Numerical simulations on the onset and the growth of viscous fingering during the miscible displacement due to the radial source flow were conducted. With introduction of a new stability criterion, the critical log-viscosity ratio, Rc, was found as a function of the Peclet number, Pe. Similar to the previous linear stability analyses, Pe made the system unstable, i.e., accelerated the onset of instability. For a large Pe system, the present numerical simulation yielded much stable results than the previous theoretical predictions This discrepancy was commonly encountered in the comparison between the theoretical prediction and the experimental finding. Additionally, the difference between the rectilinear system and the present one was also discussed. The present system was found more insensitive to the Peclet number than the rectilinear one.

Viscous fingering, Radial source flow, Stability analysis, Numerical simulation

1. Homsy, G. M., “Viscous Fingering in Porous Medium,” Ann. Rev.
Fluid Mech. 19, 271(1987).
2. Tan, C. T. and Homsy, G. M., “Stability of Miscible Displacements
in Porous Media: Radial Source Flow,” Phys. Fluids 30,
1239(1987).
3. Pritchard, D., “The instability of Thermal and fluid Fronts During
Radial Injection in a Porous Medium,” J. Fluid Mech. 508, 133
(2004)
4. Kim, M. C., “Onset of the Radial Viscous Fingering in a Hele-
Shaw Cell,” Korean J. Chem. Eng. 29, 1688(2012).
5. Kim, M. C., “Some Analytical Aspects of the Radial Fingering
in Porous Medium,” Z. Angew. Math. Phys. 63, 713(2012).
6. Sharma, V., Nand, S., Pramanik, S., Chen, C.-Y. and Mishra, M.,
“Control of Radial Viscous Fingering,” J. Fluid Mech. 862, 624
(2020).
7. Nand, S., Sharma, V., Das, S. K., Padhee, S. S. and Mishra, M.,
“Effect of Hele-Shaw Cell Gap on Radial Viscous Fingering,”
Scientific Reports 12, 18967(2022).
8. Kim, M. C., “Effect of Nonlinear Drag on the Onset and the
Growth of the Miscible Viscous Fingering in a Porous Medium,”
Korean J. Chem. Eng. 39, 548(2022).
9. COMSOL AB, Comsol multiphysics v. 5.4 reference manual
(2019).
10. Kim, M. C., Pramanik, S., Sharma, V. and Mishra, M., “Unstable
Miscible Displacements in Radial flow with Chemical Reactions,”
J. Fluid Mech. 917, A25(2021).
11. Sharma, V., Othman, H. B., Nagatsu, Y. and Mishra, M. “Viscous
Fingering of Miscible Annular Ring,” J. Fluid Mech. 916, A14
(2021).
12. Kim, M. C., “Theoretical and Numerical Studies on the Interface
Movement and the Onset of Gravitational Instability During
the Carbon Dioxide Dissolution Into Oil,” Phys. Fluids 34, 024102
(2022).
13. Kim, M. C. and Pramanik, S., “Miscible Viscous Fingering in a
Packed Cylindrical Column: Theory and Numerics,” Phys. Rev.
Fluids 8, 013901(2023).
14. Hong, J. S. and Kim, M. C., “Theoretical and Numerical Studies
on Dissolution of Horizontal Salt Body and Pattern Formation
Incorporated with a Dynamic Moving Interface,” J. Fluid
Mech. 960, A6(2023).
15. Hong, J. S. and Kim, M. C., “Gravitational Fingering and Droplet
Formation During the Phase Separation of a Partially Miscible
Binary Mixture in a Vertical Hele-Shaw Cell,” Int. J. Heat Mass
Transfer 201, 123665(2023).
16. Kim, M. C., “Numerical Simulations on Thermocapillary Flow on
Heated Sinusoidal Topography,” Korean J. Chem. Eng. (2024).
https://doi.org/10.1007/s11814-024-00109-1
17. Jha, B., Cueto-Felgueroso, L. and Juanes, R., “Fluid Mixing from
Viscous Fingering,” Phys. Rev. Lett. 106, 194502(2011).
18. Verma, P., Sharma, V. and Mishra, M., “Radial Viscous Fingering
Induced by An Infinitely Fast Chemical Reaction,” J. Fluid Mech.
945, A19(2022).
19. Ryoo, W. S. and Kim, M. C., “Linear and Non-linear Analyses
on the Onset of Miscible Viscous Fingering in a Porous Medium,”
Korean J. Chem. Eng. 35, 1423(2018).
20. Perkins, T. E., Johnston, O. E. and Hoffman, R. N., “Mechanics
of Viscous Fingering in Miscible Systems,” Soc. Pet. Eng. J., 5,
301(1965).