DRAFT Journal of Mechanical Science and Technology 00 (2010) 0000~0000 www.springerlink.com/content/1738-494x
submitted manuscript under review
Numerical Investigation of Dynamic Behavior of a Compound Drop in Shear Flow Truong V. Vu1,*, Luyen V. Vu2, Binh D. Pham1 and Quan H. Luu1,* 1
School of Transportation Engineering, Hanoi University of Science and Technology, 01 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam 2
Mokpo National University, Chonnam 534-729, South Korea
(Manuscript Received 000 0, 2017; Revised 000 0, 2017; Accepted 000 0, 2017) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Abstract We present a numerical investigation of the deformation and breakup of a compound drop in shear flow. The numerical method used in this study is a two-dimensional front-tracking/finite difference technique for representing the interface separating two fluids by connected elements. The compound drop with the initially circular and concentric inner and outer fronts is placed at the center of a domain whose top and bottom boundaries move in the opposite direction. Because of the shear rate, the compound drop deforms and can break up into drops, depending on the flow conditions based on the Reynolds number Re, the Capillary number Ca and the interfacial tension ratio 21 of the outer to inner interfaces. We vary Re in the range of 0.1–3.16, Ca in the range of 0.05–0.6 and 21 in the range of 0.8–3.2 to reveal the transition from the non-breakup to breakup regimes. Numerical results indicate that the compound drop breaks up into drops when there's an increase in Re or Ca or a decrease in 21 beyond the corresponding critical values. We also propose a phase diagram of Ca versus Re that shows the region in which the compound drop changes from the deformation mode to the breakup mode. Keywords: Compound drop; Numerical investigation; Front-tracking; Breakup; Shear flow ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1. Introduction Compound drops that contain single or multiple smaller drops inside find applications in many fields of science and technology [1–5]. In comparison with simple drops (i.e., single-phase drops) [6], the hydrodynamics of compound drops is much more complex due to the presence of two or more interfaces with multi-fluids [7]. The compound drop can be formed through either compound jet breakup [8,9], or intense shearing in a mixer [10]. For applications in, for example, labon-chip systems [11] or biology engineering [12], the compound drop can experience deformation and/or breakup during movements. Accordingly, understanding of the compound droplet’s rheological behaviors (for example, deformation and breakup) in shear flow plays an important role not only in the fluid dynamics of the compound drop but also in its industrial applications. Here, we just focus on single-core compound drops that has only one single core inside [9,13]. Concerning the numerical works related to the compound drop dynamics in shear flow, a few studies have been carried [7,14,15]. Hua et al. [14] performed two- and three- dimensional computations, based on the immersed boundary method, to study the effects of the radius ratio of the inner to outer interfaces, the interfacial tension ratio of the inner to the outer, and the inner drop location on the deformation of the compound drop. However, the authors have not investigated how these parameters affect the breakup of the compound drop. †
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Luo et al. [15] used a front-tracking method to reveal the underlying mechanisms for the deformation of the compound drop in shear flow with the consideration of the effects of the radius ratio. However, breakup have not been considered in this work [15]. In consideration with breakup, Chen et al. [7] used a volume of fluid method to investigate the deformation and breakup behaviors of compound drops in shear flow. The phase diagrams of Ca, dynamic viscosity ratios and radius ratio were proposed. Four types of breakup modes via three mechanisms (i.e., necking, end pinching, and capillary instability) were introduced. However, the effects of some other parameters such as the Reynolds number and the interfacial tension ratio have not been considered. Finishing this gap is the main purpose of the present study. In the present study, we present a numerical investigation of the compound droplet deformation and breakup in shear flow. The method used is a two-dimensional front tracking/finite difference technique [9,16] to track the evolution and breakup of the compound drop interface. This method has been widely used in multiphase and multi-fluid flows [17,18]. We examine the effects of some non-dimensional parameters, such as the Reynolds number, the Capillary number, and the ratio of the interfacial tensions of the outer to the inner, on the transition from non-breakup to breakup. This investigation is important both academically and industrially, and is still lacking in the literature.
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Fig. 1. Compound drop deformation in shear flow.
