Numerical simulations of vesicle and bubble dynamics in two-dimensional four-roll mill flows

We use a computational technique based on the immersed boundary method to construct a four-roll mill device with which we can generate a broad spectrum of flow types from an extensional flow to a rotational one. We put a vesicle or a bubble in the constructed four-roll mill device to investigate their interaction with the surrounding fluid. The vesicle dynamics are determined by its bending rigidity, inextensibility, and hydrodynamical force, whereas the bubble dynamics is governed by the surface tension and the hydrodynamic interaction. Depending on the type of the flow, these suspended objects go through either a tank-treading motion or a tumbling motion. We validate our numerical method by a convergence study and discuss the transition between tank-treading and tumbling motions for the vesicles and bubbles.

Figure 1

Schematic diagrams of four-roll mill device in Ref. [1] (left) and the microfluidic four-roll mill device designed here and in Ref. [14] (right). A broad spectrum of flow types can be generated by adjusting the angular velocity ratio ${\omega }_1/{\omega }_2$ (left) or the flux ratio $Q_1/Q_2$ (right).

Figure 2

The flow generator analogous to the four-roll mill device. Poiseuille flows are derived with the normal fluxes inward to the domain being $-Q_1$ in ${\mathrm{\Omega }}_1$ and $Q_2$ in ${\mathrm{\Omega }}_2$. The resulting velocity vector fields in the whole domain (middle panel) and in the central region (enclosed by the dashed line in the left panel) containing vesicle or bubble (right panel).

Figure 3

Schematic view of wall target boundary, wall immersed boundary, and the interface immersed boundary. The wall IB points are connected to the target points by a system of stiff springs with zero rest length, and the interface IB points represent the elastic boundary.

Figure 4

The contours of the flow-type parameters (first row), the streamlines of the velocity field in $[-130,130]\times [-130$,130] $\mu {\mathrm{m}}^2$ (second row), and the streamlines with some indications of the velocity vectors in $[-60,60]\times [-60$,60] $\mu {\mathrm{m}}^2$ (third row). The columns from left to right show an extensional flow with the flux ratio $f=1.0$, a mixed shear flow with $f=-0.585$, and a rotational flow with $f=-1.0$, respectively.

Figure 5

Flow-type parameter $\overline{\xi }$ averaged over a central disk with the radius $12.5\mu \mathrm{m}$ in terms of the flux ratio $f$. The flow type varies from purely rotational ($\xi =-1$) to purely extensional ($\xi =1$) by changing the flux ratio over the range of $-1\le f\le 1$. When the flow-type ratio $f=-0.585$, the flow-type parameter $\xi$ is almost 0.

Figure 6

Convergence ratios of the computed velocity field $\mathbf{u}(\mathbf{x},t)$ of the fluid. The convergence ratios (defined in the text) are plotted as functions of time for three different flux ratios: $f=1.0$ (left), $-0.585$ (middle), and $-1.0$ (right). In each panel, the line with “+” is the convergence ratio obtained using the grids $N=128,256,512$, and the line with “$\circ$” is the ratio obtained with $N=256,512,1024$. The convergence ratio is near two (first-order accuracy) for all the different cases.

Figure 7

The motion of the vesicle with the reduced area $V=0.7$, together with the streamlines of the velocity field at $t=0.4$ s. The flows are generated with different values of the flux ratio: $f=0.9$ (upper-left), 0.4 (upper-middle), 0.0 (upper-right), $-0.4$ (lower-left), $-0.6$ (lower-middle), and $-0.67$ (lower-right). Each vesicle reaches a stable tank-treading (TT) motion with some fixed inclination angle between the long axis of the vesicle and the negative $x$ axis.

Figure 8

The inclination angles (upper) and the angular velocities (lower) of the TT motion of the vesicles in terms of the flux ratio $f$ for various reduced areas $V$ when the vesicles are in the steady TT motions.

Figure 9

The critical flux ratio $f_c$ for the transition between TT and TU in terms of the reduced area $V$.

Figure 10

The configuration of a vesicle with the reduced area $V=0.7$ suspended in a flow with the flux ratio $f=-0.8$ at some chosen times (left panels) and the time evolution of the inclination angle (right panel) of the vesicle with the reduced area $V=0.7$ in flows with three different flux ratios: $f=-0.7$ (dotted line), $-0.8$ (dashed line), and $-0.9$ (solid line).

Figure 11

The rotational frequency of the tumbling motion of the vesicle in terms of the flux ratio $f$ for various reduced areas $V$.

Figure 12

The motion of a bubble with the surface tension $\gamma =0.0004$ dyne at $t=0.8$ s (upper-left), and $t=2.4$ s (upper-middle), together with the streamlines of the velocity field. The upper-right panel describes the velocity vectors on the bubble. The flow is purely rotational with the flux ratio $f=-1$. The lower panel shows the rotational angle, which indicates that the bubble rotates uniformly with the angular frequency $\omega =0.8928$ Hz.

Figure 13

The motion of bubbles with the surface tensions $\gamma =0.0004$ dyne (upper) and $\gamma =0.004$ dyne (lower) at $t=0.032$ s (left), 0.064 s (middle), and 0.096 s (right). The flow is purely extensional ($\xi =1$) with the flux ratio $f=1$. The bubble with a small surface tension cannot resist the extensional deformation of the flow and elongates in time, gets thinner, and finally breaks up (top panels). The bubble with a large surface tension approaches to a stable elliptical shape.

Figure 14

The length of the long axis of the elliptical bubble in terms of time for various surface tensions (upper) and the constant value of the long axis in the stable state of the bubble in terms of the surface tensions (lower).

Figure 15

The motion of bubbles with the surface tensions $\gamma =0.0004$ dyne (upper) and $\gamma =0.004$ dyne (lower) at $t=0.4$ s (left), 0.8 s (middle), and 1.2 s (right). The flow is composite with extensional and rotational flows with the flux ratio $f=-0.6$.

Figure 16

The length of the long axis of the elliptical bubble (upper) and the rotational angle (lower) in terms of time for various surface tensions. The flow is composite with extensional and rotational flows with the flux ratio $f=-0.6$.

Figure 17

The length of the long axis of the elliptical bubble (upper) and the rotational angle (lower) in terms of the surface tension for various surface tensions when the bubbles are in steady rotational states.