a swinging mode), when varying the shear rate. Between tumbling and TT with a swinging mode, the intermittent region has been obtained for the capsule with a biconcave rest shape. The estimated critical value of the swelling ratio for having the intermittent region is 0.6, it is almost impossible to capture the intermittent region computationally since the size of the capillary number range for such region is about zero if it exists. Our results are consistent with the results obtained by Tsubota and Wada (2010) since the cells used in their simulations have the swelling ratio of 0.7. To link our cases of one dimensional (1D) membrane in two-dimensional (2D) flow with those of 2D membrane in three-dimensional (3D) flow, we have adapted the concept of
The dynamics of inextensible capsules in shear flow
the energy barrier associated with the membrane mentioned, e.g., in Finken et al. (2011). Their concept about the cell motion is really the matter about whether or not the elastic energy barrier associated with the membrane can be overcome by the fluid flow hydrodynamical forces to the membrane. If it is not enough, the membrane performs tumbling motion. If it is more than enough, the membrane performs TT. The energy barrier can be associated with either nonuniform membrane meshes (e.g., in Fedosov et al. 2010; Sui et al. 2008) or the extra elastic energy terms like ours in this article and the one in Skotheim and Secomb (2007). To validate the point of the energy barrier associated with nonuniform membrane meshes, we have considered the capsule motion with a nonuniform mesh and obtain the tumbling, TT with a swinging mode and a mixture of tumbling and TT. The contents of this article are as follows: We discuss the models and numerical methods briefly in Sect. 2. In Sect. 3, the dynamics of an inextensible capsule under the effect of the natural state in shear flow are studied at different values of the capillary number. Then, the details on the dynamics in the intermittent region are investigated. The conclusions are summarized in Sect. 4.
2 Models and numerical methods A capsule with non-spherical rest shape is suspended in a domain Ω filled with a fluid which is incompressible and Newtonian as in Fig. 1. The inclination angle θ and phase angle φ are defined as in Fig. 1b, c, respectively. For some T > 0, the governing equations for the fluid–capsule system are given as follows ρf
∂u + u · ∇u = −∇ p + μu + f B in Ω × (0, T ), ∂t (1)
∇ · u = 0 in Ω × (0, T ).
(2)
°
φ=45
°
°
°
°
φ=0
°
φ=45
with the following boundary and initial conditions: umax on the top of Ω, u= −umax on the bottom of Ω, u(x, 0) = u0 (x) in Ω, and u is periodic in the x direction,
°
°
°
φ=90 /−90 φ=−45
(3)
(4)
where u and p are the fluid velocity and pressure, respectively, ρf is the fluid density, and μ is the fluid viscosity, which is assumed to be constant for the entire computational domain. In (1), f B accounts for the force acting on the fluid/capsule interface. The boundary condition in (3) is umax = (U, 0)t for simple shear flow. In (4), u0 (x) is the initial fluid velocity. 2.1 One-dimensional elastic spring models An One-dimensional elastic spring model similar to the one used in Tsubota and Wada (2010) is considered in this article to describe the deformable behavior and elasticity of capsules. Based on this model, the capsule membrane can be viewed as membrane particles connecting with the neighboring membrane particles by springs, as shown in Fig. 2. Energy stores in the spring due to the change of the length l of the spring with respect to its reference length l0 and the change in angle θ between two neighboring springs. The total energy per unit thickness of the capsule membrane, E = E l + E b , is the sum of the one for stretch and compression and the one for the bending which, in particular, are N kb 2 Eb = tan 2 i=1
θi − θi0 2
. (5)
(b)
φ=90 /−90 φ=−45
Fig. 2 The elastic spring model of the capsule membrane
i=1
(a) °
l
N k l li − l 0 2 , El = 2 l0
Ω
φ=0
θ
°
φ=0
(c) Fig. 1 Schematic diagram of (a) an inextensible biconcave capsule in shear flow with the computational domain Ω, b the inclination angle θ, and c the phase angle φ
In Eq. (5), N is the total number of the spring elements, and kl and kb are spring constants for changes in length and bendN ing angle, respectively. The set of reference angles {θi0 }i=1 0 corresponds to a preset natural state, where θi = constant for all i corresponds to a uniform natural state, and otherwise a nonuniform natural state.
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ks Γs = 2
s − se se
2 .
(6)
In Eq. 6, s and se are the time-dependent area enclosed by the capsule membrane and the targeted area enclosed by the capsule membrane, respectively. The total energy of the onedimensional spring network per unit thickness is modified as E + Γs . Based on the principle of virtual work, the force per unit thickness acting on the ith membrane particle now is Fi = −
∂(E + Γs ) ∂ri
(7)
where ri is the position of the ith membrane particle. When the area is reduced, each membrane particle moves on the basis of the following equation of motion: d2 ri m 2 dt
+γ
dri = Fi . dt
1 0.9 0.8
α b
α b
E /max(E )
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Scaled bending energy due to the change of phase angle
To obtain a specified initial shape for simulating the fluid– capsule interaction and the shapes served as the reference shape at rest for the natural state, we have used the E l and E b given in Eq. (5) with θi0 = 0 for i = 1, . . . , N . The capsule is assumed to be, initially, a circle of radius R0 = 2.8 µm and then it is discretized into N = 76 membrane particles so that 76 springs of the same length l0 are formed by connecting the neighboring particles. Then, the shape change is obtained by reducing the total area enclosed by the membrane through a penalty function
Scaled bending energy due to the change of phase angle
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Fig. 3 The comparison of the normalized elastic energy term used in Skotheim and Secomb (2007) (solid line) and the discrete energy in (5) versus the phase angle φ for two different initial shapes at rest: swelling ratio s ∗ = 0.481 (left) and 0.9 (right). The elastic spring model of the capsule membrane
sin2 φ = 0.5 − 0.5 cos 2φ used in Skotheim and Secomb (2007) versus the phase angle φ are shown in Fig. 3, in which the graph of the normalized term E b is obtained by rotating the mass nodes on the membrane of a given fixed shape in a clockwise manner. Both of them have preferred phase angles at zero and 180◦ and the energy barrier occurs at the phase angle of 90◦ and 270◦ . We also modify the bending energy per unit thickness to a weighted sum of both uniform and nonuniform natural states to have a weaker effect of the nonuniform natural state: kb Eb = 2
(1 − α)
N i=1
tan
2
θi 2
+α
N i=1
tan
2
θi − θi0 2
.
