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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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Figures

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Fig. 1

One ellipsoidal subdomain Ω embedded in the infinite domain D

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Fig. 2

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

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Fig. 5

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

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Fig. 6

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

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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

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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

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Fig. 7

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

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