Elsevier

Chemical Engineering Science

Volume 204, 31 August 2019, Pages 27-34
Chemical Engineering Science

A Taylor analogy model for droplet dynamics in planar extensional flow

https://doi.org/10.1016/j.ces.2019.04.015Get rights and content

Highlights:

  • A Taylor analogy model has been developed to predict the droplet deformation.

  • The proposed model has been verified by present simulation and previous experiments.

  • The effects of capillary number and viscosity ratio have been studied.

  • The viscosity ratio affects transient and steady behaviors of droplet deformation.

Abstract

The dynamics of a droplet suspended in a medium fluid is determined by the hydrodynamic forces and surface tension force, and the Taylor analogy had been successfully employed to predict droplet deformation and breakup in a spray. This paper aims to extend the Taylor analogy for prediction of droplet dynamics in planar extensional flow which has great significance in droplet-based microfluidic systems. Performance of the proposed model is compared with the three-dimensional numerical simulation results over a wide range of capillary number and viscosity ratio. Experimental data available in the literature is also compared with the prediction results for verification purpose. The proposed model could describe both the time scale and the magnitude of droplet deformation accurately.

Introduction

Microfluidics offers advantages of quick analysis, small volume consumption of samples and reagents, and it has been employed for development of Lab-on-a-Chip (LoC) and micro total analysis systems (microTAS) for applications in biomedical, pharmaceutical and environmental fields (Castillo-León and Svendsen, 2015). The flow regime in a microfluidic system is characterized by low Reynolds number (Re), so one can control the flow for precise manipulation of chemicals and reagents (Park and Anderson, 2012, Brimmo and Qasaimeh, 2017). In addition, surface tension becomes more dominant in the microfluidic systems, which gives rise to the development of droplet-based microfluidics (Baroud et al., 2010, Seemann et al., 2012, Liu et al., 2014, Liu and Zhang, 2015). In this droplet-based microfluidics, chemicals and reagents can be controlled and manipulated more precisely in a digitized manner by using droplets (Liu et al., 2012, Yan et al., 2012, Fu and Ma, 2015).

In droplet-based microfluidic systems, extensional flow is commonly used to generate, trap, mix and manipulate droplets precisely in a digitized manner (Brimmo and Qasaimeh, 2017). Lee et al. (2007) developed a microfluidic device producing planar extensional flow which can be used for many applications. Hu and Lips (2001) could obtain the droplet viscosity by measuring the drop deformation in planar extensional flow. Similarly, there are also many studies where cells or vesicles are subjected to the extensional flow. For instance, Tanyeri et al. (submitted for publication) utilized planar extensional flow to trap and manipulate cells for long time scales, and Guillou et al. (2016) measured cellular mechanical behavior in a similar manner. Dahl et al. (2016) studied dynamics of vesicles in planar extensional flow by using a microfluidic cross-slot device. Bae et al. (2016) used planar extensional flow to quantitatively assess the mechanical damage of cells occurring in bioreactors.

In order to understand the details of droplet dynamics in extensional flow, there have been many experimental, theoretical and numerical studies. A pioneering work on the droplet deformation and breakup was first given by Taylor (1934) who could generate simple shear flow and elongational flow by using a parallel band apparatus and four roller apparatus, respectively. Following the pioneering work, there have been many similar experimentations to understand more details of the droplet dynamics for a wider range of flow conditions (Giesekus, 1962, Fuller and Leal, 1981, Rumscheidt and Mason, 1961, Bentley and Leal, 1986). Also, non-Newtonian effects of the medium or the droplet had been investigated since practical materials in chemical and polymer processing commonly have complicated rheological properties (Ha and Leal, 2001, Hsu and Leal, 2009). Three-dimensional drop shapes at steady and transient states in planar extensional flow were experimentally studied by Hu and Lips (2003).

As far as theoretical study is concerned, Taylor (1934) proposed a small deformation theory which could predict the droplet deformation at steady state in low capillary number regime. A theoretical model for transient droplet dynamics was first proposed by Cox (1969) and then improved by Barthès-Biesel and Acrivos (1973), where the droplet surface profile is expanded in powers of capillary number. Especially for large deformation of the droplet, Acrivos and Lo (1978) developed another approximate theory to predict the breakup of a slender droplet. Maffettone and Minale (1998) proposed a phenomenological model to describe droplet deformation in arbitrary flow.

