Research Papers: Fundamental Issues and Canonical Flows

Effect of Slip on Circulation Inside a Droplet

[+] Author and Article Information
Joseph J. Thalakkottor

Department of Mechanical and
Aerospace Engineering,
University of Florida,
Gainesville, FL 32611
e-mail: tjjoseph@ufl.edu

Kamran Mohseni

Department of Mechanical and
Aerospace Engineering;
Department of Electrical and
Computer Engineering,
University of Florida,
Gainesville, FL 32611
e-mail: mohseni@ufl.edu

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received December 23, 2014; final manuscript received June 12, 2015; published online August 4, 2015. Assoc. Editor: John Abraham.

J. Fluids Eng 137(12), 121201 (Aug 04, 2015) (8 pages) Paper No: FE-14-1784; doi: 10.1115/1.4030915 History: Received December 23, 2014

Internal recirculation in a moving droplet, enforced by the presence of fluid–fluid interfaces, plays an important role in several droplet-based microfluidic devices as it could enhance mixing, heat transfer, and chemical reaction. The effect of slip on droplet circulation is studied for two canonical steady-state problems: two-phase Couette, boundary-driven, and Poiseuille, pressure/body force-driven, flows. A simple model is established to estimate the circulation in a droplet and capture the effect of slip and aspect ratio on the droplet circulation. The circulation in a droplet is shown to decrease with increasing slip length in the case of a boundary-driven flow, while for a body force-driven flow it is independent of slip length. Scaling parameters for circulation and slip length are identified from the circulation model. The model is validated using continuum and molecular dynamics (MD) simulations. The effect of slip at the fluid–fluid interface on circulation is also briefly discussed. The results suggest that active manipulation of velocity slip, e.g., through actuation of hydrophobicity, could be employed to control droplet circulation and consequently its mixing rate.