(8)
(9)
Here, m and γ represent the membrane particle mass and the membrane viscosity of the capsule. The position ri of the ith membrane particle is solved by discretizing Eq. (8) via a second order finite difference method. The total energy stored in the membrane decreases as the time elapses. The final shape of the capsule is obtained as the total energy is minimized and such shape is at a stress-free state. Based on the final shape, as the reference shape (and the initial shape for fluid–capsule interaction), the angle between two neighboring springs at the ith node is θi0 for i = 1, . . . , N . Our bending term in Eq. (5) is similar to the bending energy used by (e.g., Fedosov et al. 2010) which is based on a twodimensional spring network with the natural state as follows
Here, α indicates the weight of the nonuniform natural state. In Eqs. (5), (6) and (9), E l , E b , and Γs represent energies [N m] per unit thickness of 1 m, and thus the units of kl , kb , and ks are Newton [N]. In this article, we only consider the capillary number instead of shear rate, because it is the key number to determine the behavior of capsules in two-dimensional flow. The value of the swelling ratio of a capsule is s ∗ = se /(π R02 ). The values of parameters for modeling capsules are as follows: The spring constant is kl = 5 × 10−8 N, the penalty coefficient is ks = 10−5 N, and the bending constant is kb = 5 × 10−10 N. Using the above chosen parameter values, the area of the initial shape of the capsule has 3.242, which gives a smaller range of the capillary number for the intermittent region (see Fig. 11b). The embedded sub-figure in Fig. 11b also show the capsule behaves similarly at the intermittent region in a narrower channel. An interesting observation is that the periods of tumbling and TT with a swinging mode have a sharp raise when the capsule motion is close to the thresholds as in Fig. 11c. For the capillary number right above the intermittent region, one of the explanations is that while the capsule changes its shape, the membrane particle leaves its natural state position during the TT motion. But the force caused by the bending energy is tending to pull the membrane back to its original natural state, which obviously is against the viscous force of flow which would like to push the membrane particles moving along the membrane. Hence, when the capillary number is just right above the intermittent region, the TT motion is slower since the contrast between these two forces is not significant when comparing with those of higher capillary number. For the capillary number right below the intermittent region, the capsule just tumbles and has a shape of long body. In the non-Stokes flow regime, a neutrally buoyant rigid particle of elliptic shape in shear flow has a transition from the tumbling motion to the state with a fixed inclination angle (i.e., no rotation at all). As the shear rate increases, the circulation before and after the long body becomes stronger and then the long body can be held by the fluid flow with a fixed inclination angle (Ding and Aidun 2000; Chen et al. 2012). In this article, the capsule of a long body shape suspended in shear flow is actually a neutrally buoyant entity. Similarly the capsule slows down its tumbling rotation when the capillary number is less than and closer to the threshold for the transition to the intermittent region. Figure 12 shows the circulation of the velocity field before and after the capsule and the history of the inclination angle of the capsule in a wider channel at Ca = 4.274 s in which the tumbling rotation of the capsule slows down at the inclination angle around 5◦ . Thus the period of tumbling increases as the capillary number increases since the strength of the flow field circulation before and after the capsule is increased. For the effect of the swelling ratio at α = 1, we have tested the values of s ∗ = 0.6, 0.7, 0.8 and 0.9 and obtained that for the capsule of swelling ratio >0.6, it is almost impossible to capture the intermittent region computationally since the size
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inclination angle θ (in degrees)
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Fig. 12 History of the inclination angle (left) and a velocity filed snapshot (right) of the capsule in a wider channel at Ca = 4.274
of the range of the capillary number for such region is about zero if it exists. In Fig. 13, results similar to those in Fig. 11 are shown for the case of the capsule of the swelling ratio s ∗ = 0.9, whose shape is about an elliptical shape. The capsule of the swelling ratio s ∗ can be characterized by its excess 1 circumference c = 2π( √ − 1). For the biconcave shape s∗ of s ∗ = 0.481, its c is 2.77638; but the one for an elliptical shape of s ∗ = 0.9 is 0.33987. The excess circumference c is similar to the excess area used by Vlahovska et al. (2011). For the small values of c, we do not expect to obtain the intermittent region. Our result is consistent with the results obtained by Tsubota and Wada (2010) since the cells used in their simulations have the swelling ratio of 0.7. 3.2.1 The effect of the nonuniform natural state As mentioned in the previous subsection, with lower values of α, a capsule does not tend to change shape as much as capsule with α = 1. Even in the intermittent region, the capsule prefers keeping the biconcave shape (see Fig. 14 for an example), which is unlike the one shown in Fig. 9 for α = 1. Figure 15 shows that the critical value of the capillary number for the transition from tumbling to the intermittent region is proportional to the value of α, this can be explained by defining a weighted bending capillary number with respect to α : Caα = μR03 γ˙ /(αk B l0 ) which is similar to the bending capillary number defined by Tsubota and Wada (2010). This number represents the relative effect of fluid viscous force versus surface tension which comes from the memory of the reference angles {θi0 }, and therefore a scale of α should be added. The capsule has tumbling motion when the viscous
