Lattice Boltzmann simulations of drop deformation and breakup in shear flow
Introduction
When drops of one liquid dispersed in another immiscible liquid are subjected to shear flow, they start to deform. If the local shear rate is sufficiently large, the drops might break up into fragments. The study of the dynamics and mechanisms of drop breakup in shear flow is of fundamental importance in dispersion science and mixing processes. Experimental and theoretical investigations in this area focus on analyzing how strong the flow should be to break the drop, what the necessary energy input is to create the required intensity of the flow, and what the resulting drop size distribution (DSD) and rheology of the mixture are (Rallison, 1984). The results obtained in such studies can be applied to the formation of dispersions and emulsions and in particular the design of efficient mixing devices (Rallison, 1984). The application of shear to a premixed emulsion of various drop sizes is a technique for the production of monodisperse droplets (Cristini and Renardy, 2006).
Stirred tank reactors are widely used to obtain liquid–liquid dispersions under turbulent flow conditions. Turbulent flows contain a spectrum of eddies of different size, intensity, and lifetime (Pope, 2000). Drops continuously interact with these eddies. Large eddies convect small droplets with little deformation. When the droplet size is comparable to the eddy size, the drop can be significantly deformed and subsequently broken. Even though the randomness of turbulent flow implies complex drop/eddy interactions, simpler interactions can be identified. For example, a drop in a simple shear flow represents drop interaction with two co-rotating eddies in turbulent flow. The investigation of drop behavior in simple shear flow is more reproducible both experimentally and numerically than behavior in turbulent flow. The results obtained in such studies are helpful when it comes to engineering applications. To demonstrate that, consider a water-based turbulently agitated liquid–liquid system. Let the size of the drop be comparable to the Kolmogorov length scale. The kinematic viscosity of the continuous phase is of the order of . The local energy dissipation rate in the impeller region (Davies, 1987) may be up to . Based on these parameters the Kolmogorov time scale is . Suppose the resulting distribution of drop radii is in the range a = (1–100) μm. Assume that turbulent eddies interacting with the drop create a shear rate of the order of . Then the range of drop Reynolds number defined as is from 0.01 to 100. This implies that even in fully-developed turbulence, drops experience interactions with eddies at low to moderate Reynolds numbers. Therefore, a study of binary systems in simple shear flow has direct relevance to complex turbulently flowing systems. One can, for example, check if the local energy dissipation rate is high enough to break drops of certain sizes and eventually obtain liquid–liquid dispersions with desired characteristics.
Starting with experiments performed by Taylor, 1932, Taylor, 1934, a wide range of studies has been carried out on drop deformation and breakup. These studies have been reviewed by Rallison, 1984, Stone, 1994, Cristini and Renardy, 2006. The “retractive end pinching” breakup mechanism was outlined by Bentley and Leal (1986). Marks (1998) investigated “elongative end pinching” by applying a strong shear to a single drop. Recent experiments have been performed by Zhao (2007) where a dilute emulsion was subjected to a simple shear flow. A map of drop breakup mechanisms in simple shear flow as a function of viscosity ratio and capillary number was presented.
A viscous drop under shear flow has also been intensively investigated by means of numerical simulations. Most of the numerical studies have been performed with the boundary integral method (Kennedy et al., 1994, Kwak and Pozrikidis, 1998, Cristini et al., 2003, Janssen and Anderson, 2007). The method has been successfully applied for drop deformation studies. However, the implementation of the boundary integral method for drop breakup and coalescence poses a major obstacle because it is very difficult to handle merging and folding interfaces: the interface points should be reconstructed, which requires significant logical programming and results in computational overhead (Li et al., 2000). The mathematical implication of the boundary integral method such as singularity of the free-space Green’s kernels is discussed by Pozrikidis (1992). A way to overcome this issue is suggested by Bazhlekov et al. (2004): a higher accuracy in the vicinity of the singular point is gained, however, the performance is about an order of magnitude slower compared to a standard surface integration. An alternative numerical technique widely used to investigate drop breakup is the volume-of-fluid (VOF) method. Numerical simulation of breakup of a viscous drop in simple shear flow was carried out by Li et al. (2000). The same technique has been applied by Renardy and Cristini, 2001b, Renardy et al., 2002, Khismatullin et al., 2003, Cristini and Renardy, 2006. The topological changes of the interface are treated more naturally compared to the boundary integral method. The VOF method has been generalized to three-dimensional cases. However, the reconstruction of the interface requires significant computational effort that increases with the number of drops involved.
