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A study of pressure-driven displacement flow of two immiscible liquids using a multiphase lattice Boltzmann approach
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10.1063/1.4760257
/content/aip/journal/pof2/24/10/10.1063/1.4760257
http://aip.metastore.ingenta.com/content/aip/journal/pof2/24/10/10.1063/1.4760257

Figures

Image of FIG. 1.
FIG. 1.

Schematic showing the geometry (not to scale) and initial flow configuration. The inlet and outlet are located at x = 0 and x = L, respectively. The aspect ratio of the channel, L/H, is 48. Initially, the channel is filled with fluids “1” and “2” from 0 ⩽ x ⩽ 5 and 5 ⩽ xL of the channel, respectively.

Image of FIG. 2.
FIG. 2.

(a) Variation of volume fraction of the displaced fluid (M t /M 0) with time obtained using different mesh densities, and (b) the spatio-temporal diagram of in time versus x plane for At = 0.2, m = 2, Fr = 2.236, Re = 100, Ca = 0.263, and θ = 45°. The dotted line in panel (a) represents the analytical solution of the plug-flow displacement, given by M t /M 0 = 1 − tH/L.

Image of FIG. 3.
FIG. 3.

Spatio-temporal evolution of the contours of the index function, ϕ for At = 0.2, m = 2, Fr = 2.236, Re = 100, Ca = 0.263, and θ = 45°.

Image of FIG. 4.
FIG. 4.

Variation of volume fraction of the displaced fluid (M t /M 0) with time for different values of viscosity ratio, m for (a) At = 0 and Fr = ∞, and (b) At = 0.3, Fr = 0.577. The rest of the parameters are Re = 100, Ca = 0.263, and θ = 0°. The dotted lines in panels (a) and (b) represent the analytical solution of plug-flow displacement, given by M t /M 0 = 1 − tH/L.

Image of FIG. 5.
FIG. 5.

Variations of velocities of (a) the leading (V l ) and (b) the trailing (V t ) fronts with viscosity ratio, m. The rest of the parameter values are Re = 100, Ca = 0.263, and θ = 0°.

Image of FIG. 6.
FIG. 6.

The contours of the index function, ϕ at t = 15 and t = 50 in a horizontal channel (θ = 0°) for two different values of viscosity ratio, m for (a) At = 0 and Fr = ∞ and (b) At = 0.3 and Fr = 0.577. The rest of the parameter values are Re = 100 and Ca = 0.263.

Image of FIG. 7.
FIG. 7.

The contours of the index function, ϕ at t = 30 for different inclination angles. The rest of the parameter values are m = 10, At = 0.2, Fr = 1, Re = 100, and Ca = 0.263.

Image of FIG. 8.
FIG. 8.

The effects of angle of inclination on the variation of the volume fraction of the displaced fluid, (M t /M 0) with time. The parameters used are m = 10, At = 0.2, Fr = 1, Re = 100, and Ca = 0.263. The dotted line represents the analytical solutions of the variation of M t /M 0 for plug-flow displacement, given by M t /M 0 = 1 − tH/L.

Image of FIG. 9.
FIG. 9.

The spatio-temporal diagram of in time versus x plane for (a) θ = 0°, (b) θ = 5°, (c) θ = 60°, and (d) θ = 85°. The rest of the parameter values are the same as those in Fig. 7.

Image of FIG. 10.
FIG. 10.

The vorticity (first panel) and velocity vectors (second panel) fields for the same parameter values as those used to generate Fig. 7. The color-maps for the vorticity contours are shown as third panel.

Image of FIG. 11.
FIG. 11.

Spatio-temporal evolution of contours of the index function, ϕ for (a) m = 0.8, (b) m = 10, and (c) m = 30. The rest of the parameters are At = 0.2, Fr = 1, Re = 100, θ = 45°, and Ca = 0.263.

Image of FIG. 12.
FIG. 12.

The spatio-temporal diagram of in time versus x plane for (a) m = 0.8, (b) m = 10, and (c) m = 30. The rest of the parameter values are the same as those used to generate Fig. 11.