The dynamics of inextensible capsules in shear flow 22
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c
Fig. 13 Histories of period of tumbling and TT of the capsule of s ∗ = 0.9 (left) and the enlargement of the intermittent region (right) in (a) a wider channel and (b) a narrower channel
Fig. 16 The mixed dynamical behavior of the capsule with effect of shape memory α varies from 0.05 to 1, in a narrower channel
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Fig. 14 Snapshots of a TT motion of a capsule with α = 0.05 at the intermittent region in shear flow at Ca = 0.163 −3
4
x 10
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Fig. 17 An example of the initial mesh on the capsule membrane: (i) uniform lengths (top) and (ii) nonuniform lengths (bottom)
3 2.5 2 1.5 1 0.5 0
0
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0.4
α
0.6
0.8
1
Fig. 15 The intermittent region versus the shape memory coefficient α: the plot of the critical value of the capillary number for the transition from tumbling to the intermittent region (left) and the size of the capillary number range for the intermittent region (right)
force from outer fluid is less than the force pulling it back to initial position, and TT with a swinging mode when the viscous force is larger. Intermittent behavior occurs when Caα ∼ k, where k, at least, depends on swelling ratio and degree of confinement. This implies that γ˙ ∼ αk B l0 /(μR03 ), i.e. the capillary number of the intermittency is proportional to the value of α (because the capillary number is proportional to the shear rate). The size of the capillary number range for the intermittent region is increasing as α increases. When α is small, the rate of increasing is almost linear. But as α raising up, this rate slows down as the capsule becomes more deformable and can change shape accordingly to achieve
periodic tumbling or TT with a swinging mode, which is more preferred because the energy change is more stable than in intermittent states. The mixed dynamical behavior of the capsule in the intermittent region with respect to Caα is plotted in Fig. 16, where the y-axis interprets the number of TT with a swinging mode over the number of tumbling in each cycle, and x-axis is the difference between 1/Caα and its value at the boundary of pure TT regime and intermittency. In the “more TT than tumbling per cycle” regime, the tendency is almost linear with the slope of −0.45 until slope becomes a little sharper when close to the “one TT and one tumbling alternatively” regime, which is consistent with the results obtained by Skotheim and Secomb (2007). One major observation through Fig. 16 is that the “one tumbling and one TT alternatively” regime is not eliminated when reducing the effect of nonuniform natural state, even with α values as small as 0.05, where the capsule is only little deformable, this suggests this is a relative stable state in the intermittent region.
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Fig. 18 Histories of the inclination angle (left) and the phase angle (right) at the capillary number Ca = 0.4183 (top), 0.4456 (middle), and 0.4638 (bottom) associated with the tumbling (top), a mixture of one tumbling and one TT (middle) and TT with swinging mode (bottom), respectively
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Fig. 19 Histories of the inclination angle (left) and the phase angle (right) at the capillary number Ca = 0.4547 for the case of springs with same initial length
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3.3 The effect of nonuniform mesh In this section, we have used the E l and E b given in Eq. (10) with the springs of nonuniform lengths as shown in Fig. 17. In the nonuniform mesh on our one-dimensional membrane used in simulation, the spring length is increasing from the rim to the dimple. We have considered a capsule of the swelling ratio s ∗ = 0.481 suspended in shear flow at the capillary numbers Ca = 0.4183, 0.4456 and 0.4638 in a channel of the length 20 µm and width 20 µm. All other parameters are the same as in the previous sections. These capillary numbers give rise to, as in Fig. 18, (i) a tumbling motion, (ii) a mixture of one tumbling and one TT, and (iii) a TT motion with a swinging mode, respectively. But for the case of springs with uniform initial length, it always tank-treads as in Fig. 19 (same results also obtained in Tsubota and Wada 2010). From the histories of the energy per unit thickness presented in Fig. 20, the energy behaves differently for two cases. For the nonuniform case, the hydrodynamic forces provided by the fluid flow is not enough to overcome the energy barrier required for the tank-trading at Ca = 0.4183. Hence
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Fig. 20 Histories of the energy per unit thickness: nonuniform initial lengths (top) and uniform initial length (bottom)
the capsule tumbles. At Ca = 0.4456, the strength of the forces is not there yet at the first time, but it goes through at the second time. At the higher value Ca = 0.4638, the fluid flow provides enough hydrodynamic forces to push the membrane through the energy barrier and the capsule keeps TT. For the uniform case, the energy barrier is almost none since each spring can be identified as others. Hence, the energy is almost a constant. Based on the results in this section, we believe that the energy barrier associated with the skeleton structure of the RBC and the surface mesh of 2D membrane in 3D flow simulation does matter on the RBC dynamics.