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Reyes, D. R. , Iossifidis, D. , Auroux, P.-A. , and Manz, A. , 2002, “Micro Total Analysis Systems. 1. Introduction, Theory, and Technology,” Anal. Chem., 74(12), pp. 2623–2636. [CrossRef] [PubMed]
Auroux, P.-A. , Iossifidis, D. , Reyes, D. R. , and Manz, A. , 2002, “Micro Total Analysis Systems. 2. Analytical Standard Operations and Applications,” Anal. Chem., 74(12), pp. 2637–2652. [CrossRef] [PubMed]
Koh, W. H. , Lok, K. S. , and Nguyen, N. T. , 2013, “A Digital Micro Magnetofluidic Platform for Lab-on-a-Chip Applications,” ASME J. Fluids Eng., 135(2), p. 021302. [CrossRef]
Burns, J. , and Ramshaw, C. , 2001, “The Intensification of Rapid Reactions in Multiphase Systems Using Slug Flow in Capillaries,” Lab Chip, 1(1), pp. 10–15. [CrossRef] [PubMed]
Dummann, G. , Quittmann, U. , Gröschel, L. , Agar, D. W. , Wörz, O. , and Morgenschweis, K. , 2003, “The Capillary-Microreactor: A New Reactor Concept for the Intensification of Heat and Mass Transfer in Liquid–Liquid Reactions,” Catal. Today, 79–80, pp. 433–439. [CrossRef]
Jaritsch, D. , Holbach, A. , and Kochmann, N. , 2014, “Counter-Current Extraction in Microchannel Flow: Current Status and Perspectives,” ASME J. Fluids Eng., 136(9), p. 091211. [CrossRef]
Mohseni, K. , 2005, “Effective Cooling of Integrated Circuits Using Liquid Alloy Electrowetting,” Semiconductor Thermal Measurement, Modeling, and Management Symposium (SEMI-Therm), IEEE, San Jose, CA, Mar. 15–17, pp. 20–25.
Baird, E. , and Mohseni, K. , 2008, “Digitized Heat Transfer: A New Paradigm for Thermal Management of Compact Micro-Systems,” IEEE Trans. Compon. Packag. Technol., 31(1), pp. 143–151. [CrossRef]
Hosokawa, K. , Fujii, T. , and Endo, I. , 1999, “Handling of Picoliter Liquid Samples in a Poly(dimethylsiloxane)-Based Microfluidic Device,” Anal. Chem., 71(20), pp. 4781–4785. [CrossRef]
Song, H. , Tice, J. D. , and Ismagilov, R. F. , 2003, “A Microfluidic System for Controlling Reaction Networks in Time,” Angew. Chem. Int. Ed., 42(7), pp. 768–772. [CrossRef]
Navier, C. , 1823, “Memoire sur les lois du mouvement des fluides,” Mem. Acad. R. Sci. Inst. Fr., 6, pp. 389–440.
Maxwell, J. , 1890, The Scientific Papers of James Clerk Maxwell, Vol. V2, Cambridge University Press, Cambridge, pp. 703–711.
Thompson, P. , and Troian, S. , 1997, “A General Boundary Condition for Liquid Flow at Solid Surfaces,” Nature, 389, pp. 360–362. [CrossRef]
Thalakkottor, J. , and Mohseni, K. , 2013, “Analysis of Slip in a Flow With an Oscillating Wall,” Phys. Rev. E, 87, p. 033018. [CrossRef]
Koplik, J. , and Banavar, J. , 1995, “Continuum Deductions From Molecular Hydrodynamics,” Annu. Rev. Fluid Mech., 27, pp. 257–292. [CrossRef]
Koplik, J. , Banavar, J. , and Willemsen, J. , 1989, “Molecular Dynamics of Fluid Flow at Solid Surfaces,” Phys. Fluids A, 1(5), pp. 781–794. [CrossRef]
Koplik, J. , Banavar, J. , and Willemsen, J. , 1988, “Molecular Dynamics of Poiseuille Flow and Moving Contact Lines,” Phys. Rev. Lett., 60(13), pp. 1282–1285. [CrossRef] [PubMed]
Thompson, P. , and Robbins, M. , 1989, “Simulations of Contact Line Motion: Slip and the Dynamic Contact Angle,” Phys. Rev. Lett., 63, pp. 766–769. [CrossRef] [PubMed]
Paik, P. , Pamula, V. K. , and Fair, R. B. , 2003, “Rapid Droplet Mixers for Digital Microfluidic Systems,” Lab Chip, 3(2), pp. 253–259. [CrossRef] [PubMed]
DeVoria, A. C. , and Mohseni, K. , 2005, “Droplets in an Axisymmetric Microtube: Effects of Aspect Ratio and Fluid Interfaces,” Phys. Fluids, 27(1), pp. 80–101.
Taheri, P. , Torrilhon, M. , and Struchtrup, H. , 2009, “Couette and Poiseuille Microflows: Analytical Solutions for Regularized 13-Moment Equations,” Phys. Fluids, 21(1), p. 017102. [CrossRef]
Popinet, S. , 2009, “An Accurate Adaptive Solver for Surface-Tension-Driven Interfacial Flow,” J. Comput. Phys., 228(16), pp. 5838–5866. [CrossRef]
Plimpton, S. , 1995, “Fast Parallel Algorithms for Short-Range Molecular Dynamics,” J. Comput. Phys., 117(1), pp. 1–19. [CrossRef]
Kistler, S. F. , 1993, Wettability, J. C. Berg , ed., Marcel Dekker, New York.
Baroud, C. N. , Gallaire, F. , and Dangla, R. , 2010, “Dynamics of Microfluidic Droplets,” Lab Chip, 10(16), pp. 2032–2045. [CrossRef] [PubMed]
Young, T. , 1805, “An Essay on the Cohesion of Fluids,” Philos. Trans. R. Soc. London, 95, pp. 65–87. [CrossRef]
Laplace, P. , 1805, Traité de mécanique céleste/par, PS Laplace, tome quatrieme, Vol. 4, de l'Imprimerie de Crapelet, Paris.
Fuerstman, M. J. , Lai, A. , Thurlow, E. , Shevkoplyas, S. S. , Stone, H. A. , and Whitesides, G. M. , 2007, “The Pressure Drop Along Rectangular Microchannel Containing Bubbles,” Lab Chip, 7(11), pp. 1479–1489. [CrossRef] [PubMed]
Mohseni, K. , and Baird, E. , 2007, “A Unified Velocity Model for Digital Microfluidics,” Nanoscale Microscale Thermophys. Eng., 11(1–2), pp. 109–120.
Ou, J. , Perot, B. , and Rothstein, J. P. , 2004, “Laminar Drag Reduction in Microchannels Using Ultrahydrophobic Surfaces,” Phys. Fluids, 16(12), pp. 4635–4643. [CrossRef]
Choi, C. H. , and Kim, C. J. , 2006, “Large Slip of Aqueous Liquid Flow Over a Nanoengineered Superhydrophobic Surface,” Phys. Rev. Lett., 96(6), p. 066001. [CrossRef] [PubMed]
Koplik, J. , and Banavar, J. R. , 2006, “Slip, Immiscibility, and Boundary Conditions at the Liquid–Liquid Interface,” Phys. Rev. Lett., 96(4), p. 044505. [CrossRef] [PubMed]


