Capsule distributions and flow properties in curved tubes

The distribution of capsules and rheological properties of suspensions in curved tubes are investigated by using an immersed-boundary lattice Boltzmann method. We mainly focus on the effective suspension viscosity and equilibrium positions of capsules as functions of Reynolds number $\left(\text{Re}\right)$, Capillary number $\left(\text{Ca}\right)$ and volume fraction of capsules $\left(\varphi \right)$. We found that at limited inertia ($\mathrm{Re}O\left(10\right)$), the effective viscosity decreases with increasing $\mathrm{Re}$, which is different from the variation trend in straight tubes. Dean's vortices play an important role. When the fluid inertia increases, the vortices are strengthened. They greatly promote the capsules' circumferential transportation by trapping the capsules into their centers and making the location of maximum azimuthal velocity close to them. The curvature effect of the torus vessel is also investigated. When the curvature $\kappa$ is large enough, e.g., $\kappa >0.2$, a scaling law for the effective viscosity as a function of a redefined $\mathrm{Re}$ is proposed. Furthermore, the distribution feature of multiple capsules in the torus vessel is revealed. Generally, for the semidilute regime, the capsules concentrate on the symmetrical plane at low $\mathrm{Re}$ but the center of Dean's vortex at high $\mathrm{Re}$. In addition, for both the dilute and semidilute regime, the scaling law connecting the effective viscosity and the average location of capsules is proposed. Our data support the scaling well. This study may be useful in the design of tubes for capsule transportation.

Figure 1

Geometry for a torus vessel, (a) top view, (b) side view. The tube's radius is $R_t$. Here $R_i$ and $R_o$ are the radii of the inner and outer walls, respectively. $Z=0$ denotes the symmetrical plane. Curvature radius $R_c$ is defined as the distance between the origin $O$ and the vessel's centerline.

Figure 2

Equilibrium profiles of an initially spherical capsule with SK membrane ($\mathrm{Re}=0.125,E_b=0,\mathrm{Ca}=0.1,C=1$) in a square section channel. (a) Profile in the XZ plane, (b) Cross profile in the YZ plane. The solid lines represent our results. The diamond symbols denote the results obtained by Hu et al. [60] using boundary element method.

Figure 3

(a) Azimuthal velocity (scaled by $U_m$) contours in the case of $\mathrm{Re}=0.05,R_c=3,R_t=2$. The dashed line represents the symmetrical line $Z=0$. I and O represent the innermost wall ($R_i$) and outermost wall ($R_o$) of the vessel, respectively. (b) Distribution of azimuthal velocity along the symmetrical line. The solid line represents the location of $U_m$. (c) Location of the maximum azimuthal velocity ($R_{U_{\max }}^{\prime }$) on the symmetrical line in cases of $\kappa :0.2,0.357,0.417,0.5,0.625,0.714,0.833,0.9091$ at $\mathrm{Re}=0.05$.

Figure 4

(a) Radial location of the capsule as a function of time (grid independence study). The key parameters are $\mathrm{Re}=10$, $Ca=1.0$, $R_c=5$, and $R_t=3$. (b) Results of the cases of $\mathrm{Re}=4$, 20. The other parameters are $Ca=1.0$, $R_c=5$, and $R_t=4$. Lines represent our results and symbols represent the results of Ref. [40].

Figure 5

(a) Typical configuration of capsules ($\varphi =20.11\%,Ca=0.2$) in the computational domain where three lengths are $10a\times 10a\times 10a$. (b) ${\mu }_a$ as a function of ${\varphi }_{\mathrm{eff}}$ defined in Eq. (3.9) in Aouane et al. [61] at $\mathrm{Re}=0.01$. The mechanical properties of the capsules are $C=1,5,10,\mathrm{Ca}=0.2,\varphi =0.42\%,3.77\%,7.54\%,11.31\%,15.08\%,20.11\%$, where the cases of $\varphi =0.42\%$ contain one capsule. The symbols are our results and the black line is the scaling law [Eq. (3.10)] of Aouane et al. [61].

Figure 6

(a) Evolution of instantaneous ${\mu }_a$ for thecases with different $\varphi$ in torus vessel and straight tube ($\mathrm{Re}=5$). (b) ${\mu }_a$ as a function of $\mathrm{Re}$ for the cases of $\mathrm{Ca}=0.02$, 0.3, 1.0 at $\varphi =0.47\%$. ${\mathrm{Re}}_c\approx 3.0$.