A droplet in a quiescent fluid was investigated by Van der Sman and van der Graaf (2008) using a free energy lattice Boltzmann model (LBM). The authors further analyzed the numerical criteria for a correct description of emulsions and applied the model to study drop deformation and breakup. All simulated cases were two-dimensional.
Three-dimensional numerical simulations of the classical Taylor experiment on droplet deformation in a simple shear flow have been performed by Xi and Duncan (1999). The authors applied the lattice Boltzmann method in conjunction with the interface force model of Shan and Chen (1993). Good agreement with theoretical predictions was demonstrated for small deformations. The ability of the method to capture larger deformations and breakup events was also shown.
In the present study, the free energy lattice Boltzmann method originally proposed by Swift et al. (1996) is adopted to perform three-dimensional simulations of a single liquid drop suspended in another liquid under simple shear flow. The goal of the study is to check the capability of the method to capture the physics of drop deformation and breakup in a wide range of flow conditions: starting from near Stokes flow up to drop Reynolds numbers of 50 where inertia plays a significant role. Also the ability of the method to handle liquids with different viscosities is tested.
Diffuse interface numerical techniques require an explicit specification of the interface thickness which essentially is a numerical artifact. It is necessary to examine how this impacts the simulations, what parameters determine this additional degree of freedom, and what values of these parameters should be set for physically realistic results. In addition, it is important to outline the resolution that is sufficient to capture the physics of drop breakup while keeping a reasonable simulation time. To validate the numerical approach, its results are compared to existing experimental results and findings of numerical simulations using other methods. The present study can be considered as a development towards a numerical tool to investigate the behavior of drops in shear flow and as a verification and validation step for further applications in more complex flows. For instance, the developed code would be extended to perform Direct Numerical Simulations (DNS) of turbulent dispersion formation with hundreds of breaking and merging droplets.
The rest of the paper is organized as follows. The problem statement is outlined in Section 2. Section 3 contains the details of the numerical technique. The results of simulations are presented in Section 4. First, the choice of the numerical parameters that determine drop behavior in shear flow is discussed in Section 4.1, with additional details in Appendix A. Drop deformation and breakup in Stokes flow is presented in Section 4.2, the influence of inertia on drop deformation is shown in Section 4.3 and Section 4.4 presents the joint influence of viscosity ratio and inertia on drop deformation and breakup scenarios. Finally, conclusions are drawn in Section 5.
Section snippets
Problem statement
A liquid drop of dynamic viscosity is suspended in another liquid of viscosity . The ratio of drop viscosity to surrounding liquid viscosity is denoted as . The interfacial tension between the liquids is . The liquids are of equal density . At time , the drop is a sphere with radius a. The entire system undergoes simple shear flow between two parallel plates located a distance H apart (Fig. 1). The two plates translate in opposite directions with velocity so that the shear
Numerical method
In the present study, a diffuse interface method is used to simulate the behavior of a drop in shear flow. In diffuse interface (or phase field) methods (Jacqmin, 1999, Yue et al., 2004, Ding et al., 2007, Magaletti et al., 2013) the sharp interface between fluids is represented by a thin transition region with finite thickness where fluids may mix. At any given time, the state of the system is described by the order parameter of the phase field which is the relative concentration of the two
Simulations of a single drop under simple shear flow
A computer code for three-dimensional simulations is developed using Fortran 90 in both serial and parallel versions. The parallel code uses domain decomposition and MPI (Message Passing Interface). The simulation domain is decomposed into slabs in the x direction, one for each CPU. The number of CPUs used depends on the domain size, starting from one for low resolution drops and up to eight CPUs for the highest resolution drops. Depending on the drop size, the duration of the simulations
Conclusions
Numerical simulations of a single liquid drops suspended in another liquid and subjected to simple shear flow have been presented. The free energy lattice Boltzmann method was used to perform three-dimensional simulations of the binary systems in order to determine the drop deformation and breakup conditions. During this study the numerical tool has been implemented, verified and validated with available reference data.
The full physical description of the problem requires three physical
Acknowledgements
This research has been enabled by the use of computing resources provided by WestGrid and Compute/Calcul Canada. O.S. is grateful for the support of an Alexander Graham Bell Canada Graduate Scholarship from NSERC. A.E.K. would like to thank Schlumberger for financial support of the research.
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