Research Papers

Equivalent Inclusion Method for the Stokes Flow of Drops Moving in a Viscous Fluid

[+] Author and Article Information
H. M. Yin, P.-H. Lee, Y. J. Liu

Department of Civil Engineering
and Engineering Mechanics,
Columbia University,
610 Seeley W. Mudd Building,
500 West 120th Street,
New York, NY 10027

Manuscript received October 1, 2013; final manuscript received March 25, 2014; accepted manuscript posted March 28, 2014; published online April 25, 2014. Assoc. Editor: Kenji Takizawa.

J. Appl. Mech 81(7), 071010 (Apr 25, 2014) (12 pages) Paper No: JAM-13-1415; doi: 10.1115/1.4027312 History: Received October 01, 2013; Revised March 25, 2014; Accepted March 28, 2014

The equivalent inclusion method is presented to derive the Stokes flow of multiple drops moving in a viscous fluid at a small Reynolds number. The drops are replaced by inclusions with the same viscosity as the fluid, but an eigenstrain rate field that is a fictitious nonmechanical strain rate field is introduced to represent the viscosity mismatch between each drop and the matrix fluid. The velocity and pressure fields can be solved by considering the body force and eigenstrain rate on the inclusions with the Green's function technique. When one spherical drop is considered, the solution recovers the closed-form classic solution. This method is versatile and can be used in the simulation of a many-body system with different drop size, elongation ratio, and viscosity. Numerical examples demonstrate the capability and accuracy of the proposed formulation and illustrate particles' rotation and motion caused by particle interactions.

