# A numerical study on reaction-induced radial fingering instability

## TL;DR

This study numerically investigates how chemical reactions influence viscous fingering patterns in radial flows, revealing conditions under which instabilities occur or are suppressed, and highlighting geometric effects on the onset of fingering.

## Contribution

It provides a detailed numerical analysis of reaction-induced viscous fingering in radial geometry, identifying stability regions and critical parameters, and compares these with existing rectilinear results.

## Key findings

- Chemical reactions can suppress or promote fingering depending on viscosity contrasts.
- A stability diagram in the $Da-R_c$ parameter space delineates stable and unstable regions.
- Geometry significantly affects the onset and pattern of fingering instabilities.

## Abstract

The dynamics of $A + B \rightarrow C$ fronts is analyzed numerically in a radial geometry. We are interested to understand miscible fingering instabilities when the simple chemical reaction changes the viscosity of the fluid locally and a non-monotonic viscosity profile with a global maximum or minimum is formed. We consider viscosity-matched reactants $A$ and $B$ generating a product $C$ having different viscosity than the reactants. Depending on the effect of $C$ on the viscosity relative to the reactants, different viscous fingering (VF) patterns are captured which are in good qualitative agreement with the existing radial experiments. We have found that for a given chemical reaction rate, an unfavourable viscosity contrast is not always sufficient to trigger the instability. For every fixed $Pe$, these effects of chemical reaction on VF are summarized in the $Da-R_c$ parameter space that exhibits a stable region separating two unstable regions corresponding to the cases of more and less viscous product. Fixing $Pe$, we determine $Da$-dependent critical log-mobility ratios $R_c^+$ and $R_c^-$ such that no VF is observable whenever $R^-_c \leq R_c \leq R^+_c$. The effect of geometry is observable on the onset of instability, where we obtain significant differences from existing results in the rectilinear geometry.