Impact of Vertical Magnetic Field on Onset of Instability of a Casson Fluid Saturated Porous Layer: A Nonlinear Theory
Abstract
1. Introduction
2. Basic Equations
- Brinkmann’s law is used to characterize the flow of the fluid.
- Oberbeck–Boussinesq approximation, i.e., variation of density considered with body force term only.
- Density varies linearly with temperature and concentration, i.e.,Density of the solute is greater than the density of the solvent.
- Local thermal equilibrium is assumed between solid and fluid phases.
- Viscous dissipation is negligible.
Basic State
3. Linear Instability
One Term Galerkin Method
4. Nonlinear Instability
Solution Methodology
5. Discussion
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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| Linear | Nonlinear | Linear | Nonlinear | Linear | Nonlinear | |
|---|---|---|---|---|---|---|
| 1 | ||||||
| Instability | Instability | |||||
|---|---|---|---|---|---|---|
| 0 | 1045.046 | 3529.793 | Stationary | 1047.935 | 3538.485 | Stationary |
| 100 | 1095.046 | 3635.714 | Stationary | 1097.935 | 3644.391 | Stationary |
| 200 | 1145.046 | 3741.635 | Stationary | 1147.935 | 3750.297 | Stationary |
| 300 | 1195.046 | 3847.556 | Stationary | 1197.935 | 3856.203 | Stationary |
| 400 | 1245.046 | 3953.477 | Stationary | 1247.935 | 3962.109 | Stationary |
| 500 | 1295.046 | 4059.398 | Stationary | 1297.935 | 4068.015 | Stationary |
| 600 | 1345.046 | 4165.319 | Stationary | 1347.935 | 4173.921 | Stationary |
| 700 | 1395.046 | 4271.240 | Stationary | 1397.935 | 4279.827 | Stationary |
| 800 | 1445.046 | 4377.161 | Stationary | 1447.935 | 4385.733 | Stationary |
| 900 | 1495.046 | 4483.082 | Stationary | 1497.935 | 4491.639 | Stationary |
| 1000 | 1545.046 | 4589.003 | Stationary | 1547.935 | 4597.545 | Stationary |
| Instability | Instability | |||||
|---|---|---|---|---|---|---|
| 0 | 1045.046 | 2221.647 | Stationary | 1047.935 | 2227.434 | Stationary |
| 100 | 1145.046 | 2321.647 | Stationary | 1147.935 | 2327.434 | Stationary |
| 200 | 1245.046 | 2421.647 | Stationary | 1247.935 | 2427.434 | Stationary |
| 300 | 1345.046 | 2521.647 | Stationary | 1347.935 | 2527.434 | Stationary |
| 400 | 1445.046 | 2621.647 | Stationary | 1447.935 | 2627.434 | Stationary |
| 500 | 1545.046 | 2721.647 | Stationary | 1547.935 | 2727.434 | Stationary |
| 600 | 1645.046 | 2821.647 | Stationary | 1647.935 | 2827.434 | Stationary |
| 700 | 1745.046 | 2921.647 | Stationary | 1747.935 | 2927.434 | Stationary |
| 800 | 1845.046 | 3021.647 | Stationary | 1847.935 | 3027.434 | Stationary |
| 900 | 1945.046 | 3121.647 | Stationary | 1947.935 | 3127.434 | Stationary |
| 1000 | 2045.046 | 3221.647 | Stationary | 2047.935 | 3227.434 | Stationary |
| Instability | Instability | |||||
|---|---|---|---|---|---|---|
| 550 | 2145.046 | 2150.618 | Stationary | 2147.938 | 2155.001 | Stationary |
| 551 | 2147.046 | 2151.589 | Stationary | 2149.938 | 2155.970 | Stationary |
| 552 | 2149.0464 | 2152.559 | Stationary | 2151.938 | 2156.941 | Stationary |
| 553 | 2151.0464 | 2153.530 | Stationary | 2153.938 | 2157.911 | Stationary |
| 554 | 2153.046 | 2154.501 | Stationary | 2155.938 | 2158.882 | Stationary |
| 555 | 2155.046 | 2155.471 | Stationary | 2157.938 | 2159.852 | Stationary |
| 556 | 2157.046 | 2156.441 | Oscillatory | 2159.938 | 2160.823 | Stationary |
| 557 | 2159.046 | 2157.411 | Oscillatory | 2161.938 | 2161.793 | Oscillatory |
| 558 | 2161.046 | 2158.382 | Oscillatory | 2163.938 | 2162.7642 | Oscillatory |
| 559 | 2163.046 | 2159.352 | Oscillatory | 2165.938 | 2163.734 | Oscillatory |
| 560 | 2165.046 | 2160.322 | Oscillatory | 2167.938 | 2164.705 | Oscillatory |
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Share and Cite
Raju, S.S.K.; Mukahal, F.H.H.A.; Mulki, H.; Mahmoud, S. Impact of Vertical Magnetic Field on Onset of Instability of a Casson Fluid Saturated Porous Layer: A Nonlinear Theory. Mathematics 2025, 13, 3550. https://doi.org/10.3390/math13213550
Raju SSK, Mukahal FHHA, Mulki H, Mahmoud S. Impact of Vertical Magnetic Field on Onset of Instability of a Casson Fluid Saturated Porous Layer: A Nonlinear Theory. Mathematics. 2025; 13(21):3550. https://doi.org/10.3390/math13213550
Chicago/Turabian StyleRaju, S. Suresh Kumar, Fatemah H. H. Al Mukahal, Hasan Mulki, and Saleh Mahmoud. 2025. "Impact of Vertical Magnetic Field on Onset of Instability of a Casson Fluid Saturated Porous Layer: A Nonlinear Theory" Mathematics 13, no. 21: 3550. https://doi.org/10.3390/math13213550
APA StyleRaju, S. S. K., Mukahal, F. H. H. A., Mulki, H., & Mahmoud, S. (2025). Impact of Vertical Magnetic Field on Onset of Instability of a Casson Fluid Saturated Porous Layer: A Nonlinear Theory. Mathematics, 13(21), 3550. https://doi.org/10.3390/math13213550

