Dynamics of a contracting fluid compound filament with a variable density ratio

Nội dung text: Dynamics of a contracting fluid compound filament with a variable density ratio

1. Science & Technology Development Journal, 24(2):1909-1917 Open Access Full Text Article Research Article Dynamics of a contracting fluid compound filament with a variable density ratio Vinh T. Nguyen1,2, Truong V. Vu1,*, Phan H. Nguyen2, Nang X. Ho1, Binh D. Pham1, Hoe D. Nguyen1, Hung V. Vu1 ABSTRACT Introduction: Compound fluid filaments appear in many applications, e.g., drug delivery andpro- cessing or microfluidic systems. This paper focuses on the numerical simulation of an incompress- Use your smartphone to scan this ible, immiscible, and Newtonian fluid for the contraction process of a fluid compound filament by QR code and download this article solving the Navier-Stokes equations. The front-tracking method is used to solve this problem, which uses connected segments (Lagrangian grid) that move on a fixed grid (Eulerian grid) to represent the interface between the liquids. Methods: The interface points are advected by the velocity in- terpolated from those of the fixed grid using the area weighting function. The coordinates ofthe interface points are used to construct the indicators specifying the different fluids and compute the interfacial tension force. Results: The simulation results show that under the effects of the inter- facial tension, the capsule-shaped filament can transform into a spherical compound droplet (i.e., non-breakup) or can break up into smaller spherical compound and simple droplets (i.e., breakup). When the density ratio of the outer to middle fluids increases, the filament changes from non- breakup to breakup upon contraction. Conclusion: Increasing the density ratio enhances the breakup of the compound filament during contraction. The breakup is also promoted by increasing the initial length of the filament. Key words: Compound fluid filament, compound droplet, front-tracking 1Faculty of Vehicle and Energy Engineering, Phenikaa University, Hanoi, Vietnam INTRODUCTION filament when it contracts in an initially passive envi- 2 ronment undergoing the influences of the fluid prop- Graduate University of Science and Nowadays, compound droplets have been used in Technology, Vietnam Academy of various applications, such as drug distribution, 1 mi- erties via numerical simulations. Two considered pa- Science and Technology, 18 Hoang Quoc 2 3 rameters for the deformation and breakup of fluid Viet, Cau Giay, Hanoi, Vietnam crofluidic system, food processing technology, etc. During the breakup of the droplets, the fluid filaments compound filament include the density ratio between Correspondence can be formed with primary droplets. The contraction the outer fluid and the middle fluid and the initial as- Truong V. Vu, Faculty of Vehicle and of these filaments creates more droplets. The breakup pect ratio of the filament. This is a new problem that Energy Engineering, Phenikaa has not been explored in the literature. The method University, Hanoi, Vietnam processes of the single fluid filament have been inves- tigated for a long time. Thus, its knowledge has be- used in this work is a front-tracking method for multi- Email: 4 fluid and multi-phase flows. 10 truong.vuvan@phenikaa-uni.edu.vn come mature. Concerning compound droplets, many studies have This paper is structured as follows: the introduction History section gives an overview of research papers related • Received: 2021-02-05 been done, such as numerical simulation of dynami- 5,6 • Accepted: 2021-05-06 cal behaviors of compound droplets, the collision to fluid compound filaments, the numerical model • Published: 2021-05-12 of compound droplets, 7,8 and the deformation and and method section gives the mathematical formu- 9 lation, the numerical method, and the dimensionless DOI : 10.32508/stdj.v24i2.2515 breakup of compound droplets in shear flow. The initial conditions of the problem are significantly im- parameters governing the problem. The result section portant, depending on the surface tension of the in- presents some typical results. The discussion section terface separating different fluids, the properties of the provides discussion about the numerical results. The fluids such as viscosity and density. conclusion and future development are presented in Copyright Unlike compound droplets, compound filaments the conclusion section. © VNU-HCM Press. This is an open- have not been considered so far. Accordingly, nu- access article distributed under the NUMERICAL MODEL AND METHOD terms of the Creative Commons merical studies of a contracting compound filament Attribution 4.0 International license. still lack in the literature. Thus, to provide such lack- The entire physical domain of an axisymmetric com- ing knowledge, the present study aims to enhance un- pound fluid filament is shown in Figure 1a. To save derstanding of dynamical behaviors of the compound computation time, we consider the upper half of the Cite this article : Nguyen V T, Vu T V, Nguyen P H, Ho N X, Pham B D, Nguyen H D, Vu H V. Dynamics of a contracting fluid compound filament with a variable density ratio. Sci. Tech. Dev. J.; 24(2):1909-1917. 1909