4 Conclusions In this article, we have analyzed the dynamics of an inextensible capsule suspended in two-dimensional shear flow
The dynamics of inextensible capsules in shear flow
under the effect of the nonuniform natural state. Besides the viscosity ratio of the internal fluid and external fluid of the capsule, the natural state effect also plays a role for having the transition between two well- known motions, tumbling and TT with the long axis oscillating about a fixed inclination angle, when varying the shear rate. The intermittent region between tumbling and TT with a swinging mode of the capsule with a biconcave rest shape has been obtained in a narrow range of the capillary number. In such Transition region, the intermittent dynamics of the capsule is a mixture of tumbling and TT with a swinging mode and when having the tumbling motion, the membrane tank-tread backward and forward within a small range. As the capillary number is very close to and below the threshold for the pure TT with a swinging mode, the capsule tumbles once after several TT periods in each cycle. The number of TT periods in one cycle decreases with respect to the decreasing of the capillary number, until the capsule has one tumble and one TT period alternatively and such alternating motion exists over a range of the capillary number; and then the capsule performs more tumbling between two consecutive TT periods when reducing the capillary number further, and finally shows pure tumbling. Surprisingly, the “one tumbling and one TT alternatively” mode is very persistent. The critical value of the swelling ratio for having the intermittent behavior has been estimated. For those >0.6 (c = 1.82837), it is almost impossible to capture the intermittent behavior computationally since the size of the range of the capillary number for such behavior is about zero if it exists. For the small values of c (associated with large swelling ratio s ∗ and the rest shape closer to a full disk), we do not expect to obtain the intermittent behavior. An interesting observation is that the period has a sharp raise when the capsule motion is close to the intermittent region. For the capillary number right above the intermittent region, the force caused by the bending energy is tending to pull the membrane back to its original natural state, which obviously is against the viscous force of flow which would like to push the membrane particles moving along the membrane. Hence, when the capillary number is right above the intermittent region, the TT motion is slower since the contrast between these two forces is not significant. For the capillary number right below the intermittent region, the capsule is a neutrally buoyant entity and slows down its tumbling rotation when the capillary number is less than and closer to the threshold for the intermittent region, which is closely related to the case of a neutrally buoyant rigid particle in shear flow. Another observation is that in the intermittent region, the change of contrast between numbers of tumbling and TT is not linear, especially when the weighted bending capillary number is close to or in the range of a relatively stable “one tumbling and one TT per cycle” mode, we discussed this phenomenon may result from the deformability of capsule,
even with the value of α as small as 0.05, the capsule may still have some invisible shape changes and is able to perform one tumbling one TT in each cycle within a range of weighted bending capillary number. To link our cases of one dimensional (1D) membrane in two-dimensional (2D) flow with those of 2D membrane in three-dimensional (3D) flow, we have adapted the concept of the energy barrier associated with the membrane. The energy barrier can be associated with either nonuniform membrane meshes or the extra elastic energy terms. We believe that the energy barrier associated with the skeleton structure of the RBC (in 3D simulations, it is the surface mesh of 2D membrane) does matter on the dynamics of the capsule motion. It is worth for further study in this direction, especially the mesh dependency of the capsule motion in 3D simulation. In Levant and Steinberg (2012), they have found thermal noise can drastically modify the vesicle dynamics in general linear flow and obtained a time-dependent state, called trembling. We have not addressed the effect of the thermal noise in our current model. However, it is a topic for future work since the vesicle shape perturbations due to the effect of thermal noise could change the dynamics of the capsule in its intermittent state. Other related future directions are the effect of the viscosity contrast of the fluid inside and outside of the membrane and that of the ratio between membrane and fluid viscosity on the capsule motion. Acknowledgments This work is supported by an NSF grant DMS0914788. We acknowledge the helpful comments of James Feng, MingChih Lai, Sheldon X. Wang and reviewers.
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Pozrikidis C (2003) Numerical simulation of the flow-induced deformation of red blood cells. Ann Biomed Eng 31:1194. doi:10.1114/ 1.1617985 Seifert U (1999) Fluid membranes in hydrodynamic flow fields: formalism and an application to fluctuating quasispherical vesicles in shear flow. Eur Phys J B 8:405–415. doi:10.1007/s100510050706 Shi L, Pan TW, Glowinski R (2012a) Deformation of a single blood cell in bounded Poiseuille flows. Phys Rev E 85:016307. doi:10.1103/ PhysRevE.85.016307 Shi L, Pan TW, Glowinski R (2012b) Lateral migration and equilibrium shape and position of a single red blood cell in bounded Poiseuille flows. Phys Rev E 86:056308. doi:10.1103/PhysRevE.86.056308 Shi L, Pan TW, Glowinski R (2012c) Numerical simulation of lateral migration of red blood cells in Poiseuille flows. Int J Numer Methods Fluids 68:1393. doi:10.1002/fld.2455 Skotheim JM, Secomb TW (2007) Red blood cells and other nonspherical capsules in shear flow: oscillatory dynamics and the tanktreading-to-tumbling transition. Phys Rev Lett 98:078301. doi:10. 