Lattice Boltzmann simulations of droplet formation in confined channels with thermocapillary flows

Based on mesoscale lattice Boltzmann simulations with the “Shan-Chen” model, we explore the influence of thermocapillarity on the breakup properties of fluid threads in a microfluidic T-junction, where a dispersed phase is injected perpendicularly into a main channel containing a continuous phase, and the latter induces periodic breakup of droplets due to the cross-flowing. Temperature effects are investigated by switching on-off both positive-negative temperature gradients along the main channel direction, thus promoting a different thread dynamics with anticipated-delayed breakup. Numerical simulations are performed at changing the flow rates of both the continuous and dispersed phases, as well as the relative importance of viscous forces, surface tension forces, and thermocapillary stresses. The range of parameters is broad enough to characterize the effects of thermocapillarity on different mechanisms of breakup in the confined T-junction, including the so-called “squeezing” and “dripping” regimes, previously identified in the literature. Some simple scaling arguments are proposed to rationalize the observed behavior, and to provide quantitative guidelines on how to predict the droplet size after breakup.

Figure 1

Numerical simulations in the confined T-junction geometry in presence of thermocapillarity. The importance of the thermocapillary stresses is quantified with the Marangoni number $[\mathrm{Ma}$; see (27)] at changing also the temperature gradient $\nabla T$. (a), (f), (k) Temperature across the T-junction for three representative cases: $\mathrm{Ma}=0$ (a); $\mathrm{Ma}=1.26,\nabla T>0$ (f); $\mathrm{Ma}=1.26,\nabla T<0$ (k). High-temperature regions appear as light (light green), and low-temperature regions are dark (dark green). The other panels report density contours during the droplet formation process overlaid on the hydrodynamic viscous stress in the continuous phase: $\mathrm{Ma}=0$ (b)–(e); $\mathrm{Ma}=1.26,\nabla T>0$ (g)–(j); $\mathrm{Ma}=1.26,\nabla T<0$ (l)–(o). Low-stress regions appear as light (light blue), and high-stress regions are dark (dark blue). In all the cases the capillary number $[\mathrm{Ca}$; see (25)] is fixed to $\mathrm{Ca}=0.001$ and the flow-rate ratio $[\varphi$; see (26)] is fixed to $\varphi =1.0$. Notice that we have used the characteristic shear time ${\tau }_{\mathrm{shear}}=H/v_{\mathrm{c}}$ as a unit of time.

Figure 2

Quantitative analysis for the dimensionless droplet volume as a function of the flow-rate ratio $\varphi$ in the small $\mathrm{Ca}$ regime. Left: The scaling laws for the dimensionless droplet volume versus the flow-rate ratio $\varphi$ in the squeezing [25, 31] regime at $\mathrm{Ca}=0.0014$, for different values of $\mathrm{Ma}$. Both negative and positive temperature gradients $\nabla T$ are considered. Linear fits are drawn to guide the eyes. Right: Analysis of the linear scaling law (28) for the data in the left panel. We report the scaling factor ${\alpha }_2$ versus $sgn\left(\nabla T\right)\left[\mathrm{Ma}\right]$. We also provide two slope fits for the data at $\nabla T\le 0$ (red circles) and $\nabla T\ge 0$ (blue squares). The inset shows the plot of ${\alpha }_1$ versus $sgn\left(\nabla T\right)\left[\mathrm{Ma}\right]$.

Figure 3

Quantitative analysis for the dimensionless droplet volume as a function of $\mathrm{Ca}$, for flow-rate ratio $\varphi =1.0$ (left) and $\varphi =0.5$ (right). For $\mathrm{Ma}$, we have chosen the reference case $\mathrm{Ma}=0$ (red circles), and the value $\mathrm{Ma}=1.26$ for both negative (black squares) and positive (blue triangles) temperature gradients $\nabla T$. The two insets show the plot of the dimensionless droplet volume versus the rescaled ${\mathrm{Ca}}_{\mathrm{m}}$ defined in (29), where a characteristic velocity $v_{\mathrm{m}}$ is introduced to account for the thermocapillary effects.

Figure 4

Quantitative analysis for the dimensionless droplet volume as a function of $\mathrm{Ca}$, for different values of $\mathrm{Ma}$. Both negative and positive temperature gradients $\nabla T$ are considered. Left: We report the dimensionless droplet volume versus $\mathrm{Ca}$ for different values of $\mathrm{Ma}$ and $\nabla T$. In all cases, the flow-rate ratio is kept fixed to $\varphi =1.0$. Right: Data in the left panel are plotted against the rescaled ${\mathrm{Ca}}_{\mathrm{m}}$ defined in (29). Notice that the value of the thermocapillary velocity $v_{\mathrm{m}}$ is changing with $\mathrm{Ma}$ and is determined by simply rescaling the value of $v_{\mathrm{m}}$ found for $\mathrm{Ma}=1.26$ in Fig. 3.

Figure 5

We report the evolution in time of the center of mass of a droplet (relative to its initial value) steadily translating in a temperature gradient. The droplet radius $R$ is resolved with a variable number of lattice cells, and the temperature gradient is consequently adjusted to match the same Marangoni number $\mathrm{Ma}=1.25$. Other simulation details are described in the text.

Figure 6

The velocity profiles in the reference frame of the moving droplet steadily translating in a temperature gradient. The droplet moves from top to bottom. Results from numerical simulations at $\mathrm{Ma}=5.0$ (left) are compared with the analytical profiles (right; see Appendix pp2).

Figure 7

The velocity profiles in the reference frame of the moving droplet steadily translating in a temperature gradient at changing $\mathrm{Ma}$: $\mathrm{Ma}=20$ (left); $\mathrm{Ma}=50$ (right).

Figure 8

Left: $\mathrm{Ma}$ (27) as a function of the parameter $G_T$ (22). Ma is computed from the steady velocity of migration of a droplet in a constant temperature gradient described in Sec. pp1. Inset: We report the steady velocity (A1). Right: The black line stands for the RHS of Eq. (A1), with the jump in surface tension $\mathrm{\Delta }{\sigma }_{2R}$ estimated from the modified pressure tensor, as described in Sec. 2a. The points represent the steady velocity of migration obtained from the numerical simulations.

Figure 9

The analysis shown in the right panel of Fig. 8 is extended to fluids with dissimilar thermal conductivities: ${\kappa }_{\mathrm{c}}^{\left(T\right)}=0.06,{\kappa }_{\mathrm{d}}^{\left(T\right)}=0.166$ lbu (left) and ${\kappa }_{\mathrm{d}}^{\left(T\right)}=0.06,{\kappa }_{\mathrm{c}}^{\left(T\right)}=0.166$ lbu (right).