Image of FIG. 13.
FIG. 13.

Effects of viscosity ratio, m on variation of the volume fraction of the displaced fluid, (M t /M 0) with time. The rest of the parameter values are the same as those used to generate Fig. 11. The dotted line represents the analytical solution for plug-flow displacement, given by M t /M 0 = 1 − tH/L.

Image of FIG. 14.
FIG. 14.

The contours of the index function, ϕ for different values of capillary number at t = 30. The rest of the parameter values are m = 5, At = 0.2, Fr = 1, Re = 100, and θ = 45°.

Image of FIG. 15.
FIG. 15.

Variation of the normalized front velocity, FrV l with Froude number, Fr for (a) m = 1, (b) m = 10. The rest of the parameter values are Re = 100, Ca = 0.263, and θ = 5°. The dashed line is the best fitted polynomial, given by FrV l = 0.38 + 0.665Fr + 0.3534Fr 2. The points 1, 2, and 3 in panel (a) correspond to Fr = 0.604, Fr = 0.671, and Fr = 2.24, respectively.

Image of FIG. 16.
FIG. 16.

The spatio-temporal diagram of in time versus x plane for (a) Fr = 0.604, (b) Fr = 0.671, and (c) Fr = 2.24 for the parameters At = 0.2, m = 1, Re = 100, θ = 5°, and Ca = 0.263. The white dashed lines represent the initial location of the interface (x = 5).

Image of FIG. 17.
FIG. 17.

Normalized front velocity as a function of normalized Fr, for (a) m = 1 and (b) m = 10 where χ = 2Resinθ/AtFr 2. The rest of the parameters are Re = 100, θ = 5°, and Ca = 0.263. The solid line represents the line V l = V 0. The filled circle in (a) corresponds to Fr = 0.671, the squares to the left and the right of this filled circle correspond to Fr = 0.604 and Fr = 2.24, respectively.

Image of FIG. 18.
FIG. 18.

Schematic showing the geometry and initial condition of the flow. Also shown here is profile of the steady, streamwise velocity component generated with m = 10 and h 0 = 0.5.

Image of FIG. 19.
FIG. 19.

Variation of (a) maximal growth rate, ω i,max , (b) axial wavenumber associated with the most dangerous mode, β max with viscosity ratio, m. The rest of the parameter values are Re = 100, At = 0.1, Fr = 1, θ = 0°, and Ca = 0.263.

Image of FIG. 20.
FIG. 20.

Variation of (a) maximal growth rate, ω i,max , (b) axial wavenumber associated with the most dangerous mode, β max with Atwood number At. The rest of the parameter values are Re = 100, m = 10, Fr = 1, θ = 0°, and Ca = 0.263.

Image of FIG. 21.
FIG. 21.

Variation of (a) maximal growth rate, ω i,max , (b) axial wavenumber associated with the most dangerous mode, β max with Froude number, Fr. The rest of the parameter values are Re = 100, m = 10, At = 0.1, θ = 0°, and Ca = 0.263.

Image of FIG. 22.
FIG. 22.

Variation of (a) maximal growth rate, ω i, max , (b) axial wavenumber associated with the most dangerous mode, β max with capillary number, Ca. The rest of the parameter values are Re = 100, m = 10, At = 0.1, θ = 0°, and Fr = 1.

Tables

Generic image for table
Table I.

The velocity of the leading (V l ) and trailing (V t ) fronts for different grid densities. The rest of the parameter values are At = 0.2, m = 2, Fr = 2.236, Re = 100, Ca = 0.263, and θ = 45°.

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/content/aip/journal/pof2/24/10/10.1063/1.4760257
2012-10-19
2014-04-21
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: A study of pressure-driven displacement flow of two immiscible liquids using a multiphase lattice Boltzmann approach
http://aip.metastore.ingenta.com/content/aip/journal/pof2/24/10/10.1063/1.4760257
10.1063/1.4760257
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