1103/PhysRevLett.98.078301 Sui Y, Chew YT, Roy P, Cheng YP, Low HT (2008) Dynamic motion of red blood cells in simple shear flow. Phys Fluids 20:112106. doi:10.1063/1.3026569 Tran-Son-Tay R, Sutera SP, Rao PR (1984) Determination of red blood cell membrane viscosity from rheoscopic observations of tank-treading motion. Biophys J 46:65. doi:10.1016/ S0006-3495(84)83999-5 Tsubota K, Wada S (2010) Effect of the natural state of an elastic cellular membrane on tank-treading and tumbling motions of a single red blood cell. Phys Rev E 81:011910. doi:10.1103/PhysRevE.81. 011910 Tsubota K, Wada S, Liu H (2013) Elastic behavior of a red blood cell with the membrane’s nonuniform natural state: equilibrium shape, motion transition under shear flow, and elongation during tank-treading motion. Biomech Model Mechanobiol 13:735–746. doi:10.1007/s10237-013-0530-z Vlahovska PM, Gracia RS (2007) Dynamics of a viscous vesicle in linear flows. Phys Rev E 75:016313. doi:10.1103/PhysRevE.75. 016313 Vlahovska PM, Young YN, Danker G, Misbah C (2011) Dynamics of a non-spherical with incompressible interfacemicrocapsule in shear flow. J Fluid Mech 678:221. doi:10.1017/jfm.2011.108 Zhao H, Shaqfeh ESG (2011) The dynamics of a vesicle in simple shear flow. J Fluid Mech 674:578. doi:10.1017/S0022112011000115
ORIGINAL PAPER
The dynamics of inextensible capsules in shear flow under the effect of the natural state Xiting Niu · Tsorng-Whay Pan · Roland Glowinski
Received: 30 July 2014 / Accepted: 9 December 2014 © Springer-Verlag Berlin Heidelberg 2014
Abstract The effect of the natural state on the motion of an inextensible capsule in two-dimensional shear flow has been studied numerically. The energy barrier based on such natural state plays a role for having the transition between two well-known motions, tumbling and tank-treading (TT) with the long axis oscillating about a fixed inclination angle (a swinging mode), when varying the shear rate. Between tumbling and TT with a swinging mode, the intermittent region has been obtained for the capsule with a biconcave rest shape. The estimated critical value of the swelling ratio for having the intermittent region is 0.6, it is almost impossible to capture the intermittent region computationally since the size of the capillary number range for such region is about zero if it exists. Our results are consistent with the results obtained by Tsubota and Wada (2010) since the cells used in their simulations have the swelling ratio of 0.7. To link our cases of one dimensional (1D) membrane in two-dimensional (2D) flow with those of 2D membrane in three-dimensional (3D) flow, we have adapted the concept of
The dynamics of inextensible capsules in shear flow
the energy barrier associated with the membrane mentioned, e.g., in Finken et al. (2011). Their concept about the cell motion is really the matter about whether or not the elastic energy barrier associated with the membrane can be overcome by the fluid flow hydrodynamical forces to the membrane. If it is not enough, the membrane performs tumbling motion. If it is more than enough, the membrane performs TT. The energy barrier can be associated with either nonuniform membrane meshes (e.g., in Fedosov et al. 2010; Sui et al. 2008) or the extra elastic energy terms like ours in this article and the one in Skotheim and Secomb (2007). To validate the point of the energy barrier associated with nonuniform membrane meshes, we have considered the capsule motion with a nonuniform mesh and obtain the tumbling, TT with a swinging mode and a mixture of tumbling and TT. The contents of this article are as follows: We discuss the models and numerical methods briefly in Sect. 2. In Sect. 3, the dynamics of an inextensible capsule under the effect of the natural state in shear flow are studied at different values of the capillary number. Then, the details on the dynamics in the intermittent region are investigated. The conclusions are summarized in Sect. 4.
2 Models and numerical methods A capsule with non-spherical rest shape is suspended in a domain Ω filled with a fluid which is incompressible and Newtonian as in Fig. 1. The inclination angle θ and phase angle φ are defined as in Fig. 1b, c, respectively. For some T > 0, the governing equations for the fluid–capsule system are given as follows ρf
∂u + u · ∇u = −∇ p + μu + f B in Ω × (0, T ), ∂t (1)
∇ · u = 0 in Ω × (0, T ).
(2)
°
φ=45
°
°
°
°
φ=0
°
φ=45
with the following boundary and initial conditions: umax on the top of Ω, u= −umax on the bottom of Ω, u(x, 0) = u0 (x) in Ω, and u is periodic in the x direction,
°
°
°
φ=90 /−90 φ=−45
(3)
(4)
where u and p are the fluid velocity and pressure, respectively, ρf is the fluid density, and μ is the fluid viscosity, which is assumed to be constant for the entire computational domain. In (1), f B accounts for the force acting on the fluid/capsule interface. The boundary condition in (3) is umax = (U, 0)t for simple shear flow. In (4), u0 (x) is the initial fluid velocity. 2.1 One-dimensional elastic spring models An One-dimensional elastic spring model similar to the one used in Tsubota and Wada (2010) is considered in this article to describe the deformable behavior and elasticity of capsules. Based on this model, the capsule membrane can be viewed as membrane particles connecting with the neighboring membrane particles by springs, as shown in Fig. 2. Energy stores in the spring due to the change of the length l of the spring with respect to its reference length l0 and the change in angle θ between two neighboring springs. The total energy per unit thickness of the capsule membrane, E = E l + E b , is the sum of the one for stretch and compression and the one for the bending which, in particular, are N kb 2 Eb = tan 2 i=1
θi − θi0 2
. (5)
(b)
φ=90 /−90 φ=−45
Fig. 2 The elastic spring model of the capsule membrane
i=1
(a) °
l
N k l li − l 0 2 , El = 2 l0
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θ
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(c) Fig. 1 Schematic diagram of (a) an inextensible biconcave capsule in shear flow with the computational domain Ω, b the inclination angle θ, and c the phase angle φ
In Eq. (5), N is the total number of the spring elements, and kl and kb are spring constants for changes in length and bendN ing angle, respectively. The set of reference angles {θi0 }i=1 0 corresponds to a preset natural state, where θi = constant for all i corresponds to a uniform natural state, and otherwise a nonuniform natural state.
