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Radial viscous fingering induced by an infinitely fast chemical reaction

Published online by Cambridge University Press:  20 July 2022

Priya Verma Open the ORCID record for Priya Verma [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]
Priya Verma
Affiliation:
Department of Mathematics, Indian Institute of Technology Ropar, 140001 Rupnagar, India
Vandita Sharma
Affiliation:
Department of Mathematics, Indian Institute of Technology Ropar, 140001 Rupnagar, India
Manoranjan Mishra*
Affiliation:
Department of Mathematics, Indian Institute of Technology Ropar, 140001 Rupnagar, India
*
†Email address for correspondence: manoranjan@iitrpr.ac.in
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Abstract

A numerical insight into the radial displacement of two viscously stable reactants undergoing an infinitely fast chemical reaction is obtained. This work broadens the numerical knowledge about the interaction between chemical reaction and hydrodynamic instability. A suitable transformation is utilised to deal with an infinite dimensionless parameter and reduce computational cost. Viscous fingering instability originates when the product has a different viscosity than the reactants. We calculate the onset time $t_{on}$ when the instability begins to appear for different reactants by varying the log mobility ratio $R_c$ and the Péclet number $Pe$. Based on $t_{on}$, we divide the time domain into stable and unstable zones with respect to instability. This helps to characterise reactants so as to have a flow with/without instability. For a fixed $Pe$ and $\vert R_c \vert$, it is found that $t_{on}$ is early for $R_c > 0$ in contrast to the result for rectilinear displacement. This results in a wider stable zone for $R_c < 0$. Interestingly, it is found that the stable zone can be made independent of $Pe$ by using a careful rescaling and we obtain the dependence of the onset time of instability on $R_c$ and $Pe$ as $t_{on} \propto (\vert R_c\vert Pe^{\beta _1}-\beta _2)^{\beta _3}$ where the constant of proportionality and $\beta _i$, $i=1,2,3$, depend upon the sign of $R_c$. In the unstable zone, we find that the length of the fingers depends upon the sign of $R_c$, which is not observed for any radial displacement of reactants undergoing a slow or moderately fast chemical reaction.

JFM classification

Interfacial Flows (free surface): Fingering instability Low-Reynolds-number flows: Hele-Shaw flows Low-Reynolds-number flows: Porous media
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 945 , 25 August 2022 , A19
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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