Mesoscopic model for soft flowing systems with tunable viscosity ratio

We propose a mesoscopic model of binary fluid mixtures with tunable viscosity ratio based on a two-range pseudopotential lattice Boltzmann method, for the simulation of soft flowing systems. In addition to the short-range repulsive interaction between species in the classical single-range model, a competing mechanism between the short-range attractive and midrange repulsive interactions is imposed within each species. Besides extending the range of attainable surface tension as compared with the single-range model, the proposed scheme is also shown to achieve a positive disjoining pressure, independently of the viscosity ratio. The latter property is crucial for many microfluidic applications involving a collection of disperse droplets with a different viscosity from that of the continuum phase. As a preliminary application, the relative effective viscosity of a pressure-driven emulsion in a planar channel is computed.

Figure 1

The discrete lattice used in this work. The fluid lives in the D2Q9 lattice, while the interactions extend to the full D2Q25.

Figure 2

A sketch of different interactions on component $A$. Left panel: short-range attractive ($G_{A,1}<0$) and mid-range repulsive ($G_{A,2}>0$) interactions within component $A$. Right panel: short-range repulsive ($G_{AB}>0$) interaction between components $A$ and $B$.

Figure 3

A sketch for one and two interfaces between components $A$ and $B$. Top panel: one interface case for the calculation of the surface tension $\gamma$ in Eq. (15). Bottom panel: two interfaces separated by a distance $h$ for the calculation of the overall line tension ${\gamma }_f\left(h\right)$ in Eq. (16). When the distance $h$ is large enough, ${\gamma }_f\left(h\right)$ is convergent to $2\gamma$. By setting a series of $h$, the disjoining pressure can be calculated through Eq. (18).

Figure 4

Sketch of the two-component Poiseuille flow.

Figure 5

Steady velocity profiles for two-component Poiseuille flow with different viscosity ratios, $M=\left\{1/50,1,50\right\}$ (from left to right); the profiles are normalized by the centerline velocity $u_0=0.05$ (lbu). The symbols are numerical results by the present two-range pseudopotential model, and the lines are the analytical solutions.

Figure 6

The density profile along the horizontal centerline for a steady droplet (component $A$) with radius $R=42\mathrm{\Delta }x$ in the center of a periodic box: we measure the interface thickness as $W=({\rho }_{A,max}-{\rho }_{A,min})\mathrm{\Delta }x/[{\rho }_A\left(M\right)-{\rho }_A\left(N\right)]$, where $M$ and $N$ are the two points near the location where ${\overline{\rho }}_A=({\rho }_{A,max}+{\rho }_{A,min})/2$. The interface thickness is $W=(1.041-0.008)\mathrm{\Delta }x/(0.644-0.336)=3.35\mathrm{\Delta }x$.

Figure 7

Dimensionless disjoining pressure ${\Pi }^{*}$ as a function of dimensionless distance $h^{*}$ for different cases. For the case $G_{A,1}=0,G_{A,2}=0$(without two-range competing interactions), the disjoining pressure decreases with $h^{*}$, thus it cannot support a stable thin film between two interfaces. For other cases (with two-range competing interactions), with the decrease of $h^{*}$, the disjoining pressure increases gradually to a peak and then goes down, which stabilizes the thin film. The peak value of disjoining pressure increases with decreasing $G_{A,1}$.

Figure 8

Comparison of the obtained disjoining pressure ${\Pi }^{*}$ at different viscosity ratio $M$ by the original (left) and present (right) two-range models. For the original model [23, 24], ${\Pi }^{*}$ changes substantially and even changes the sign when $M$ varies. For the present model, ${\Pi }^{*}$ is independent of $M$. It may be noted that the profiles for the two models at $M=1$ are approximately the same.

Figure 9

Snapshots $(t^{*}=tU/D)$ of two equal-sized droplets collision at viscosity ratio $M=1$: (a) the original two-range model and (b) the present two-range model.

Figure 10

Snapshots $(t^{*}=tU/D)$ of two equal-sized droplets collision at viscosity ratio $M=3$: (a) the original two-range model and (b) the present two-range model.

Figure 11

Snapshots of pressure-driven emulsion in a planar channel at volume fraction $\mathrm{\Phi }=0.64$ with $M={\mu }_B/{\mu }_A=1$ (left panel), and $M=1/10$ (right panel).

Figure 12

Rheological feature of pressure-driven emulsion in a planar channel at ${\overline{F}}_b=37$. Left panel: $x$-direction average velocity profile as a function of $y$ at different droplets volume fractions $\mathrm{\Phi }$. Right panel: measured ${\mu }_r$ (symbols) as a function of droplets volume fraction $\mathrm{\Phi }$, where the solid line is a fit by a model based on the differential effective medium theory [58].

Figure 13

Relative effective viscosity ${\mu }_r$ as a function of ${\overline{F}}_b$ at different volume fractions $\mathrm{\Phi }$.

Figure 14

Relative effective viscosity ${\mu }_r$ as a function of viscosity ratio $M={\mu }_B/{\mu }_A$ at different volume fractions $\mathrm{\Phi }$ and ${\overline{F}}_b=37$. The left branch of the figure, $M<1$, corresponds to more viscous, hence less deformable, droplets ($M\to 0$ is the solid sphere limit), while in the right branch $M>1$ they are less viscous than the matrix, and correspondingly more deformable ($M\gg 1$ is the bubble case). This figure can be also seen then as the effective viscosity of a suspension of soft deformable particles, as a function of their stiffness. Inset: Plot of ${\mu }_r$ versus $\mathrm{\Phi }$ for $M=1/10$, 1/6, and 1/3.

Figure 15

Maximum spurious currents $u_s$ measured from a steady droplet in a continuous matrix at different viscosity ratios $M$ and strength coefficients $G_{A,1}$.