Grahic Jump Location
Fig. 1

Schematic of the problem of a 2D: (a) Couette flow and (b) body force-driven flow. The problem simulates a shear-driven and body force-driven flow with two immiscible fluids in a microchannel. Here, z is the out of plane axis.

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Fig. 6

Circulation versus slip length for droplets with different AR. For the case of a Poiseuille flow, the aspect ratio of vortex is ARΓ=2 AR.

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Fig. 2

Circulation versus slip length for droplets with different ARΓ. Here, vortex aspect ratios equals the droplet aspect ratio, ARΓ=AR. (a) The unscaled data and (b) the scaled. Results from continuum simulations show the decrease in circulation with increasing slip length and decreasing vortex AR. The scaled results for different droplet AR collapse, except for ARΓ=1. Here, Γ*=Γ/UL,Ls*=Ls/H.

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Fig. 3

Percentage error of nondimensional droplet circulation in continuum simulations. The error is computed as (Γmodel*-Γsimulation*)/Γsimulation*, where Γ* = Γ/UL. The error increases with decrease in droplet AR and an increase in slip length. Unless specified, the parameters for different cases were (surface tension) γ = 1, (static contact angle) θs = 90 deg, and (viscosity ratio) μ12 = 100. For high AR, the relative error is less than 20%. Here, is Γ = 0.6, θ = 120, and †† is μ12 = 50.

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Fig. 4

Variation of slip length on the wall while moving away from the TCP. Two cases with hydrophilic and hydrophobic wall are shown. The results are obtained using MD simulations, where σ is the LJ parameter corresponding to the diameter of the molecule and having units of an Å. The details of the two cases are listed in Table 1.

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Fig. 5

(a) Axial velocity profile across the droplet center (x/L = 0.5) for AR=1 and 4. Droplet with AR=4 exhibits a linear velocity profile while for AR=1, deviation from the linear velocity profile of a Couette flow is seen which also suggests the formation of two circulatory flows in a droplet. (b) The velocity field for two droplets of AR=1 has a viscosity of μ1 = 0.01μ2, where μ1 is the viscosity of the droplet on the left, and μ2 is the viscosity of the droplet on the right. These results are for the case with no-slip at the wall.

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Fig. 11

Percentage error of nondimensional droplet circulation versus interfacial tension, where centerline velocity is (a) directly obtained from simulation, U˜direct sim., and (b) predicted using model presented in Eq. (17)U˜from model*

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Fig. 7

Percentage error of nondimensional droplet circulation. The error is computed as (Γmodel*-Γsimulation*)/Γsimulation*, where Γ*=Γ/U˜L. For the most part, the relative error is within 10%. The model breaks down for a ARΓ=1 due to the deviation of its velocity profile from a single-phase Poiseuille flow.

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Fig. 8

Axial velocity profile at the droplet center (x/L = 0.5) for different droplet aspect ratios. Velocity profile for single phase Poiseuille flow is shown in solid line.

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Fig. 9

Scaled droplet circulation is plotted against the nondimensional slip length, where Γ*=Γ/U˜L and Ls*=Ls/(H/2)

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Fig. 10

Percentage error of nondimensional droplet circulation versus viscosity ratio, where centerline velocity is (a) directly obtained from simulation, U˜direct sim., and (b) predicted using model presented in Eq. (17)U˜from model*



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