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Unstable miscible displacements in radial flow with chemical reactions

Published online by Cambridge University Press:  26 April 2021

Min Chan Kim ,
Satyajit Pramanik Open the ORCID record for Satyajit Pramanik [Opens in a new window] ,
Vandita Sharma Open the ORCID record for Vandita Sharma [Opens in a new window]  and
Manoranjan Mishra Open the ORCID record for Manoranjan Mishra [Opens in a new window]
Min Chan Kim
Affiliation:
Department of Chemical Engineering, Jeju National University, Jeju63243, Republic of Korea
Satyajit Pramanik*
Affiliation:
Department of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar382355, India
Vandita Sharma
Affiliation:
Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar140001, India
Manoranjan Mishra
Affiliation:
Department of Mathematics, Indian Institute of Technology Ropar, Rupnagar140001, India
*
†Email address for correspondence: satyajit.pramanik@iitgn.ac.in
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Abstract

The effects of the $A + B \rightarrow C$ chemical reaction on miscible viscous fingering in a radial source flow are analysed using linear stability theory and numerical simulations. This flow and transport problem is described by a system of nonlinear partial differential equations consisting of Darcy's law for an incompressible fluid coupled with nonlinear advection–diffusion–reaction equations. For an infinitely large Péclet number ($Pe$), the linear stability equations are solved using spectral analysis. Further, the numerical shooting method is used to solve the linearized equations for various values of $Pe$ including the limit $Pe \rightarrow \infty$. In the linear analysis, we aim to capture various critical parameters for the instability using the concept of asymptotic instability, i.e. in the limit $\tau \rightarrow \infty$, where $\tau$ represents the dimensionless time. We restrict our analysis to the asymptotic limit $Da^{\ast }$$(= Da \tau ) \rightarrow \infty$ and compare the results with the non-reactive case ($Da = 0$) for which $Da^{\ast } = 0$, where $Da$ is the Damköhler number. In the latter case, the dynamics is controlled by the dimensionless parameter $R_{Phys} = -(R_{A} - \beta R_{B})$. In the former case, for a fixed value of $R_{Phys}$, the dynamics is determined by the dimensionless parameter $R_{Chem} = -(R_{C} - R_{B} - R_{A})$. Here, $\beta$ is the ratio of reactants’ initial concentration and $R_{A}$, $R_{B}$ and $R_{C}$ are the log-viscosity ratios. We perform numerical simulations of the coupled nonlinear partial differential equations for large values of $Da$. The critical values $R_{Phys, c}$ and $R_{Chem, c}$ for instability decrease with $Pe$ and they exhibit power laws in $Pe$. In the asymptotic limit of infinitely large $Pe$ they exhibit a power-law dependence on $Pe$ ($R_{Chem, c} \sim Pe^{-1/2}$ as $Pe \rightarrow \infty$) in both the linear and nonlinear regimes.

JFM classification

Interfacial Flows (free surface): Fingering instability Low-Reynolds-number flows: Porous media
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 917 , 25 June 2021 , A25
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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