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ks Γs = 2
s − se se
2 .
(6)
In Eq. 6, s and se are the time-dependent area enclosed by the capsule membrane and the targeted area enclosed by the capsule membrane, respectively. The total energy of the onedimensional spring network per unit thickness is modified as E + Γs . Based on the principle of virtual work, the force per unit thickness acting on the ith membrane particle now is Fi = −
∂(E + Γs ) ∂ri
(7)
where ri is the position of the ith membrane particle. When the area is reduced, each membrane particle moves on the basis of the following equation of motion: d2 ri m 2 dt
+γ
dri = Fi . dt
1 0.9 0.8
α b
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E /max(E )
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Scaled bending energy due to the change of phase angle
To obtain a specified initial shape for simulating the fluid– capsule interaction and the shapes served as the reference shape at rest for the natural state, we have used the E l and E b given in Eq. (5) with θi0 = 0 for i = 1, . . . , N . The capsule is assumed to be, initially, a circle of radius R0 = 2.8 µm and then it is discretized into N = 76 membrane particles so that 76 springs of the same length l0 are formed by connecting the neighboring particles. Then, the shape change is obtained by reducing the total area enclosed by the membrane through a penalty function
Scaled bending energy due to the change of phase angle
X. Niu et al.
7
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Phase angle φ
Fig. 3 The comparison of the normalized elastic energy term used in Skotheim and Secomb (2007) (solid line) and the discrete energy in (5) versus the phase angle φ for two different initial shapes at rest: swelling ratio s ∗ = 0.481 (left) and 0.9 (right). The elastic spring model of the capsule membrane
sin2 φ = 0.5 − 0.5 cos 2φ used in Skotheim and Secomb (2007) versus the phase angle φ are shown in Fig. 3, in which the graph of the normalized term E b is obtained by rotating the mass nodes on the membrane of a given fixed shape in a clockwise manner. Both of them have preferred phase angles at zero and 180◦ and the energy barrier occurs at the phase angle of 90◦ and 270◦ . We also modify the bending energy per unit thickness to a weighted sum of both uniform and nonuniform natural states to have a weaker effect of the nonuniform natural state: kb Eb = 2
(1 − α)
N i=1
tan
2
θi 2
+α
N i=1
tan
2
θi − θi0 2
.
(8)
(9)
Here, m and γ represent the membrane particle mass and the membrane viscosity of the capsule. The position ri of the ith membrane particle is solved by discretizing Eq. (8) via a second order finite difference method. The total energy stored in the membrane decreases as the time elapses. The final shape of the capsule is obtained as the total energy is minimized and such shape is at a stress-free state. Based on the final shape, as the reference shape (and the initial shape for fluid–capsule interaction), the angle between two neighboring springs at the ith node is θi0 for i = 1, . . . , N . Our bending term in Eq. (5) is similar to the bending energy used by (e.g., Fedosov et al. 2010) which is based on a twodimensional spring network with the natural state as follows
Here, α indicates the weight of the nonuniform natural state. In Eqs. (5), (6) and (9), E l , E b , and Γs represent energies [N m] per unit thickness of 1 m, and thus the units of kl , kb , and ks are Newton [N]. In this article, we only consider the capillary number instead of shear rate, because it is the key number to determine the behavior of capsules in two-dimensional flow. The value of the swelling ratio of a capsule is s ∗ = se /(π R02 ). The values of parameters for modeling capsules are as follows: The spring constant is kl = 5 × 10−8 N, the penalty coefficient is ks = 10−5 N, and the bending constant is kb = 5 × 10−10 N. Using the above chosen parameter values, the area of the initial shape of the capsule has 3.242, which gives a smaller range of the capillary number for the intermittent region (see Fig. 11b). The embedded sub-figure in Fig. 11b also show the capsule behaves similarly at the intermittent region in a narrower channel. An interesting observation is that the periods of tumbling and TT with a swinging mode have a sharp raise when the capsule motion is close to the thresholds as in Fig. 11c. For the capillary number right above the intermittent region, one of the explanations is that while the capsule changes its shape, the membrane particle leaves its natural state position during the TT motion. But the force caused by the bending energy is tending to pull the membrane back to its original natural state, which obviously is against the viscous force of flow which would like to push the membrane particles moving along the membrane. Hence, when the capillary number is just right above the intermittent region, the TT motion is slower since the contrast between these two forces is not significant when comparing with those of higher capillary number. For the capillary number right below the intermittent region, the capsule just tumbles and has a shape of long body. In the non-Stokes flow regime, a neutrally buoyant rigid particle of elliptic shape in shear flow has a transition from the tumbling motion to the state with a fixed inclination angle (i.e., no rotation at all). As the shear rate increases, the circulation before and after the long body becomes stronger and then the long body can be held by the fluid flow with a fixed inclination angle (Ding and Aidun 2000; Chen et al. 2012). In this article, the capsule of a long body shape suspended in shear flow is actually a neutrally buoyant entity. Similarly the capsule slows down its tumbling rotation when the capillary number is less than and closer to the threshold for the transition to the intermittent region. Figure 12 shows the circulation of the velocity field before and after the capsule and the history of the inclination angle of the capsule in a wider channel at Ca = 4.274 s in which the tumbling rotation of the capsule slows down at the inclination angle around 5◦ . Thus the period of tumbling increases as the capillary number increases since the strength of the flow field circulation before and after the capsule is increased. For the effect of the swelling ratio at α = 1, we have tested the values of s ∗ = 0.6, 0.7, 0.8 and 0.9 and obtained that for the capsule of swelling ratio >0.6, it is almost impossible to capture the intermittent region computationally since the size