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Batchelor, G. K., 1967, An Introduction to Fluid Dynamics, Cambridge University, Cambridge, UK.
Lamb, H. S., 1975, Hydrodynamics, Cambridge University, Cambridge, UK.
Subramanian, R. S., and Balasubramaniam, R., 2001, The Motion of Bubbles and Drops in Reduced Gravity, Cambridge University, Cambridge, UK.
Yin, H., Yang, D., Kelly, G., and Garant, J., 2013, “Design and Performance of a Novel Building Integrated PV/Thermal System For Energy Efficiency of Buildings,” Solar Energy, 87, pp. 184–195. [CrossRef]
Yang, D. J., Yuan, Z. F., Lee, P.-H., and Yin, H. M., 2012, “Simulation and Experimental Validation Of Heat Transfer in a Novel Hybrid Solar Panel,” Int. J. Heat Mass Transfer, 55(4), pp. 1076–1082. [CrossRef]
Yang, D. J., and Yin, H. M., 2011, “Energy Conversion Efficiency of a Novel Hybrid Solar System for Photovoltaic, Thermoelectric, and Heat Utilization,” IEEE Trans. Energy Convers., 26(2), pp. 662–670. [CrossRef]
Lee, P.-H., and Yin, H., 2014, “Size Effect on Functionally Graded Material Fabrication by Sedimentation,” J. Nanomech. Mircomech., p. A4014008 (in press) [CrossRef].
Zhu, H. P., Zhou, Z. Y., Yang, R. Y., and Yu, A. B., 2007. “Discrete Particle Simulation of Particulate Systems: Theoretical Developments,” Chem. Eng. Sci., 62(13), pp. 3378–3396. [CrossRef]
Wörner, M., 2012, “Numerical Modeling of Multiphase Flows in Microfluidics and Micro Process Engineering: A Review of Methods and Applications,” Microfluid. Nanofluidics, 12(6), pp. 841–886. [CrossRef]
Brady, J. F., and Bossis, G., 1988, “Stokesian Dynamics,” Annu. Rev. Fluid Mech., 20(1), pp. 111–157. [CrossRef]
Ladd, A. J. C., and Verberg, R., 1988, “Lattice–Boltzmann Simulations of Particle-Fluid Suspensions,” J. Stat. Phys., 104(5–6), pp. 1191–1251. [CrossRef]
Mura, T., 1987, Micromechanics of Defects in Solids, Kluwer Academic Publishers, Dordrecht, Netherlands.
Eshelby, J., 1957, “The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems,” Proc. Roy. Soc. London Ser. A, Math. Phys. Sci., 241(1226), pp. 376–396. [CrossRef]
Eshelby, J. D., 1959, “The Elastic Field Outside An Ellipsoidal Inclusion,” Proc. Roy. Soc. London Ser. A, Math. Phys. Sci., 252(1271), pp. 561–569. [CrossRef]
Yin, H. M., Buttlar, W. G., Paulino, G. H., and Di Benedetto, H., 2008, “Assessment of Existing Micro-Mechanical Models for Asphalt Mastics Considering Viscoelastic Effects,” Road Mater. Pave. Des., 9(1), pp. 31–57. [CrossRef]
Huang, J. H., 1996, “Equivalent Inclusion Method for the Work-Hardening Behavior of Piezoelectric Composites,” Int. J. Solids Struct., 33(10), pp. 1439–1451. [CrossRef]
Hatta, H., and Taya, M., 1986, “Equivalent Inclusion Method for Steady State Heat Conduction in Composites,” Int. J. Eng. Sci., 24(7), pp. 1159–1172. [CrossRef]
Yin, H. M., Paulino, G. H., Buttlar, W. G., and Sun, L. Z., 2007, “Micromechanics-Based Thermoelastic Model for Functionally Graded Particulate Materials With Particle Interactions,” J. Mech. Phys. Solids, 55(1), pp. 132–160. [CrossRef]
Dunn, M. L., and Taya, M., 1993, “Micromechanics Predictions of the Effective Electroelastic Moduli of Piezoelectric Composites,” Int. J. Solids Struct., 30(2), pp. 161–175. [CrossRef]
Bilby, B. A., Eshelby, J. D., and Kundu, A. K., 1975, “The Change of Shape of a Viscous Ellipsoidal Region Embedded in a Slowly Deforming Matrix Having a Different Viscosity,” Tektonophysics, 28(4), pp. 265–274. [CrossRef]
Wetzel, E. D., and Tucker, C. L., 2001, “Droplet Deformation in Dispersions With Unequal Viscosities and Zero Interfacial Tension,” J. Fluid Mech., 426(1), pp. 199–228. [CrossRef]
Wu, Y., Zinchenko, A. Z., and Davis, R. H., 2002, “General Ellipsoidal Model for Deformable Drops in Viscous Flows,” Ind. Eng. Chem. Res., 41(25), pp. 6270–6278. [CrossRef]
Yu, W., and Bousmina, M., 2003, “Ellipsoidal Model for Droplet Deformation in Emulsions,” J. Rheol., 47(4), pp. 1011–1039. [CrossRef]
Jackson, N. E., and Charles, L. T., III, 2003, “A Model for Large Deformation of an Ellipsoidal Droplet With Interfacial Tension,” J. Rheol., 47(3), pp. 659–682. [CrossRef]
Jiang, D., 2012, “A General Approach for Modeling the Motion of Rigid and Deformable Ellipsoids in Ductile Flows,” Comput. Geosci., 38(1), pp. 52–61. [CrossRef]
Rudnicki, J., 2002, “Eshelby Transformations, Pore Pressure and Fluid Mass Changes, and Subsidence,” Second Biot Conference on Poromechanics (Poromechanics II), Grenoble, France, August 26–28, pp. 307–312.
Kröner, E., 1990, “Modified Green Functions in the Theory of Heterogeneous and/or Anisotropic Linearly Elastic Media,” Micromechanics and Inhomogeneity, G. Weng, M. Taya, and H. Abé, eds., Springer, New York, pp. 197–211. [CrossRef]
Yin, H. M., Sun, L. Z., and Chen, J. S., 2006, “Magneto-Elastic Modeling of Composites Containing Chain-Structured Magnetostrictive Particles,” J. Mech. Phys. Solids, 54(5), pp. 975–1003. [CrossRef]
Kim, S., and Karrila, S. J., 1991, Microhydrodynamics: Principle and Selected Applications, Butterworth-Heinemann, Stoneham, MA.
Dhont, J. K. G., 1996, An Introduction to Dynamics of Colloids, Elsevier Science, Oxford, UK.
Zapryanov, Z., and Tabakova, S., 1998, Dynamics of Bubbles, Drops and Rigid Particles, Springer, New York.
Yin, H. M., and Sun, L. Z., 2006, “Magnetoelastic Modelling of Composites Containing Randomly Dispersed Ferromagnetic Particles,” Philos. Mag., 86(28), pp. 4367–4395. [CrossRef]
Liu, Y. J., and Yin, H. M., 2014, “Equivalent Inclusion Method-Based Simulation of Particle Sedimentation Toward Functionally-Graded Material Manufacturing,” Acta Mech., 225(4–5), pp. 1429–1445 [CrossRef].
Pozrikidis, C., 1992, Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge Texts in Applied Mathematics, Cambridge Press, New York.
Pozrikidis, C., 1997, Introduction to Theoretical and Computational Fluid Dynamics, Oxford University Press, Oxford, UK.
Magnaudet, J., 2011, “A ‘Reciprocal’ Theorem for the Prediction of Loads on a Body Moving in an Inhomogeneous Flow at Arbitrary Reynolds Number,” J. Fluid Mech., 689, pp. 564–606. [CrossRef]
Claeyst, I. L., and Brady, J. F., 1993, “Suspensions of Prolate Spheroids in Stokes Flow. Part 1. Dynamics of a Finite Number of Particles in an Unbounded Fluid,” J. Fluid Mech., 251, pp. 411–442. [CrossRef]
Mo, G., and Sangani, A. S., 1994, “A Method for Computing Stokes Flow Interactions Among Spherical Objects,” Phys. Fluids, 6(5), pp. 1637–1652. [CrossRef]
Batchelor, G. K., 1972, “Sedimentation in a Dilute Dispersion of Spheres,” J. Fluid Mech., 52(2), pp. 245–268. [CrossRef]
Nemat-Nasser, S., and Hori, M., 1999, Micromechanics: Overall Properties of Heterogeneous Materials, North-Holland, Amsterdam.


Grahic Jump Location
Fig. 1

One ellipsoidal subdomain Ω embedded in the infinite domain D

Grahic Jump Location
Fig. 2

Many spherical drops Ωm(m=1,2,3,…) embedded in the infinite domain D

Grahic Jump Location
Fig. 3

The distribution of v3 along (a) x1 and (b) x3 axes when the drop moves along the long axis of x3 direction with assumption of a1 = 1, a2 = 0.5 and body force fi0=1

Grahic Jump Location
Fig. 4

The distribution of v3 along (a) x1, (b) x2 and (c) x3 axis when the drop moves along the short axis of x3 direction with assumption of a1 = 1, a2 = 0.5 and body force fi0=1

Grahic Jump Location
Fig. 5

Variation of average velocity of two solid particles with center-center distance and orientation

Grahic Jump Location
Fig. 6

Distribution of v3 along x1 axis with particle interactions (a = 1)

Grahic Jump Location
Fig. 7

The particle average velocity versus the particle number for a particle center-center distance at 4 (a = 1)




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