Lattice Boltzmann simulations of droplet breakup in confined and time-dependent flows

We study droplet dynamics and breakup in generic time-dependent flows via a multicomponent lattice Boltzmann algorithm, with emphasis on flow startup conditions. We first study droplet breakup in a confined oscillatory shear flow via two different protocols. In one setup, we start from an initially spherical droplet and turn on the flow abruptly (“shock method”); in the other protocol, we start from an initially spherical droplet as well, but we progressively increase the amplitude of the flow, by allowing the droplet to relax to the steady state for each increase in amplitude, before increasing the flow amplitude again (“relaxation method”). The two protocols are shown to produce substantially different breakup scenarios. The mismatch between these two protocols is also studied for variations in the flow topology, the degree of confinement, and the inertia of the fluid. All results point to the fact that under extreme conditions of confinement the relaxation protocols can drive the droplets into metastable states, which break only for very intense flow amplitudes, but their stability is prone to external perturbations, such as an oscillatory driving force.

Figure 1

Snapshots of a droplet in a confined oscillatory shear flow with a nondimensionalized oscillation frequency ${\omega }_f{t}_d$. Snapshots of the droplet in the velocity field are shown for $\text{Ca}<{\text{Ca}}_{\text{cr}}$ and $\text{Ca}>{\text{Ca}}_{\text{cr}}$. The plots on the right panel show the time evolution of the normalized droplet length $L\left(t\right)/R$. The degree of confinement of the system is given by $\alpha =2R/L_z$, where $R$ is the droplet radius of the undeformed droplet and $L_z$ is the wall separation.

Figure 2

Planar cut of a droplet in a shear flow, featuring an ellipsoidally deformed droplet and large droplet deformation before breakup. The droplet contours are shown in black and the velocity field is visualized by streamlines coloured according to the velocity magnitude.

Figure 3

Critical capillary number ${\text{Ca}}_{\text{cr}}$ at varying frequencies ${\omega }_f{t}_d$. There is a mismatch between the predictions of the two breakup protocols. Whereas droplet breakup is largely independent in the case of the shock method, except for the asymptotic behavior in the high frequency region, the relaxation method in the low-frequency limit predicts a higher ${\text{Ca}}_{\text{cr}}$ than the ones of the shock method. This mismatch is investigated in the article. The error bars are estimated via steps in the critical capillary number $\mathrm{\Delta }\text{Ca}$. Both curves are interpolated via bezier curves.

Figure 4

${\text{Ca}}_{\text{cr}}$ vs. Reynolds number $\text{Re}$. The mismatch between the shock and relaxation breakup protocols does not depend on inertia. This is especially clear in the case of the Stokes solution, for which $\text{Re}\text{=}\text{0}$. The error bars are estimated via steps in the critical capillary $\mathrm{\Delta }\text{Ca}$ and Reynolds number $\mathrm{\Delta }\text{Re}$.

Figure 5

Critical capillary number ${\text{Ca}}_{\text{cr}}$ for different confinement ratios $\alpha =2R/L_z$. We compare the values obtained by the LB simulations with the shock method and the ones obtained by the relaxation method. Since the relaxation method is dependent on the startup conditions of the outer flow and the droplet, we provide a range of different increments $\mathrm{\Delta }\text{Ca}$, where smaller $\mathrm{\Delta }\text{Ca}$ denote a slower and flow build up and vice versa. The error bars are estimated via steps in the critical capillary number $\mathrm{\Delta }\text{Ca}$. For each simulation run of the relaxation method with a given $\text{Ca}$ we gave the droplet a sufficiently long time to relax to its stationary state.

Figure 6

Normalized droplet major axis $L\left(t\right)/R$ against simulation time $t$ given in units of the droplet relaxation time $t_d$. The droplet breaks up shortly after its maximum elongation for the shock method. Breakup in the relaxation method is dependent on the shear rate and thus capillary number increase: (a) for a rate with increment $\mathrm{\Delta }\text{Ca}=0.30$ the droplet relaxes after reaching its maximum elongation for the first time to breakup at a longer length at a higher ${\text{Ca}}_{\text{cr}}$ later on; (b) for a smaller capillary number increase $\mathrm{\Delta }\text{Ca}=0.24$ the droplet length at ${\text{Ca}}_{\text{cr}}$ increases even further and the $L\left(t\right)$ contains more full extensions and subsequent retractions.

Figure 7

The flow layout of a droplet in an elongational (uniaxial extensional) flow. The image is a planar cut, with the flow being rotational symmetric around the elongated droplet axis in the image. The streamlines are coloured according to the velocity magnitude.

Figure 8

Critical capillary number ${\text{Ca}}_{\text{cr}}$ against different frequencies ${\omega }_f{t}_d$ for a droplet in an unconfined elongational flow. We consider the two breakup protocols, the shock method and the relaxation method. Even though the droplet breakup is dependent on the shear rate frequency ${\omega }_f{t}_d$, a protocol mismatch does not occur, contrary to the case of the confined shear flow topology. The error bars are estimated via steps in the critical capillary number $\mathrm{\Delta }\text{Ca}$.