Figure 7

(a) Contours of secondary velocity $U_s=\sqrt{(u_R^2+u_Z^2)}$ inside a cross section, where $u_R$, $u_Z$ are $R$ and $Z$-component velocities. Streamlines are also shown. (b) Contours of the azimuthal velocity $U$. It is perpendicular to the paper inward. (c) Contours of the dissipation $D$ which is proportional to the product of the local shear rate $\dot{\gamma }$ and fluid shear stress ${\sigma }_f$, $D\propto \dot{\gamma }·{\sigma }_f$ [33]. Here $U_s$, $U$ are scaled by $U_m$, $D$ is scaled by $D_m$ (maximum of $D$). Squares represent the locations of $U_m$ in (b) and the locations of the lowest $D$ in (c). The solid dots represent the central locations of the Dean's vortices. I and O represent the inner and outer walls of the vessel.

Figure 8

(a) Trajectories and equilibrium locations of different cases in the cross-section. Upper semicircle: the trajectories of the capsule in the cases of $\mathrm{Re}\ge 1$ ($\mathrm{Re}=1$, 5 and 30, $\mathrm{Ca}=0.3$). The capsules are all released from the initial position a. Point b denotes the center of Dean's vortex at $\mathrm{Re}=30$. Lower semicircle: equilibrium locations of the capsule in the cases with $\mathrm{Re}\le 1$. At the same $\mathrm{Re}$, from left to right, $\mathrm{Ca}=1.0$, 0.3, 0.02, respectively (the case $\mathrm{Re}=1$ and $\mathrm{Ca}=0.1$ is also presented). The intersection of the two dotted lines denotes the location of $U_m$ when $\mathrm{Re}\le 1$. Ebrahimi et al. [40] also got similar results. (b) The normalized azimuthal velocity $U/U_m$ at the equilibrium position of capsules as a function of $\mathrm{Re}$. In this figure, all cases are calculated with one capsule.

Figure 9

Equilibrium position in $z$ direction ($\left|Z\right|$) as a function of $\mathrm{Ca}$ at $\kappa =0.6,{\mathrm{Re}}_c=3.0$. Red squares are our data, blue line is the exponential fitting formula $\left|Z\right|=0.35e^{-Ca/0.2}$. The y axis is in log scale.

Figure 10

(a) Configurations of the curved tubes with curvature $\kappa =0.6,0.3,0.2$ and straight tube (from left to right). (b) Intrinsic viscosity as a function of $\mathrm{Re}$ in torus vessels with various curvatures and the straight tubes. Each case contains one capsule.

Figure 11

(a) Transition Reynolds number ${\mathrm{Re}}_c$ as a function of $\kappa$. All cases contains one capsule with $\mathrm{Ca}=0.3$. The orange squares are our results, blue line is the fitting line ${\mathrm{Re}}_c=3.30+227.45e^{-17.48\kappa }$. (b) Intrisic viscosity $\left[\mu \right]$ as a function of ${\mathrm{Re}}^{\prime }$ at various $\mathrm{Ca}$ and $\varphi$. The line represents a data fitting curve $\left[\mu \right]=-0.013{\mathrm{Re}}^{\prime }+1.3$.

Figure 12

Typical snapshots of configurations of capsules at equilibrium states ($\varphi =7.55\%,\mathrm{Ca}=0.3$). Top view (a), (b) and front view (c), (d) for the cases of $\mathrm{Re}=1.0(a,c)$ and $\mathrm{Re}=30(b,d)$. Dashed lines represent the symmetrical plane.

Figure 13

The temporal and spatial-averaged probability density of capsules $P$ in a cross-section. In all cases $\varphi =11.325\%$. The dashed line in (a) and (b) is the symmetrical line. The dashed circle in (c) and annulus in (d) denote the regions where the capsules are concentrated. The solid dot and cross represent the equilibrium position in the case of a single capsule and the average position of multiple capsules, respectively.

Figure 14

(a) ${\mu }_a$ as a function of $\varphi$($0.47\%,1.89\%,3.77\%,5.66\%,7.55\%$) in torus vessel (solid lines) and straight tube (dotted lines) at $\mathrm{Re}=1$, 5, 30 and $\mathrm{Ca}=0.3$. (b) Intrinsic viscosity $\left[\mu \right]$ as a function of $d$. The directions of the arrows represent an increase of $\mathrm{Re}$. The dashed line represents a transitional regime $\mathrm{Re}\approx {\mathrm{Re}}_c$.