Multiplicity of stable orbits for deformable prolate capsules in shear flow

This work investigates the orbital dynamics of a fluid-filled deformable prolate capsule in unbounded simple shear flow at zero Reynolds number using direct simulations. The motion of the capsule is simulated using a model that incorporates shear elasticity, area dilatation, and bending resistance. Here the deformability of the capsule is characterized by the nondimensional capillary number $\mathrm{Ca}$, which represents the ratio of viscous stresses to elastic restoring stresses on the capsule. For a capsule with small bending stiffness, at a given $\mathrm{Ca}$, the orientation converges over time toward a unique stable orbit independent of the initial orientation. With increasing $\mathrm{Ca}$, four dynamical modes are found for the stable orbit, namely rolling, wobbling, oscillating-swinging, and swinging. On the other hand, for a capsule with large bending stiffness, multiplicity in the orbit dynamics is observed. When the viscosity ratio $\lambda \lesssim 1$, the long axis of the capsule always tends toward a stable orbit in the flow-gradient plane, either tumbling or swinging, depending on $\mathrm{Ca}$. When $\lambda \gtrsim 1$, the stable orbit of the capsule is a tumbling motion at low $\mathrm{Ca}$, irrespective of the initial orientation. Upon increasing $\mathrm{Ca}$, there is a symmetry-breaking bifurcation away from the tumbling orbit, and the capsule is observed to adopt multiple stable orbital modes including nonsymmetric precessing and rolling, depending on the initial orientation. As $\mathrm{Ca}$ increases further, the nonsymmetric stable orbit loses existence at a saddle-node bifurcation, and rolling becomes the only attractor at high $\mathrm{Ca}$, whereas the rolling state coexists with the nonsymmetric state at intermediate values of $\mathrm{Ca}$. A symmetry-breaking bifurcation away from the rolling orbit is also found upon decreasing $\mathrm{Ca}$. The regime with multiple attractors becomes broader as the aspect ratio of the capsule increases, while it narrows as the viscosity ratio increases. We also report the particle contribution to the stress, which also displays multiplicity.

Figure 1

Schematic of the 3D orientation of a prolate capsule in unbounded simple shear flow.

Figure 2

Evolution of $\overline{\theta }$ of a prolate capsule (AR = 2.0) with ${\widehat{\kappa }}_B=0.02$ and $\lambda =1$ at (a) $\mathrm{Ca}=0.08$, (b) $\mathrm{Ca}=0.2$, (c) $\mathrm{Ca}=0.7$, and (d) $\mathrm{Ca}=1.5$. Here the spontaneous shape of the capsule is assumed to be the same as its rest shape.

Figure 3

Bifurcation behavior for a prolate capsule (AR = 2.0) with ${\widehat{\kappa }}_B=0.02$ and varying $\lambda$. The black dashed lines at $\theta =0^{\circ }$ and ${90}^{\circ }$ represent the orbits for a capsule aligned with the $z$ axis and in the shear plane, respectively.

Figure 4

Time sequence images (side and front views) of a prolate capsule (AR = 2.0) with ${\widehat{\kappa }}_B=0.02$ and $\lambda =1$ taking a rolling [(a) $\mathrm{Ca}=0.08$], wobbling [(b) $\mathrm{Ca}=0.2$], oscillating-swinging [(c) $\mathrm{Ca}=0.7$], and swinging [(d) $\mathrm{Ca}=1.5$] motion, respectively. The initial orientation of the capsule is $\alpha =5^{\circ }$. The markers indicate the positions of a membrane point initially on the equator of the capsule for (a), and a membrane point initially on the major axis of the capsule for (b), (c), and (d).

Figure 5

Evolution of $\overline{\theta }$ of a prolate capsule (AR = 2.0) with ${\widehat{\kappa }}_B=0.02$ and $\lambda =1$ at (a) $\mathrm{Ca}=0.08$, (b) $\mathrm{Ca}=0.2$, (c) $\mathrm{Ca}=0.8$, and (d) $\mathrm{Ca}=1.5$. Here the spontaneous curvature is assumed to be zero $\left(c_0=0\right)$ everywhere on the membrane surface.

Figure 6

Intrinsic viscosity (a) and dimensionless normal stress differences (b,c) predicted for a dilute suspension (with volume fraction $\mathrm{\Phi })$ of identical prolate capsules (AR = 2.0) with ${\widehat{\kappa }}_B=0.02$ taking the same corresponding stable orbit at varying $\mathrm{Ca}$.