2. Science & Technology Development Journal, 24(2):1909-1917 fluid compound filament and use the axisymmet- To solve the problem, we use the front-tracking with ric configuration with a W×H domain and proper finite difference approximations. A finite number of boundary conditions, as illustrated in Figure 1b. The Lagrangian grid points is used to represent the inter- cylindrical coordinate system, in which r and z de- face among different fluids. These Lagrangian points note the radial and axial directions, is used for analy- are laid on a fixed staggered grid, i.e., Eulerian back- sis. The following initial conditions are assumed: Lo, ground grid. Integrating the equation below to deter- Li is the lengths of one-half of the outer and inner fluid mine the positions of the discrete interface points (xf) filaments, respectively; Ro and Ri are the radius of the moving with the velocity of Vn. Vn is determined by outer and inner fluid filaments, respectively. The vis- interpolation from the velocities of the four nearest cosity and density of the outer, middle, and inner flu- background grid points. ids are (µo, ro), (µm, rm), and (µi, ri), respectively. Here, gravity is ignored as it plays a very small role. dx f = n f Vn (4) These fluids are assumed to be incompressible, immis- dt cible, and Newtonian. Hence, the interfacial tension Instead of solving the density directly, we first move coefficient at the interface between the inner and mid- the interface between the different fluids and recon- σ dle fluids is i, and that at the interface between the struct indicator functions from its position. Using σ outer and middle fluids is o. these indicator functions is to specify the material It is assumed in this paper that the material properties properties at every location in the domain. 12 In par- are constant in each fluid. With numerical computa- ticular, we here build two indicator functions Ii and tion, three fluids are considered one fluid with viscos- Io, from the inner and outer interfaces, respectively. ity µ and density ρ changing when traveling through Ii equals zero in the inner fluid and one in the other the interface. The Navier – Stokes equations are valid fluids. Similarly, Io equals one in the outer fluid and for fluid domains, and a set of equations can be de- zero in the other fluids. 9,11,13 Accordingly, for exam- fined for the entire domain, provided that the viscos- ple, the density at every location is given as ity and density jumps are computed accurately and the interfacial tension is included. Therefore, the Navier ρ = ρi (1 − Ii) + [ρoIo + (1 − Io)ρm]Ii (5) – Stokes equations are represented as below: 10,11 The positions of these points are also used to calculate ∂ ρ [ ( )] the interfacial tension force. To do so, we first com- ( u) + ∇.ρuu = −∇p + ∇. µ ∇u + ∇uT (1) ∂t ∫ ( ) pute the force on each front element connecting two + σκδ x − x f n f dS f adjacent points using the tangents at these points, and Where u is the velocity vector, p is the pressure, k is we then sum these forces on the entire interface ele- twice mean curvature, t is time, and superscript T is ments. The interfacial tension force is transferred to the transpose. The interfacial tension force is the force the background grid using the area weighting func- caused by the interfacial tension between two fluids tion. 10 The continuity and momentum equations are σ is the interfacial tension coefficient, nf denotes the approximated by the second-order centered finite dif- unit normal vector to the interface, f denotes the in- ference method for spatial derivatives. Time integra- terface of the compound filament, s denotes the arc tion is solved with an explicit, second-order predictor- length. d is the Dirac delta function that has a value corrector scheme. of zero at all points except for the interface point xf In this study, the middle fluid is chosen as the based where the function is equal to one. The fluids are fluid because such choice is to better understand and not mixed, and the material properties are constant refer to simple filaments 4 when, for example, ignor- within each fluid. Hence, the equations of state for ing the presence of the inner fluid (i.e., inner fila- density and viscosity are: ment). The scaling length is the radius of the outer Dρ Dµ √filament Lc = Ro, and the reference time is tc = = 0; = 0; (2) ρ 3 σ Dt Dt mRo/ o. Accordingly, the reference velocity is Ro Uc = . As a result, the problem dynamics is in- Where (D/Dt) = (∂/∂t) + u.∇. Because this prob- tc fluenced by dimensionless parameters: Oh denot- lem is investigated with the assumption of incom- ing the Ohnesorge number, Ar and Ar denoting the pressible fluids, the continuity equation is represented i o initial aspect ratios of the inner and outer filaments, by: Rio denotes radius ratio between inner and outer fil- ∇.u = 0 (3) aments; rom and rim denoting the density ratios, µom 1910