2. Mathematical Formulation and Numerical Parameters Fig. 1 shows our investigated problem with a concentric circular compound drop immersed in an outer fluid (denoted by 3) and at the center of the domain. The inner and middle fluids are denoted by 1 and 2, respectively. The radii of the inner and outer interfaces of the compound drop are R1 and R2, respectively. We assume that the fluid properties of each fluid are constant, and the gravity effect is neglected. All fluids are assumed laminar, immiscible, incompressible and Newtonian. We treat all fluids as one fluid with variable properties such as density , and viscosity . In terms of the one-fluid representation, the governing equations are given as follows: u uu p (u uT ) t (1) n f (x x f )dS
f
u 0 (2) Here, u = (u, v) is the velocity vector, p is the pressure. The superscript T denotes the transpose. At the interfaces, denoted by f, is the interfacial tension coefficient. Accordingly, 1 and 2 are the interfacial tension coefficients of the inner and outer interfaces, respectively. is twice the mean curvature, and nf is the unit normal vector to the interface. The Dirac delta function δ(x − xf) is zero everywhere except at the interface xf. These above-mentioned equations are solved by the fronttracking method [9,16] on a staggered grid with second order accuracy in time and space. The interfaces are represented by connected points that move with the velocities interpolated from the background grid. The points on the inner interface is used to reconstruct an indicator function Ii that has a value of 1.0 within the interface and 0.0 other. A similar indicator function Io is built from the outer interface points. Thereby, the value of the fluid properties at every location in the domain is given as 1Ii 2 1 Ii I o 3I o (3) where stands for and
Fig. 2. Grid refinement study: compound drop profiles at = 9.0 with Re = 0.8 and Ca = 0.1 for different grid resolutions.
Fig. 3. Comparison of the compound drop deformation in shear flow between the present calculation with Hua et al. [14]. The parameters are Re = 0.25, Ca = 0.125 and 21 = 1.
The computation domain is shown in Fig. 1 with the periodic boundary conditions on the left and right. At the top and bottom the fluid moves in the opposite directions to induce a shear rate 2U H . Accordingly, the dynamics of the problem is governed by the Reynolds number Re, the Capillary number Ca, the radius ratio R21, the density ratios 21 and 31, the viscosity ratios 21 and 31, and the interfacial tension ratio
21
1 R12 R R , Ca 1 1 , R21 2 1 1 R1 2 3 2 3 21 , 31 , 21 , 31 , 21 2 1 1 1 1 1 The dimensionless time is defined as t . Re
(4) (5)
In this study, we focus on the effects of the Reynolds and Capillary numbers and the interfacial tension, and thus other parameters are kept constant, i.e., R21 = 2, and 21 = 31 = 21 = 31 = 1. Re and Ca are varied in the ranges of 0.1–3.16 and 0.05–0.6, respectively with 21 = 0.8–3.2. The values of these parameters correspond to compound drops of such materials as water and silicon, with diameter in the order of a few hundreds micrometer.
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3. Grid study and method validation To verify the numerical method, we perform a grid refinement study with Re = 0.8, Ca = 0.1, and 21 = 1.0. Fig. 2 shows the compound drop profiles at = 3 (i.e., at almost steady state) for five grid resolutions (64 16, 128 32, 256 64, 512 128, and 1024 256) with a computational domain size W H = 24d1 6d1, where d1 = 2R1. The result obtained from the 512 128 grid agrees very well with that obtained from the 1024 256 grid, while the coarser grids yield some differences. Accordingly, we use 128 grid points in the vertical direction with H/d1 = 6 for the rest of the computations presented in this paper. The number of grid points in the horizontal direction depends on the length W of the computational domain, which is varied with the flow condition. To validate the numerical method applied to simulate the deformation and breakup of the compound drop, we have compared our result to that reported in [14]. Hua et al. [14] performed the numerical calculations using the immersed boundary method proposed by Peskin [19]. Our computational
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domain size is the same as in [14] with Re = 0.25, Ca = 0.125 and 21 = 1. Fig. 3 compares the compound drop profile at steady state from our codes with that reported in Hua et al. [14], yielding complete agreement. Other validations of the front-tracking method for drops and jets can be found elsewhere, e.g. [9,16,20].
4. Results and discussion Fig. 4 shows the temporal evolution of the compound drop with Re = 0.1, Ca = 0.3 and 21 = 1. At the initial stage of deformation, i.e., at = 0.5, the compound drop slightly deforms with nearly uniform distribution of the pressure in each fluid, as shown in Fig. 4a. As time progresses, the shear flow deforms the compound drop interfaces, leading to nonuniformity in the pressure field (Fig. 4b). As comparison to other regions the pressure is higher at the two farthest ends within each interface. This high pressure is to balance with high curvatures there. Thereby, the compound drop keeps
Fig. 4. Compound drop evolution with the normalized pressure and velocity fields with Re = 0.1, Ca = 0.3 and 21 = 1. The velocity is normalized by U.
Fig. 5. Compound drop deformation (at steady state) for Re = 0.2 ( = 10), 0.4 ( = 10), 0.8 ( = 30), and 1.6 ( = 35), and compound drop breakup for Re = 3.2 ( = 95). The other parameters are Ca = 0.1 and 21 = 1.
