Abstract
In this study, we consider a viscous instability induced by injecting a fluid, containing a heterogeneously distributed species, into a two-dimensional porous medium. The initial concentration of the species is a piece wise linear function with a discontinuity. Using the quasi-steady-state approximation, the stability of the system can be reduced to a coupled system of ODEs. A dispersion equation for the initial configuration was obtained. As the species diffuses, the instantaneous growth rates evolve in time. The stability of the system is obtained numerically. Although the initial discontinuity destabilises the system, eventually, after a finite amount of time, the system becomes stable. The system becomes stable even when non-monotonic viscosity profiles are present. The linear stability predictions, around the time when the product of the instantaneous growth rate and time equal one, yields wave numbers around 17% smaller than those found in the early stages of the nonlinear simulations. The nonlinear simulations show that when a local minimum in the viscosity profile is present, the fingers tend to be slightly longer, than when a local maximum in the viscosity profile is present.
| Original language | English |
|---|---|
| Article number | 3 |
| Number of pages | 31 |
| Journal | Journal of Engineering Mathematics |
| Volume | 154 |
| Early online date | 6 Sept 2025 |
| DOIs | |
| Publication status | Published - 6 Sept 2025 |
Bibliographical note
Copyright © Met Office and Philip Trevelyan, Aleena Urooj, Sotos Generalis 2025.This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit https://creativecommons.org/licenses/by-nc-nd/4.0/.
Keywords
- Linear stability
- Non-linear simulations
- Porous media
- Viscous instabilities