Computational analysis of binary collisions of shear-thinning droplets

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Abstract

Scale-reduced models of transport processes and reactive mixing in sprays require improved closure laws, taking into account the characteristic features of elementary spray processes. The present paper investigates binary droplet collisions as such an elementary process. In the case of shear-thinning liquids considered here, this requires a profound understanding of the influence of the non-Newtonian fluid rheology on the flow inside the colliding drops and the collision complex dynamics. We employ direct numerical simulations based on the Volume-of-Fluid method to study these collisions. The results give a quantitative prediction of the resulting droplet collision diameter as well as a qualitative prediction of the complete time evolution. During collisions, extremely thin fluid lamellae appear inside the expanding complex. These have to be accounted for in a physically sound simulation and we apply a stabilization of the lamella to keep it from rupturing. The simulations show that in all considered cases an effective constant viscosity can be found a posteriori which leads to the same collision dynamics. But this effective viscosity is neither the mean nor the minimum viscosity.

Highlights

► Numerical stabilization of thin fluid lamella allows for DNS of head-on collisions. ► VOF-simulations reveal an effective viscosity for shear thinning droplet collisions. ► Decomposition of total energy in DNS allows for classification of collision stages.

Introduction

Droplet collisions are of importance in a variety of practical applications comprising dispersed two-phase flow and a large amount of studies have been devoted to the description of these processes; see e.g., [1], [10], [13], [19], [24], [25]. The background of our research is the prediction of properties of particulate products formed in spray processes. Such products are solids in powder form, manufactured in the chemical, pharmaceutical or medical technology or related industries. The process properties depend strongly on the underlying spray pattern and in applications dealing with polymers, non-Newtonian rheological behavior comes into focus. To gain a more thorough understanding of the elementary sub-processes inside a spray, we have extended the in-house Volume-of-Fluid (VOF) code FS3D to account for shear thinning flow behavior and performed direct numerical simulations of binary droplet collisions.

Collisions of droplets can generally be classified into four main categories: bouncing collisions, where the drops rebound from each other without ever coalescing; coalescing collisions, where two drops merge into one; separating collisions, where the drops temporarily form a single collision complex but then break up again; and splashing, where the impact is so strong that the drops break up into several smaller droplets [13], [19]. The outcome of a droplet collision mainly depends on the droplets’ kinetic energy, the surface tension and the displacement of the drops perpendicular to the flight direction. In the investigation of sprays, very small droplets appear. While experiments can show the evolution of the collision complex geometry, the flow processes inside the drops can hardly be assessed by means of the observed surface evolution [10], [24]. Numerical investigations of droplet collisions give rise to a much more detailed study of the local phenomena including the flow fields inside the droplets, but are so far mainly limited to Newtonian fluids.

In the present work, binary collisions of droplets of non-Newtonian liquids are investigated and compared to Newtonian flow rheology. The non-Newtonian liquid is a polymeric solution and is assumed to be of generalized Newtonian type, i.e. with negligible elastic effects but shear-thinning rheological behavior. Preliminary studies of this system where already reported in [18], where the computations were limited to small drop velocities. In the present work, the droplets have a diameter of about 300 μm and a relative velocity of 5 m/s. The high kinetic energy of the droplets leads to a large collision complex diameter, thus showing a significant difference to droplet collisions of Newtonian droplets of the same zero shear rate viscosity. Hence an approximation of the non-Newtonian behavior by a Newtonian viscosity is not straight forward because an effective viscosity is not a priori known. Somewhat surprisingly, it turns out that the collision of a Newtonian droplet pair with carefully chosen (constant) viscosity can reproduce the non-Newtonian collision process in a very accurate manner, both regarding shape and collision complex dynamics. This can in fact be explained from the numerical simulations, since only during an initial phase of the collision the viscous forces are important and the shear rate dependent viscosity found in this stage is the effective viscosity for the dynamics.

A major difficulty arises concerning the numerical simulation of the thin lamella structure which arises during the extension of the collision complex. If the lamella ruptures during the collision because of insufficient resolution, the result is a severe deterioration of the collision physics. These difficulties were first described in [29] and led to an artificial rupture of the lamella there. In contrast to the results of [29], a stabilization of the lamella in the computation of the free surface flow could be achieved in the present work which does not affect the mass and momentum balance but leads to a major improvement in the computation of the collision complex.

In the first section of this paper, the underlying numerical methods of the VOF-based direct numerical simulation (DNS) and especially the incorporation of shear-thinning viscosities are described. In the second section, a verification of the numerical scheme and an experimental validation of the simulation method are described, using experiments with non-Newtonian droplet collisions of [18]. In the third section, the distribution of the viscosity inside the droplet is investigated and compared with the prevailing flow phenomena. Finally, the non-Newtonian droplet collisions are compared to simulations with effective Newtonian fluid in the fourth section.

Section snippets

Mathematical modeling

The numerical simulations are performed with the two-phase Navier–Stokes solver FS3D (Free Surface 3D), which was originally developed at the ITLR, University of Stuttgart [23]. FS3D is based on the Volume of Fluid (VOF) method [16]. The code FS3D is specifically designed for the investigation of two-phase flows [22] and has been extended both at the ITLR and in the Mathematical Modeling and Analysis (MMA) group at the Center of Smart Interfaces, Technical University of Darmstadt. A few

Validation data from collision experiments

For the validation of our numerical simulations, experimental results on binary droplet collisions from [18] are used. The shear thinning liquid is an aqueous solution of 2.8 wt% CMC sodium salt with a density of 1000g/m3and a surface tension coefficient of 70×10-3N/m. CMC can show Newtonian, shear thinning and viscoelastic behavior. CMC solutions with concentrations between 0.2 and 3.0 wt% display shear thinning behavior with an increasing shear thinning effect for higher concentrations. Above

Setup

The simulations were performed on 16 Intel XEON cores with 4GB main memory per core using Cartesian grids having about 8.4 million cells. Based on the symmetry of a binary droplet collision of two equal sized droplets, three symmetry planes were used in the simulations to reduce the computational effort. The domain size is 1.4·10-2×1.4·10-2×7·10-3 m, the resolution is 256 × 256 × 128 cells. The number of grid cells in one direction is a power of two in order to maximize the efficiency of the

Results and discussion

The comparison of the time evolution between the DNS and the experimental results from [18] show that the numerical simulation nearly captures the maximum collision complex diameter which is also attained at the correct time, but shows deviations in the contraction phase where the retraction of the collision complex seems to be faster than in the experiments. On the other hand, the range of uncertainties in the experimental measurements given in [18] is ±10% for the initial diameter D0.

Conclusions

Given that no rupture of the collision lamella occurs in the experiments, the numerical stabilization introduced above is an appropriate means to capture the collision dynamics with significantly reduced numerical effort. The stabilization on a symmetry plane gives reliable results for a lamella thickness even below 1 μm. Thereby, the lamella is resolved with 1/3 grid cell without observing any limitations at this size. In the case of an off-center collision of droplets with different diameter,

Acknowledgements

The authors gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG) within the DFG-priority program “Process Sprays” (SPP 1423). We also thank Dr.-Ing. N. Roth from the Institute of Aerospace Thermodynamics (ITLR) at the University of Stuttgart for helpful discussions concerning flow phenomena inside colliding droplets. Finally, we thank the anonymous referees for their helpful criticism.

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