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Fig. 12 History of the inclination angle (left) and a velocity filed snapshot (right) of the capsule in a wider channel at Ca = 4.274
of the range of the capillary number for such region is about zero if it exists. In Fig. 13, results similar to those in Fig. 11 are shown for the case of the capsule of the swelling ratio s ∗ = 0.9, whose shape is about an elliptical shape. The capsule of the swelling ratio s ∗ can be characterized by its excess 1 circumference c = 2π( √ − 1). For the biconcave shape s∗ of s ∗ = 0.481, its c is 2.77638; but the one for an elliptical shape of s ∗ = 0.9 is 0.33987. The excess circumference c is similar to the excess area used by Vlahovska et al. (2011). For the small values of c, we do not expect to obtain the intermittent region. Our result is consistent with the results obtained by Tsubota and Wada (2010) since the cells used in their simulations have the swelling ratio of 0.7. 3.2.1 The effect of the nonuniform natural state As mentioned in the previous subsection, with lower values of α, a capsule does not tend to change shape as much as capsule with α = 1. Even in the intermittent region, the capsule prefers keeping the biconcave shape (see Fig. 14 for an example), which is unlike the one shown in Fig. 9 for α = 1. Figure 15 shows that the critical value of the capillary number for the transition from tumbling to the intermittent region is proportional to the value of α, this can be explained by defining a weighted bending capillary number with respect to α : Caα = μR03 γ˙ /(αk B l0 ) which is similar to the bending capillary number defined by Tsubota and Wada (2010). This number represents the relative effect of fluid viscous force versus surface tension which comes from the memory of the reference angles {θi0 }, and therefore a scale of α should be added. The capsule has tumbling motion when the viscous
The dynamics of inextensible capsules in shear flow 22
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c
Fig. 13 Histories of period of tumbling and TT of the capsule of s ∗ = 0.9 (left) and the enlargement of the intermittent region (right) in (a) a wider channel and (b) a narrower channel
Fig. 16 The mixed dynamical behavior of the capsule with effect of shape memory α varies from 0.05 to 1, in a narrower channel
Y(μm)
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0
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0.6
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range of capillary number in intermittent state
Fig. 14 Snapshots of a TT motion of a capsule with α = 0.05 at the intermittent region in shear flow at Ca = 0.163 −3
4
x 10
3.5
Fig. 17 An example of the initial mesh on the capsule membrane: (i) uniform lengths (top) and (ii) nonuniform lengths (bottom)
3 2.5 2 1.5 1 0.5 0
0
0.2
0.4
α
0.6
0.8
1
Fig. 15 The intermittent region versus the shape memory coefficient α: the plot of the critical value of the capillary number for the transition from tumbling to the intermittent region (left) and the size of the capillary number range for the intermittent region (right)
force from outer fluid is less than the force pulling it back to initial position, and TT with a swinging mode when the viscous force is larger. Intermittent behavior occurs when Caα ∼ k, where k, at least, depends on swelling ratio and degree of confinement. This implies that γ˙ ∼ αk B l0 /(μR03 ), i.e. the capillary number of the intermittency is proportional to the value of α (because the capillary number is proportional to the shear rate). The size of the capillary number range for the intermittent region is increasing as α increases. When α is small, the rate of increasing is almost linear. But as α raising up, this rate slows down as the capsule becomes more deformable and can change shape accordingly to achieve
periodic tumbling or TT with a swinging mode, which is more preferred because the energy change is more stable than in intermittent states. The mixed dynamical behavior of the capsule in the intermittent region with respect to Caα is plotted in Fig. 16, where the y-axis interprets the number of TT with a swinging mode over the number of tumbling in each cycle, and x-axis is the difference between 1/Caα and its value at the boundary of pure TT regime and intermittency. In the “more TT than tumbling per cycle” regime, the tendency is almost linear with the slope of −0.45 until slope becomes a little sharper when close to the “one TT and one tumbling alternatively” regime, which is consistent with the results obtained by Skotheim and Secomb (2007). One major observation through Fig. 16 is that the “one tumbling and one TT alternatively” regime is not eliminated when reducing the effect of nonuniform natural state, even with α values as small as 0.05, where the capsule is only little deformable, this suggests this is a relative stable state in the intermittent region.
