Dynamics of deformable straight and curved prolate capsules in simple shear flow

This work investigates the motion of neutrally buoyant, slightly deformable straight and curved prolate fluid-filled capsules in unbounded simple shear flow at zero Reynolds number using direct simulations. The curved capsules serve as a model for the typical crescent-shaped sickle red blood cells in sickle cell disease (SCD). The effects of deformability and curvature on the dynamics are revealed. We show that with low deformability, straight prolate spheroidal capsules exhibit tumbling in the shear plane as their unique asymptotically stable orbit. This result contrasts with that for rigid spheroids, where infinitely many neutrally stable Jeffery orbits exist. The dynamics of curved prolate capsules are more complicated due to a combined effect of deformability and curvature. At short times, depending on the initial orientation, slightly deformable curved prolate capsules exhibit either a Jeffery-like motion such as tumbling or kayaking, or a non-Jeffery-like behavior in which the director (end-to-end vector) of the capsule crosses the shear-gradient plane back and forth. At long times, however, a Jeffery-like quasiperiodic orbit is taken regardless of the initial orientation. We further show that the average of the long-time trajectory can be well approximated using the analytical solution for Jeffery orbits with an effective orbit constant $C_{eff}$ and aspect ratio ${\ell }_{eff}$. These parameters are useful for characterizing the dynamics of curved capsules as a function of given deformability and curvature. As the capsule becomes more deformable or curved, $C_{eff}$ decreases, indicating a shift of the orbit towards log-rolling motion, while ${\ell }_{eff}$ increases weakly as the degree of curvature increases but shows negligible dependency on deformability. These features are not changed substantially as the viscosity ratio between the inner and outer fluids is changed from 1 to 5. As cell deformability, cell shape, and cell-cell interactions are all pathologically altered in blood disorders such as SCD, these results will have clear implications on improving our understanding of the pathophysiology of hematologic disease.

Figure 1

Schematic of the initial orientation of a curved prolate capsule in unbounded simple shear flow.

Figure 2

Evolution of inclination angle $\varphi$ for a stiff prolate spheroid (Ca = 0.06) with (a) $r_p=2.0$ and (b) $r_p=4.8$, respectively. Solid blue lines are predictions by Jeffery's theory, and red dashed lines are simulation results.

Figure 3

Trajectories on the unit sphere of (a) p and (b) n for a prolate spheroid $(r_p=4.8$, Ca = 0.12) with initial orientation $\alpha =0$ (up to $\dot{\gamma }t=2000)$.

Figure 4

Trajectories on the unit sphere of p (a,b) $(y\text{−}z$ view) and n (c,d) $(x\text{−}z$ view) for a prolate spheroid $(r_p=4.8$, Ca = 0.12) with initial orientation $\alpha =\pi /36$.

Figure 5

The evolution of the $p_z$ and $n_z$ for a prolate spheroid $(r_p=4.8$, Ca = 0.12) with initial orientation $\alpha =\pi /36$.

Figure 6

Trajectories of p (a) and n (b) for a curved prolate capsule with initial orientation $[\alpha ,\beta ]$ = [0,0] at the early stage of simulation $\left(0<\dot{\gamma }t<500\right)$.

Figure 7

Early stage trajectories on the unit sphere of the p (blue) and n (red) vectors of a curved prolate capsule with initial orientation $[\alpha ,\beta ]$ = [0,$\pi /6]\left(0<\dot{\gamma }t<500\right)$ (a,d), [0,$\pi /2]\left(0<\dot{\gamma }t<100\right)$ (b,e), and $[\pi /6,\pi /3]\left(0<\dot{\gamma }t<400\right)$ (c,f), respectively.

Figure 8

Time sequence images (front view) of the early stage (a) and long-time (b) motions of a curved prolate capsule with initial orientation $[\alpha ,\beta ]$ = [0,$\pi /6]$. The dashed arrow in each image represents the p vector at the corresponding time spot.

Figure 9

Long-time trajectories on the unit sphere of the p (blue) and n (red) vectors of a curved prolate capsule with initial orientation $[\alpha ,\beta ]$ = [0,$\pi /6]$ (a,d), [0,$\pi /2]$ (b,e), and $[\pi /6,\pi /3]$ (c,f), respectively.

Figure 10

Evolution of $y$ and $z$ components $y_{cm}$ and $z_{cm}$ of the center-of-mass position of curved prolate capsules with initial orientations $[\alpha ,\beta ]$ = [0,$\pi /6]$ (a), [0,$\pi /2]$ (b), and $[\pi /6,\pi /3]$ (c).

Figure 11

The long-time trajectory (blue curve) of a curved prolate capsule in terms of $\varphi$ and $\theta$ and the average trajectory (black circles). The red curve represents the fit of the average trajectory using Eq. (11). The black dashed line corresponds to $\varphi =\pi$. $\mathrm{\Delta }{\theta }^{+}$ and $\mathrm{\Delta }{\theta }^{-}$ denote the maximum positive and negative deviations, respectively, of the instantaneous trajectory from the mean when $\varphi =\pi$. (a) Ca = 0.12, $K=0.36,\lambda =1$; (b) Ca = 0.06, $K=0.39,\lambda =5$.

Figure 12

The dependencies of (a) the effective orbit constant $C_{eff}$ of the long-time orbit and (b) the effective aspect ratio ${\ell }_{eff}$ on the deformability Ca and the degree of curvature $K$ of the curved prolate capsules.