3. Science & Technology Development Journal, 24(2):1909-1917 Figure 1: A compound filament contracting in an initially passive environment. (a) The entire physical domain and (b) the computational domain with an axisymmetric configuration. and µim denoting the viscosity ratios, and σio denot- a higher number of grid points in the axial direction ing interfacial tension ratio between inner and outer (e.g., 192×1538). filaments. To validate the method, we here compare our com- putational result with that computed by Notz and µm Lo Ri Oh = √ ; Aro = ; Rio = ; Basaran 4 for the evolution of the shape upon contrac- ρ R σ R R ρ m o o ρ o µ o ρ o ρ i µ o tion of a simple filament with Oh = 1.0, Aro= 15.0. om = ρ ; im = ρ ; om = µ ; (6) µm σ m m Other parameters for our computations include Ari= µ i σ i ρ µ ρ µ σ im = µ ; io = σ 0.0, om = om = 0.01, om = im = io = 0.0 and Rio m o = 0.0 (i.e., simple filament). The comparison is shown The dimensionless time is in Figure 2. This figure indicates that our prediction is very well agreed with that computed by Notz and τ = t/tc (7) Basaran. This comparison confirms our method ac- curacy. RESULT Because there are many parameters related to the de- formation dynamics and the breakup of the filament, × In this study, a typical grid resolution of 192 576 in this study, we pay more attention to the density ra- × × grid for a domain W H = 4Ro 12Ro is used in the tio between the outer and the middle fluids (r om) and simulation model to perform Aro = 10. Such a grid aspect ratio Aro. Therefore, other dimensionless pa- solution is selected through a grid refinement study rameters are fixed in calculations: Oh = 0.1, ρim = µom × (not shown here), and it is found that the 192 576 = µim = σio = 1, Rio = 0.6. grid gives results without variance in the compound Figure 3 compares the velocity and pressure fields of fluid filament shape as compared to those obtained the fluid filament corresponding to two ratios ρom = using finer grids. A higher Aro (e.g., Aro = 30) re- 0.8 (left) and ρom = 1.6 (right). At t = 10.08, the in- quires a larger domain (e.g., W×H = 4Ro×32Ro) with ner fluid filament tends to break up into droplets in 1911
4. Science & Technology Development Journal, 24(2):1909-1917 the case of ρom = 1.6 because the pressure difference between the inner and outer fluids is greater than that zomax − romax ρ To = (9) in the case of om = 0.8, and the velocity field at the zomax + romax neck of ρom = 1.6 has an inward direction while the Where T and T are the deformation parameter of the velocity field of rom = 0.8 has an outward direction. i o inner and outer fluid filaments, respectively. romaxand Also, in the case of ρom = 0.8, the inner and outer r are the radial coordinate of the furthest point on fluid filaments will inflate at the neck and become a imax the outer and inner interfaces. zomaxand zimax are the spherical compound droplet. In addition, due to ρm axial coordinate of the furthest point on the outer and = ρi, in the case of ρom = 1.6, ρi decreases more as inner interfaces, respectively. compared to the case of ρom = 0.8 (assuming ro re- mains unchanged). When ρ decreases, the instability The results show that when decreasing the density ra- i ρ of the separating interface increases, and so does the tio om, the retraction rate of the inner and outer fil- ρ breakup possibility of the filaments. 14 aments increases. In the non-breakup ( om = 0.2 and Figure 4 shows the change in the shape of the fluid 0.8), the deformation parameter is varied from posi- compound filament by time, from τ = 0 to τ = 25.9. tive to a negative value, and its value is equal to zero The shape of the compound fluid filament changes when the filament recoils to a spherical shape at the gradually, from the capsule shape at the beginning to end of the contraction process. In the case that the in- the bone shape at τ = 8.12. At τ = 9.38, the inner fil- ner filament breaks up into two droplets at the center ament has a similar shape to an hourglass. The inner position of the filament, the deformation parameter filament is expected to break up; however, at τ = 10.36, of the inner one vanishes after