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Fig. 18 Histories of the inclination angle (left) and the phase angle (right) at the capillary number Ca = 0.4183 (top), 0.4456 (middle), and 0.4638 (bottom) associated with the tumbling (top), a mixture of one tumbling and one TT (middle) and TT with swinging mode (bottom), respectively
10
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Fig. 19 Histories of the inclination angle (left) and the phase angle (right) at the capillary number Ca = 0.4547 for the case of springs with same initial length
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time (ms)
3.3 The effect of nonuniform mesh In this section, we have used the E l and E b given in Eq. (10) with the springs of nonuniform lengths as shown in Fig. 17. In the nonuniform mesh on our one-dimensional membrane used in simulation, the spring length is increasing from the rim to the dimple. We have considered a capsule of the swelling ratio s ∗ = 0.481 suspended in shear flow at the capillary numbers Ca = 0.4183, 0.4456 and 0.4638 in a channel of the length 20 µm and width 20 µm. All other parameters are the same as in the previous sections. These capillary numbers give rise to, as in Fig. 18, (i) a tumbling motion, (ii) a mixture of one tumbling and one TT, and (iii) a TT motion with a swinging mode, respectively. But for the case of springs with uniform initial length, it always tank-treads as in Fig. 19 (same results also obtained in Tsubota and Wada 2010). From the histories of the energy per unit thickness presented in Fig. 20, the energy behaves differently for two cases. For the nonuniform case, the hydrodynamic forces provided by the fluid flow is not enough to overcome the energy barrier required for the tank-trading at Ca = 0.4183. Hence
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Fig. 20 Histories of the energy per unit thickness: nonuniform initial lengths (top) and uniform initial length (bottom)
the capsule tumbles. At Ca = 0.4456, the strength of the forces is not there yet at the first time, but it goes through at the second time. At the higher value Ca = 0.4638, the fluid flow provides enough hydrodynamic forces to push the membrane through the energy barrier and the capsule keeps TT. For the uniform case, the energy barrier is almost none since each spring can be identified as others. Hence, the energy is almost a constant. Based on the results in this section, we believe that the energy barrier associated with the skeleton structure of the RBC and the surface mesh of 2D membrane in 3D flow simulation does matter on the RBC dynamics.
4 Conclusions In this article, we have analyzed the dynamics of an inextensible capsule suspended in two-dimensional shear flow
The dynamics of inextensible capsules in shear flow
under the effect of the nonuniform natural state. Besides the viscosity ratio of the internal fluid and external fluid of the capsule, the natural state effect also plays a role for having the transition between two well- known motions, tumbling and TT with the long axis oscillating about a fixed inclination angle, when varying the shear rate. The intermittent region between tumbling and TT with a swinging mode of the capsule with a biconcave rest shape has been obtained in a narrow range of the capillary number. In such Transition region, the intermittent dynamics of the capsule is a mixture of tumbling and TT with a swinging mode and when having the tumbling motion, the membrane tank-tread backward and forward within a small range. As the capillary number is very close to and below the threshold for the pure TT with a swinging mode, the capsule tumbles once after several TT periods in each cycle. The number of TT periods in one cycle decreases with respect to the decreasing of the capillary number, until the capsule has one tumble and one TT period alternatively and such alternating motion exists over a range of the capillary number; and then the capsule performs more tumbling between two consecutive TT periods when reducing the capillary number further, and finally shows pure tumbling. Surprisingly, the “one tumbling and one TT alternatively” mode is very persistent. The critical value of the swelling ratio for having the intermittent behavior has been estimated. For those >0.6 (c = 1.82837), it is almost impossible to capture the intermittent behavior computationally since the size of the range of the capillary number for such behavior is about zero if it exists. For the small values of c (associated with large swelling ratio s ∗ and the rest shape closer to a full disk), we do not expect to obtain the intermittent behavior. An interesting observation is that the period has a sharp raise when the capsule motion is close to the intermittent region. For the capillary number right above the intermittent region, the force caused by the bending energy is tending to pull the membrane back to its original natural state, which obviously is against the viscous force of flow which would like to push the membrane particles moving along the membrane. Hence, when the capillary number is right above the intermittent region, the TT motion is slower since the contrast between these two forces is not significant. For the capillary number right below the intermittent region, the capsule is a neutrally buoyant entity and slows down its tumbling rotation when the capillary number is less than and closer to the threshold for the intermittent region, which is closely related to the case of a neutrally buoyant rigid particle in shear flow. Another observation is that in the intermittent region, the change of contrast between numbers of tumbling and TT is not linear, especially when the weighted bending capillary number is close to or in the range of a relatively stable “one tumbling and one TT per cycle” mode, we discussed this phenomenon may result from the deformability of capsule,
even with the value of α as small as 0.05, the capsule may still have some invisible shape changes and is able to perform one tumbling one TT in each cycle within a range of weighted bending capillary number. To link our cases of one dimensional (1D) membrane in two-dimensional (2D) flow with those of 2D membrane in three-dimensional (3D) flow, we have adapted the concept of the energy barrier associated with the membrane. The energy barrier can be associated with either nonuniform membrane meshes or the extra elastic energy terms. We believe that the energy barrier associated with the skeleton structure of the RBC (in 3D simulations, it is the surface mesh of 2D membrane) does matter on the dynamics of the capsule motion. It is worth for further study in this direction, especially the mesh dependency of the capsule motion in 3D simulation. In Levant and Steinberg (2012), they have found thermal noise can drastically modify the vesicle dynamics in general linear flow and obtained a time-dependent state, called trembling. We have not addressed the effect of the thermal noise in our current model. However, it is a topic for future work since the vesicle shape perturbations due to the effect of thermal noise could change the dynamics of the capsule in its intermittent state. Other related future directions are the effect of the viscosity contrast of the fluid inside and outside of the membrane and that of the ratio between membrane and fluid viscosity on the capsule motion. Acknowledgments This work is supported by an NSF grant DMS0914788. We acknowledge the helpful comments of James Feng, MingChih Lai, Sheldon X. Wang and reviewers.
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