Figure 7

Evolution of $\overline{\theta }$ of a prolate capsule (AR = 2.0) with ${\widehat{\kappa }}_B=0.2$ and $\lambda =1$ at varying $\mathrm{Ca}$. At low $\mathrm{Ca}$ the capsule takes a stable tumbling motion at long times [(a) $\mathrm{Ca}=0.08$ and (b) $\mathrm{Ca}=0.2$]. A tumbling-to-swinging transition is observed at moderate $\mathrm{Ca}$ [(c) $\mathrm{Ca}=0.8$] before swinging becomes the attractor at high $\mathrm{Ca}$ [(d) $\mathrm{Ca}=1.5$].

Figure 8

Evolution of $\overline{\theta }$ of a prolate capsule (AR = 2.0) with ${\widehat{\kappa }}_B=0.2$ and $\lambda =5$ at (a) $\mathrm{Ca}=0.2$, (b) $\mathrm{Ca}=0.65$, and (c) $\mathrm{Ca}=2.0$.

Figure 9

Evolution of $\overline{\theta }$ of a prolate capsule (AR = 2.0) with ${\widehat{\kappa }}_B=0.2$ and $\lambda =5$ at (a) $\mathrm{Ca}=0.65$, (b) $\mathrm{Ca}=0.8$, (c) $\mathrm{Ca}=0.9$, and (d) $\mathrm{Ca}=1.0$.

Figure 10

(a) Bifurcation behavior for a prolate capsule (AR = 2.0) with ${\widehat{\kappa }}_B=0.2$ and $\lambda =5$. The values for ${\overline{\theta }}_{\text{eq}}$ corresponding to the attractors are represented by the solid blue lines with filled circles, and the estimated values for ${\alpha }_c$ by the dashed blue line with filled circles (the error bars indicating the range for ${\alpha }_c$ at each $\mathrm{Ca}$ in the multiplicity regime). The unstable solutions at $\theta =0^{\circ }$ and ${90}^{\circ }$ are indicated by the dashed black lines with crosses. (b) Evolution of $\overline{\theta }$ for a capsule with $\alpha ={60}^{\circ }$ initially at $\mathrm{Ca}=0.9$. When a stable orbit is reached, the value of $\mathrm{Ca}$ is suddenly increased to 1.0, and the capsule evolves toward a new stable orbit.

Figure 11

Effects of capsule aspect ratio [(a) $\lambda =5]$ and the viscosity ratio [(b) AR = 3.0] on the bifurcation behavior of the attractors for a prolate capsule with ${\widehat{\kappa }}_B=0.2$. Again, the black dashed lines at $\theta =0^{\circ }$ and ${90}^{\circ }$ correspond to the orbits for a capsule aligned with the $z$ axis and in the shear plane, respectively. (c) Phase diagram of stable orbital motions for a prolate capsule (AR = 3.0) with ${\widehat{\kappa }}_B=0.2$ over a range of $\mathrm{Ca}$ and $\lambda$. The multiplicity regime is shaded in blue, and only the motion modes for the attractors on the upper branch are shown here (the attractors on the lower branch always correspond to rolling and thus are not shown here). T denotes tumbling, P precessing, and R rolling.

Figure 12

Time sequence images (side and front views) of a prolate capsule (AR = 3.0) with ${\widehat{\kappa }}_B=0.2$ and $\lambda =6.0$ taking a tumbling [(a) $\mathrm{Ca}=0.6]$, precessing [(b) $\mathrm{Ca}=1.4]$, and rolling [(c) $\mathrm{Ca}=2.0]$ motion, respectively. The initial orientation of the capsule is $\alpha ={75}^{\circ }$. The markers indicate the positions of a membrane point initially on the major axis of the capsule for (a) and (b), and a membrane point initially on the equator of the capsule for (c).

Figure 13

Intrinsic viscosity (a) and dimensionless normal stress differences (b,c) predicted for a dilute suspension (with volume fraction $\mathrm{\Phi })$ of identical prolate capsules (AR = 3.0) with ${\widehat{\kappa }}_B=0.2$ taking the same corresponding stable orbit(s) at varying $\mathrm{Ca}$ for $\lambda =3.0$ and 5.0. In each case, the squares represent the pure tumbling orbits at lower $\mathrm{Ca}$, and the circles represent the pure rolling orbits at higher $\mathrm{Ca}$; the downward and upward triangles represent the attractors on the upper branch (either tumbling or precessing) and the stable rolling orbits on the lower branch of the multiplicity regime (bounded by vertical dotted lines), respectively.