Effect of droplet deformability on shear thinning in a cylindrical channel

Droplets suspended in fluids flowing through microchannels are often encountered in different contexts and scales, from oil extraction down to microfluidics. They are usually flexible and deform as a product of the interplay between flexibility, hydrodynamics, and interaction with confining walls. Deformability adds distinct characteristics to the nature of the flow of these droplets. We simulate deformable droplets suspended in a fluid at a high volume fraction flowing through a cylindrical wetting channel. We find a discontinuous shear thinning transition, which depends on the droplet deformability. The capillary number is the main dimensionless parameter that controls the transition. Previous results have focused on two-dimensional configurations. Here we show that, in three dimensions, even the velocity profile is different. To perform this study, we improve and extend to three dimensions a multicomponent lattice Boltzmann method which prevents the coalescence between the droplets.

Figure 1

Spurious velocities (arrows) around a circular interface (red) of radius $r=20$ at equilibrium for $G_{k,1}=-7.9$ and $G_{k,2}=4.9$. The largest spurious velocities are (a) 0.017 l.u. for the D3Q19 lattice and (b) 0.0092 l.u. for the D3Q41 lattice. The velocity field is on the same scale in both panels. The ratio of the density inside and outside the droplet is 1.133 for D3Q19 and 1.062 for D3Q41.

Figure 2

Initial setup. (a) Here 105 droplets are arranged in a hexagonally non-close-packed system with a volume fraction of 0.61. A force $\mathbf{F}$ is imposed in the $z$ direction and varied. The walls have a neutral wetting condition [47]. The droplets (red) are inside the cylindrical channel (gray). The system is periodic in the direction of the flow. The parameters for the interactions are $G_{k,1}=-7.9$ and $G_{k,2}=4.9$ and the viscosity ratio of the droplet and the continuous fluid is set to one. (b) Top view. (c) Front view.

Figure 3

Discontinuous shear thinning for different values of $\gamma$. (a) Discontinuous shear thinning occurs for progressively larger values of the external force which drives the flow. The black arrow indicates the threshold viscosity ${\mu }_r^{*}$, i.e., the relative viscosity before shear thinning. (b) Relative viscosity ${\mu }_r$ (normalized by the threshold viscosity ${\mu }_r^{*}$) as a function of the capillary number $\mathrm{Ca}$. The transition is at approximately 0.025. Simulations were carried out for different values of surface tension $\gamma$.

Figure 4

(a) Threshold force ${\mathbf{F}}^{*}$ required to trigger discontinuous shear thinning as a function of $\gamma$. Here ${\mathbf{F}}^{*}$ increases roughly linearly with the droplets' surface tension. (b) Corresponding threshold viscosity ${\mu }_r^{*}$ as a function of $\gamma$.

Figure 5

Velocity profile given by the dimensionless Reynolds number $\text{Re}=\frac{\langle V_z\rangle D}{\nu }$, where $D$ is the diameter of the channel, $\nu$ is the kinematic viscosity, and $\langle V_z\rangle$ is the average velocity in the $z$ direction. The capillary numbers are $\text{Ca}=0.017$ and 0.0314. The droplet and fluid velocity are plotted separately.

Figure 6

Streamlines (blue) for a central region of the channel just before shear thinning, which indicates that the continuous fluid flows between droplets (red). The system has $\text{Ca}=0.025$.

Figure 7

Relative velocity fields and droplet shape cross section (a) before and (b) after discontinuous shear thinning. The capillary numbers are (a) $\text{Ca}=0.017$ and (b) $\text{Ca}=0.0314$. The relative velocity is obtained by subtracting the velocity at the center of the droplet and it is on the same scale in both images. The cross section is obtained along the $ZY$ plane. Large black arrows indicate the direction of flow. (c) Relative position of the sample droplet in question marked as $A$ (front view, i.e., $XY$ plane).

Figure 8

Pressure difference $\mathrm{\Delta }p$ as a function of the inverse of the droplet radius $r$. We simulate a static droplet for a fixed ${\rho }_0$ (reference density) and let it reach the steady state. The symbols represent the pressure difference inside and outside the droplet in our simulations. The solid lines are linear fits corresponding to Eq. (A1), where the slope is $\gamma$.

Figure 9

Disjoining pressure $\mathrm{\Pi }$ as a function of the distance $h$ between two flat interfaces with fixed $G_{k,1}=-3.5$ and $G_{k,2}=2.5$ achieved with competing forces ${\mathbf{F}}^c$ given by Eq. (7), which provides the repulsive force to avoid coalescence. A positive value indicates repulsion, while a negative value indicates attraction. (a) Disjoining pressure for different values of the reference density ${\rho }_0$. (b) Disjoining pressure for different values of the viscosity ratio $M$ with ${\rho }_0=1.00$.

Figure 10

Snapshots of a collision between two droplets illustrating the frustrated coalescence.