Due to the development of computational fluid dynamics, the droplet dynamics could be better understood recently. For example, Ramaswamy and Leal (1999) numerically studied the deformation of a non-Newtonian droplet suspended in a Newtonian medium. The effects of rheological properties of the droplet or medium on the deformation and breakup of the droplet in simple shear flow were numerically studied by Wang et al. (2017). Yu et al. (2016) studied three-dimensional asymmetric breakup in axisymmetric extensional flow by using a commercial software. Liu et al (2018) developed a hybrid numerical method to study the effect of surfactant on the droplet dynamics.

There has been another approach to describe the droplet dynamics by considering analogy with a damped spring-mass system. Particularly, this approach has been successfully employed to predict the breakup of droplets in sprays of high Re regime, and the model is commonly termed as a Taylor Analogy Breakup (TAB) model (O’Rourke and Amsden, 1987). This TAB model has advantages in terms of simplicity and accuracy, so it has been used in many applications (Basu and Cetegen, 2008, Dos Santos and Le Moyne, 2011, Turner et al., 2012, Nishad et al., 2018, Aramendia et al., 2018).

In this study, we propose to use the Taylor analogy model (Taylor, 1963) to predict the droplet deformation in extensional flow at low Re regime, with the microfluidic applications in mind. In contrast to the TAB model, a viscous drag force is introduced as an external force term, and additional considerations have been made to capture the droplet dynamics for a wide range of flow conditions. In addition, an extensive computational simulation has been performed to examine the accuracy of the proposed model.

In the next section, a problem description on the droplet deformation in a planar extensional flow is given. In Section 3, the Taylor analogy is briefly presented, and details of the proposed model are described. Three-dimensional computational model is explained in Section 4. The results and discussions are given in Section 5. Finally, a summary and conclusions are made in Section 6.

Section snippets

Problem description

A droplet of radius R suspended in a medium fluid undergoing planar extensional flow is shown in Fig. 1(a). A planar extensional flow has the velocity field given by Equation (1) where ε̇ is extension rate and vX, vY and vZ are the velocity components in X, Y and Z directions, respectively.vX=ε̇X,vY=-ε̇Y,vZ=0

A magnified view of the droplet is shown in Fig. 1(b), where the geometry of deformed droplet is measured in terms of L (half length of droplet in X-direction) and B (half breadth of

Taylor analogy modeling

Taylor analogy was first proposed to study droplet deformation in a high-speed air stream (Taylor, 1963, O’Rourke and Amsden, 1987). According to this model, a spring-mass system is analogous to the droplet where the spring force represents the surface tension force and the pressure drag force on the droplet was introduced as an external force. Later, a damping component was introduced to describe the viscous behavior of the droplet and the damped spring-mass system was used to predict droplet

Computational model

For the computational study on the droplet deformation, a computational domain of cube shape is used in the present study. By considering the symmetry of the problem, a one-eighth of the full three-dimensional model is employed as shown in Fig. 2. In this study, the droplet radius is 0.167 times the domain edge length, so that boundary effect of the computational domain on droplet dynamics is negligible (Ioannou et al., 2016). The computational domain is discretized by 100 × 100 × 100 uniform

Results and discussion

An overview of the droplet deformation in planar extensional flow is first given by reviewing available data from the previous studies (Bentley and Leal, 1986, Hsu and Leal, 2009). In addition, the computational results of this study are provided for verification purpose. Then, the performance of the proposed model for prediction of the droplet deformation is presented in terms of steady and transient behaviors of the droplet.

Conclusion

This study has extended Taylor analogy model for predicting the droplet deformation under planar extensional flow at low Reynolds number. Particularly, appropriate models for the external force and the damping coefficient have been proposed. As a result, the effects of the capillary number and the viscosity ratio on the droplet deformation could be taken into account. Also, an extensive numerical simulation has been performed to observe the droplet dynamics over a wide range of flow conditions,

Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

Authors would like to thank the financial support from Basic Science Research Program through the National Research Foundation of Korea (NRF) Grant funded by the Ministry of Science, ICT & Future Planning (NRF-2017R1E1A1A03070443).

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