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Research Papers: Fundamental Issues and Canonical Flows

A Lattice Boltzmann Simulation of Three-Dimensional Displacement Flow of Two Immiscible Liquids in a Square Duct

[+] Author and Article Information
Kirti Chandra Sahu

e-mail: ksahu@iith.ac.in
Department of Chemical Engineering,
Indian Institute of Technology Hyderabad,
Ordinance Factory Estate, Yeddumailaram,
Andhra Pradesh, Hyderabad 502205, India

S. P. Vanka

Department of Chemical Engineering,
Indian Institute of Technology Hyderabad,
Ordinance Factory Estate, Yeddumailaram,
Andhra Pradesh, Hyderabad 502205, India;
Department of Mechanical
Science and Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61822

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received March 5, 2013; final manuscript received July 10, 2013; published online September 19, 2013. Assoc. Editor: Samuel Paolucci.

J. Fluids Eng 135(12), 121202 (Sep 19, 2013) (8 pages) Paper No: FE-13-1132; doi: 10.1115/1.4024998 History: Received March 05, 2013; Revised July 10, 2013

A three-dimensional (3D), multiphase lattice Boltzmann approach is used to study a pressure-driven displacement flow of two immiscible liquids of different densities and viscosities in a square duct. A three-dimensional, 15-velocity (D3Q15) lattice model is used. The effects of channel inclination, viscosity, and density contrasts are investigated. The contours of the density and the average viscosity profiles in different planes are plotted and compared with those obtained in a two-dimensional (2D) channel. We demonstrate that the flow dynamics in a 3D channel is quite different as compared to that of a 2D channel. We found that the flow is relatively more coherent in a 3D channel than that in a 2D channel. A new screw-type instability is seen in the 3D channel that cannot be observed in the 2D channel.

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Figures

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Fig. 1

Schematic describing the geometry and the initial configuration

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Fig. 2

Schematic diagram of D3Q15 lattice

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Fig. 3

Variation of mass fraction of the displaced liquid (Mt/M0) with time for different grids. The rest of the parameters are Re = 100, m = 10, At = 0.2, Fr = 5, Ca = 29.9, and θ = 45 deg. The dotted line represents the analytical solution of the plug-flow displacement, given by Mt/M0 = 1-tH/L.

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Fig. 4

Contours of the index function, ϕ, at (a) t = 10 and (b) t = 30 in the x-y plane at z = W/2 for different grids. The parameters are the same as those of Fig. 3.

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Fig. 5

Evolution of the isosurface of ϕ at the interface at different times (from left to right: t = 12, 18, 30, and 50) for the simulation domain of 2112 × 66 × 66 grid. The parameters are Re = 100, m = 10, At = 0.2, Fr = 5, Ca = 29.9, and θ = 45 deg. The flow is in the positive x direction.

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Fig. 6

Contours of the index function, ϕ, at t = 20 in (a) x-y plane at z = W/2 and (b) x-z plane at y = H/2. The parameters are Re = 100, At = 0.2, m = 10, Fr = 1, and Ca = 29.9.

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Fig. 7

Contours of the index function, ϕ, at t = 20 in the y-z plane at x = L/2. The parameters are Re = 100, At = 0.2, m = 10, Fr = 1, and Ca = 29.9.

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Fig. 8

Axial variation of normalized average viscosity, μ¯yz(≡μyz/μyz0) for (a) θ = 30 deg and (b) θ = 85 deg. The rest of the parameters are Re = 100, At = 0.2, m = 10, Fr = 1, and Ca = 29.9. Here, μyz=(1/HW)∫0W∫0Hμdydz and μyz0 is the value of μyz at x = 0.

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Fig. 9

Vertical variation of normalized average viscosity, μ¯xz(≡μxz/μxz0) for (a) θ = 30 deg and (b) θ = 85 deg. The rest of the parameters are the same as those used in Fig. 6. Here, μxz=1/LW∫0W∫0Lμdxdz and μxz0 is the value of μxz at y = 0.

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Fig. 10

Axial variation of normalized average viscosity, μ¯y(≡μy/μy0), for (a) θ = 5 deg and (b) θ = 85 deg in a two-dimensional channel. The rest of the parameters are the same as those used to generate Fig. 6. Here, μy=1/H∫0Hμdy and μy0 is the value of μy at x = 0.

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Fig. 11

Vertical variation of normalized average viscosity, μ¯x(≡μx/μx0), for (a) θ = 5 deg and (b) θ = 85 deg in a two-dimensional channel. The rest of the parameters are the same as those used to generate Fig. 6. Here, μx=1/L∫0Lμdx and μx0 is the value of μx at y = 0.

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Fig. 12

Spatiotemporal evolution of the contours of the index function, ϕ. (a) x-y plane at z = W/2 (three-dimensional simulation) and (b) the corresponding two-dimensional results. The parameters are Re = 100, m = 20, At = 0.2, Fr = 5, θ = 30 deg, and Ca = 29.9.

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Fig. 13

Spatiotemporal evolution of the contours of the index function, ϕ, obtained from the three-dimensional channel for m = 5 with the rest of the parameter values the same as those used to generate Fig. 12. The results shown are in x-y plane at z = W/2 of the channel.

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Fig. 14

The contours of the index function, ϕ, at t = 20 in the x-y plane at z = W/2 for different values of Atwood number. The rest of the parameter values are Re = 100, m = 10, Fr = 1, θ = 30 deg, and Ca = 29.9.

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