the breakup (the pink it inflates at the middle, and in the end, at τ = 25.9, the delta line in Figure 6 (a) and (b)). inner and outer fluid filament becomes spherical fluid ρ droplets which are called a spherical compound fluid The effect of the density ratio ( om) on the droplet. breakup of a compound fluid filament Figure 5 shows that when the density ratio between To evaluate the entire figure of the contraction process the outer and middle fluids is increased to rom = 1.6, of the compound filament. We varied the aspect ratio the breakup of the inner fluid filament happens. The of the outer filament (Aro) from 10 to 30 and the den- shape of the filament changes from capsule shape to sity ratio ρom from 0.1 to 6.4. The phase regime of the bone shape at τ = 8.1. At τ = 10.44, the inner fluid deformation and breakup of the compound filament filament nearly breaks up, and at τ = 10.62, the inner is presented in Figure 7. fluid filament performs breakup. As a result, from a We found that after contraction, the compound fila- capsule-shaped fluid filament, under the action of in- ment has three pattern modes as follows: the non- terfacial tension, contracts lead to the breakup of the breakup mode as shown in Figure 4, the inner breakup inner fluid filament and create a compound droplet mode as shown in Figure 5, and the mix-breakup τ containing two cores (at = 27.72). mode as shown in Figure 8. As indicated in Fig- Thus, based on analysis of the pressure field, velocity ure 7, on the left side of the regime diagram, the field, and deformation of the fluid filament over time, non-breakup mode represented by triangle symbols it is shown that the contraction and breakup of the is zoned by a green line. On the right side, the mixed fluid compound filament are influenced by the den- breakup mode appears with circle symbols zoned by a sity ratio of the outer and middle fluids. For example, red line. The remaining cases, represented by square ρ in the case of Aro = 10, when om changes from 0.1 symbols, correspond to the breakup of the inner fil- ρ ≥ to 6.4, the inner fluid filament breaks up when om ament while the outer filament is non-breakup. It 1.6. is clear that increasing the density ratio ρom and in- DISCUSSION creasing the initial length of the filament promote the breakup of the compound filament upon contraction. The effect of the density ratio (ρom) on the deformation parameter of a compound fluid filament CONCLUSIONS Figure 6 (a) and (b) show the effect of the density ratio Research results show that the fluid density has an es- between outer and middle fluids (ρom) on the defor- sential influence on the deformation or the breakup mation parameters of the inner and outer filaments. of the fluid compound filament. Accordingly, the lower the density ratio r of the outer and middle zimax − rimax om Ti = (8) ρ zimax + rimax fluids ( om < 0.8) and the lower aspect ratio (Aro = 1912
5. Science & Technology Development Journal, 24(2):1909-1917 Figure 2: Comparison of a contracting simple filament between the present result and that reported by Notzand Basaran4. At each time, Notz and Basaran’s prediction is shown in the upper half and our prediction is shown in the lower half. The parameters are shown in the text. 10) lead to the fluid compound filaments contracting AUTHORS CONTRIBUTION to a spherical compound droplet (i.e., non-breakup Vinh T. Nguyen: Investigation, Visualization, Formal mode). The contraction of the inner and outer fila- analysis, Writing- Original draft ments mostly breaks up when the aspect ratio of the Truong V. Vu: Conceptualization, Methodology, outer filament is high (Aro = 25-30). The remaining Writing- Reviewing and Editing, Supervision cases correspond to the breakup of the inner fluid fil- Phan H. Nguyen: Conceptualization, Supervision ament (Figure 7). All compound filaments, whether or not breakup, tend to return to the spherical shapes Nang X. Ho, Binh D. Pham, Hoe D. Nguyen and Hung at the end of the contraction process due to the effect V. Vu: Investigation, Formal analysis of interfacial tension. ACKNOWLEDGMENTS This study is very important for desirable compound droplet control in biomedical technology or some This research is funded by Vietnam National Foun- other applications. Therefore, the present study helps dation for Science and Technology Development to select the treatment process through the initial con- (NAFOSTED) under grant number 107.03-2019.307. ditions to form compound droplets containing two or more cores or prevent breakup of the filament. REFERENCES The present study is limited to the compound fil- 1. McClements DJ. Advances in fabrication of emulsions with en- ament that contains one inner filament. However, hanced functionality using structural design principles. Curr 15,16 in many applications, e.g., microfluidic systems, Opin Colloid Interface Sci. 2012 Oct 1;17(5):235-45;Available the filament can enclose two or more inner inter- from: 2. Lao K-L, Wang J-H, Lee G-B. A microfluidic platform for faces. In some other situations, the filament dynamics formation of double-emulsion droplets. Microfluid Nanoflu- 17 can be affected by contaminants or the temperature idics. 