^{1}, Kirti Chandra Sahu

^{1,a)}and S. P. Vanka

^{2}

### Abstract

The pressure-driven displacement of two immiscible fluids in an inclined channel in the presence of viscosity and density gradients is investigated using a multiphase lattice Boltzmann approach. The effects of viscosity ratio, Atwood number, Froude number, capillary number, and channel inclination are investigated through flow structures, front velocities, and fluid displacement rates. Our results indicate that increasing viscosity ratio between the fluids decreases the displacement rate. We observe that increasing the viscosity ratio has a non-monotonic effect on the velocity of the leading front; however, the velocity of the trailing edge decreases with increasing the viscosity ratio. The displacement rate of the thin-layers formed at the later times of the displacement process increases with increasing the angle of inclination because of the increase in the intensity of the interfacialinstabilities. Our results also predict the front velocity of the lock-exchange flow of two immiscible fluids in the exchange flow dominated regime. A linear stability analysis has also been conducted in a three-layer system, and the results are consistent with those obtained by our lattice Boltzmann simulations.

The support from the Indian Institute of Technology Hyderabad, India, is gratefully acknowledged. We also thank the Department of Science and Technology, India, for their partial financial support. The authors thank Dr. Aaron Shinn for assistance in the GPU implementation of the code.

I. INTRODUCTION

II. FORMULATION

A. Numerical method

III. RESULTS AND DISCUSSION

A. Grid independency test

B. Horizontal channel

C. Inclined channel

IV. LINEAR STABILITY ANALYSIS

V. CONCLUDING REMARKS

### Key Topics

- Viscosity
- 64.0
- Flow instabilities
- 29.0
- Solubility
- 25.0
- Core annular flows
- 22.0
- Lattice Boltzmann methods
- 18.0

## Figures

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 ⩽ *x* ⩽ *L* of the channel, respectively.

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 ⩽ *x* ⩽ *L* of the channel, respectively.

(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*.

(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*.

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°.

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°.

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*.

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*.

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°.

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°.

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.

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.

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.

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.

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*.

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*.

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.

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.

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.

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.

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.

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.

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.

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.

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*.

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*.

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°.

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°.

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 + 0.665*Fr* + 0.3534*Fr* ^{2}. The points 1, 2, and 3 in panel (a) correspond to *Fr* = 0.604, *Fr* = 0.671, and *Fr* = 2.24, respectively.

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 + 0.665*Fr* + 0.3534*Fr* ^{2}. The points 1, 2, and 3 in panel (a) correspond to *Fr* = 0.604, *Fr* = 0.671, and *Fr* = 2.24, respectively.

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).

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).

Normalized front velocity as a function of normalized Fr, for (a) *m* = 1 and (b) *m* = 10 where χ = 2*Re*sinθ/*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.

Normalized front velocity as a function of normalized Fr, for (a) *m* = 1 and (b) *m* = 10 where χ = 2*Re*sinθ/*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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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°.

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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