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Stability Analysis of Miscible Viscous Fingering in Bingham and Carreau Fluids

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Abstract

A linear stability analysis is carried out to examine viscous fingering in miscible flow displacement involving two non-Newtonian fluids. One of the fluids, either displacing or displaced fluid, exhibits a yield-stress behavior described using the Bingham model and the other fluid described by the Carreau model exhibits shear-rate-dependent viscosity. The role of fluid rheology in growth rate of fingering is analyzed in the early stage of instability. Two different flow arrangements are considered—Carreau fluid displacing Bingham fluid and Bingham fluid displacing Carreau fluid. The rate-dependent models primarily modify the viscosity stratification between the two miscible fluids which controls the onset of the instability. Introducing yield-stress behavior to the displaced (displacing) fluid leads to enhancement (attenuation) of the fingering growth rate, primarily due to modification in the gap-averaged viscosity. Similarly, shear-thinning behavior of the displacing (displaced) fluid amplifies (weakens) the growth of finger formation and vice versa for shear-thickening fluid. Interestingly, apart from viscosity modification, the nature of the flow curve governed by the rheological description plays an important role in the early growth of fingering. Upon fixing the effective viscosity ratio under flow condition, yield-stress and shear-thinning fluids tend to enhance the growth of fingering instability vis-à-vis Newtonian fluid regardless of the flow arrangement. This suggests that the fingering instability is controlled by not only mere modification of the fluid viscosity but also the nature of rheological description of the fluid. The interplay of both factors governs the onset and the initial kinetics of finger formation.

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Abbreviations

A :

Aspect ratio of the Hele-Shaw cell, L/W

\({\text{Bn}}\) :

Bingham number

b :

Separation gap of the Hele-Shaw cell

\({\text{Cu}}\) :

Carreau number

c :

Dimensionless concentration of the displacing fluid in fluid mixture

K :

Permeability of the porous medium

k :

Disturbance wavenumber

L :

Length of the Hele-Shaw cell

n :

Power-law index of the Carreau fluid

\({\text{Pe}}\) :

Péclet number

p :

Fluid pressure

\(R_0\) :

Log viscosity ratio (displaced/displacing fluid) at zero-shear rate

\(R_{\text {eff}}\) :

Effective log viscosity ratio (displaced/displacing fluid)

\(t_0\) :

Frozen time for the base-state profile

\(\mathbf{u }\) :

Velocity field

u :

x-component of fluid velocity

v :

y-component of fluid velocity

W :

Width of the Hele-Shaw cell

\({\dot{\gamma }}\) :

Shear-rate

\(\sigma\) :

Disturbance growth rate

\(\sigma _{\max }\) :

Maximum growth rate

\(\mu _0\) :

Zero-shear viscosity of the Carreau fluid

\(\mu _{p}\) :

Plastic viscosity of the Bingham fluid

\(\tau _0\) :

Yield-stress of the Bingham fluid

\({\chi }\) :

Generic variable (u, v, c, p)

\({\bar{\chi }}\) :

Base-state of the generic variable (\({\bar{u}},{\bar{v}},{\bar{c}},{\bar{p}}\))

\({\chi }'\) :

Disturbance of the generic variable (\({u^\prime },{v^\prime },{c^\prime },{p^\prime }\))

\({\tilde{\chi }}\) :

Disturbance eigenfunction (\({\tilde{u}},{\tilde{v}},{\tilde{c}},{\tilde{p}}\))

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  1. Department of Chemical Engineering, Indian Institute of Technology Delhi, New Delhi, 110 016, India

    Pooja Jangir, Ratan Mohan & Paresh Chokshi

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  1. Pooja Jangir
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  2. Ratan Mohan
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Jangir, P., Mohan, R. & Chokshi, P. Stability Analysis of Miscible Viscous Fingering in Bingham and Carreau Fluids. Transp Porous Med 141, 561–583 (2022). https://doi.org/10.1007/s11242-021-01732-w

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