2009 Nov 1;7(5):709-19;Available from: field. 18 Such complicated issues should be addressed 10.1007/s10404-009-0430-9. in the future studies of the contracting compound fil- 3. Maan AA, Schroën K, Boom R. Spontaneous droplet for- mation techniques for monodisperse emulsions prepara- ament. tion - Perspectives for food applications. J Food Eng. 2011 Dec 1;107(3):334-46;Available from: CONFLICT OF INTEREST jfoodeng.2011.07.008. The authors declare that they have no competing in- terests. 1913
6. Science & Technology Development Journal, 24(2):1909-1917 Figure 3: The form of the compound filament at time τ = 10.08 corresponding to two densityratios ρom = 0.8 (left) and ρom = 1.6 (right). The aspect ratio Aro equals 10. Other parameters are provided in the text.For each density 2 ratio,the left half shows the pressure normalized by 0.5roUc , and the vector on the right halfshows the velocity field normalized by Uc. The neck of theinner filament tends to increase its size for ρom = 0.8 (left) whereas itis decreased for ρom = 1.6 (right). 4. Notz PK, Basaran OA. Dynamics and breakup of a contracting of multiphase flow. J Comput Phys. 2001 May 20;169(2):708- liquid filament. J Fluid Mech. 2004;512:223-56;Available from: 59;Available from: 11. Vu T-V, Vu TV, Nguyen CT, Pham PH. Deformation and breakup 5. Vu TV, Vu LV, Pham BD, Luu QH. Numerical investigation of of a double-core compound droplet in an axisymmetric chan- dynamic behavior of a compound drop in shear flow. J Mech nel. Int J Heat Mass Transf. 2019 Jun 1;135:796-810;Available Sci Technol. 2018 May 1;32(5):2111-7;Available from: https: from: //doi.org/10.1007/s12206-018-0420-5. 12. Vu TV. Fully resolved simulations of drop solidification under 6. Ho NX, Vu TV. Numerical simulation of the deformation forced convection. Int J Heat Mass Transf. 2018 Feb;122:252- and breakup of a two-core compound droplet in an ax- 63;Available from: isymmetric T-junction channel. Int J Heat Fluid Flow. 2020 2018.01.124. Dec 1;86:108702;Available from: 13. Vu TV. Deformation and breakup of a pendant drop with solid- ijheatfluidflow.2020.108702. ification. Int J Heat Mass Transf. 2018 Feb;122:341-53;Available 7. Pham BD, Vu TV, Nguyen CT, Nguyen HD, Nguyen VT. Nu- from: merical study of collision modes of multi-core compound 14. Suryo R, Doshi P, Basaran OA. Nonlinear dynamics droplets in simple shear flow. J Mech Sci Technol. 2020 and breakup of compound jets. Phys Fluids. 2006 May 1;34(5):2055-66;Available from: Aug;18(8):082107;Available from: s12206-020-0427-6. 1.2245377. 8. Vu TV. Parametric study of the collision modes of com- 15. Abate AR, Thiele J, Weitz DA. One-step formation of multiple pound droplets in simple shear flow. Int J Heat Fluid Flow. emulsions in microfluidics. Lab Chip. 2011 Jan 21;11(2):253- 2019 Oct 1;79:108470;Available from: 8;PMID: 20967395. Available from: j.ijheatfluidflow.2019.108470. C0LC00236D. 9. Vu T-V, Vu TV, Bui DT. Numerical study of deformation and 16. Abate AR, Weitz DA. High-order multiple emulsions breakup of a multi-core compound droplet in simple shear formed in poly(dimethylsiloxane) microfluidics. Small. flow. Int J Heat Mass Transf. 2019 Mar 1;131:1083-94;Available 2009 Sep;5(18):2030-2;PMID: 19554565. Available from: from: 10. Tryggvason G, Bunner B, Esmaeeli A, Juric D, Al-Rawahi N, 17. Hansen S, Peters GWM, Meijer HEH. The effect of surfactant on Tauber W, et al. A front-tracking method for the computations the stability of a fluid filament embedded in a viscous fluid.J 1914
7. Science & Technology Development Journal, 24(2):1909-1917 Figure 4: Deformation of the compound fluid filament over time in the caseof ρom = 0.8 and Aro = 10. The other parameters are the same as those in Figure 3. Figure 5: Deformation of the compound fluid filament over time in the caseof ρom = 1.6 and Aro = 10. The other parameters are the same as those in Figure 3. 1915
8. Science & Technology Development Journal, 24(2):1909-1917 Figure 6: (a) The deformation parameter of the outer fluid filament over time To. (b) The deformation parameter of the inner fluid filament over time Ti in the case of Aro = 10. The other parameters are the same as those in Figure 3. Figure 7: The phase regime of the deformation and breakup of the compound filament. The other parameters are provided in the text. 1916
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