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such a deformed shape at its steady state [15]. Next, we consider the effects of some parameters on the deformation and transition to the breakup of the compound drop in shear flow. (a) Effect of the Reynolds number Re Fig. 5 shows the effect of Re on the deformation and breakup of the compound drop with Ca = 0.1 and 21 = 1. At Re = 0.2, the shear flow deforms the outer interface of the compound drop. Increasing the Reynolds number from 0.2 to 1.6 corresponding to increasing the shear rate results in more deformation of the compound drop. An interesting point here is that varying Re in the range of 0.2 to 1.6 has a minor effect on the inner interface. This is understandable since the deformation of the inner interface is mostly caused by vorticial flow, within the outer interface, induced by balance between the interfacial and viscous forces [15]. However, when the Reynolds number increases from 1.6 to 3.2, the shear rate causes the outer interface to break up into smaller drops [14]. After breakup, the rest of the outer interface that
still encapsules the inner drop retracts and forms a smaller compound drop, as shown in the last frame of Fig. 6. (b) Effect of the Capillary number Ca Fig. 6 shows the effect of Ca on the deformation and breakup of the compound drop with Re = 0.1 and 21 = 1. At Ca = 0.1, the outer interface of the compound drop slightly deforms due to the shear rate. Since the Capillary number is the ratio of the viscous force to the interfacial tension force, increasing Ca corresponds to decreases the force holding the drop in a spherical shape. Accordingly, the drop deforms more as Ca increases to a higher value (i.e., Ca = 0.2 or 0.3). At Ca = 0.5, the interfacial tension force is so low that the outer interface breaks up into smaller drops in an end-pinching mode [7]. (c) Effect of the interfacial tension ratio Fig. 7 shows the effect of the interfacial tension ratio 21 on the deformation and breakup of the compound drop with Re =
Fig. 6. Compound drop deformation (at steady state) for Ca = 0.1 ( = 6), 0.2 ( = 10), and 0.3 ( = 30), and drop breakup for Ca = 0.5 ( = 100). The other parameters are Re = 0.1 and 21 = 1.
Fig. 7. Compound drop deformation (at steady state) for 21 = 3.2 ( = 10), 1.6 ( = 10), and 1.0 ( = 10), and breakup for 21 = 0.8 ( = 100). The other parameters are Re = 0.1 and Ca = 0.3.
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value 0.8 (with Re = 0.1 and Ca = 0.3) results in the compound drop breakup. Moreover, a phase diagram of Ca (varied in the range of 0.05 – 0.6) versus Re (varied in the range of 0.1 – 3.16) is also proposed to indicate the transition from the non-breakup to breakup regions, in which high Re and high Ca enhance the drop breakup.
Acknowledgment This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2014.25.
References Fig. 8. Phase diagram of Ca versus Re with 21 = 1.0.
0.1 and Ca = 0.3. The interfacial tension force tends to hold the drop in a spherical shape while the shear force enhances drop deformation. Accordingly, at a high interfacial tension ratio, i.e., 21 = 3.2, the compound drop just slightly deforms. Decreasing the value of 21 corresponding to decreasing the force induced by the interfacial tension acting on the outer front results in more deformation. As a result, at a low interfacial tension ratio, i.e., 21 = 0.8, the shear force dominates over the interfacial tension force and makes the drop break up into simple drops with a smaller compound drop at the center, as shown in the last frame of Fig. 7. (d) Phase diagram of Ca versus Re As previously discussed, the transition from non-breakup to breakup of the compound drop in shear flow is strongly affected by the Capillary and Reynolds numbers. Increasing Ca or Re enhances the breakup of the compound drop outer interface. This is clearly shown in Fig. 8 where a phase diagram of Ca (varied in the range of 0.05–0.6) versus Re (at 0. 1, 0.316, 1.0, and 3.16) with 21 = 1.0. This figure indicates that the breakup occurs at Ca 0.1 for Re = 3.16. Decreasing the value of Re to 0.1 the breakup region is narrow, i.e., at Ca 0.4.
5. Conclusion We have presented a numerical investigation of the compound drop deformation and breakup by a twodimensional front-tracking/finite difference method. The method is verified and validated through the grid refinement study and comparison with the numerical prediction for the compound drop deformation reported in [14]. Various parameters including the Reynolds number Re, the Capillary number Ca and the interfacial tension ratio of the outer to the inner 21 are varied to reveal their effects on the transition from the deformation to breakup regimes. The numerical results show that starting from non-breakup the compound drop can break up into drops when Re increases to a value beyond 0.316 for Ca = 0.3 and 21 = 1.0, or Ca increases to a value beyond 0.2 for Re = 21 = 1.0. In addition, decreasing 21 to a
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Truong V. Vu received the B.E (2007) degree in mechanical engineering from Hanoi University of Science and Technology (HUST) in Vietnam, the M.E (2010) and Ph.D (2013) degrees in integrated science and engineering from Ritsumeikan University in Japan. He is a lecturer, School of Transportation Engineering, HUST. Current interests include multiphase and free surface flows, and numerical methods.
