Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 7.
Fig. 6.
Variation of critical magnetic Rayleigh numbers Nc with M3, χ and δ at fixed h1 = 3 and Das = 1
| h1 = 3 | Critical Magnetic Rayleigh Number | |||||
|---|---|---|---|---|---|---|
| χ | M3 | δ = 0.01 | δ = 0.03 | δ = 0.05 | δ = 0.07 | δ = 0.09 |
| 1 | 1 | 412.4400 (7.7909) | 420.6071 (7.7909) | 428.7742 (7.7909) | 436.9413 (7.7909) | 445.1085 (7.7909) |
| 5 | 333.0876 (7.0576) | 364.8102 (7.0576) | 396.5328 (7.0576) | 428.2554 (7.0576) | 459.9781 (7.0576) | |
| 10 | 321.2218 (6.6092) | 379.6257 (6.6092) | 438.0297 (6.6092) | 496.4336 (6.6092) | 554.8376 (6.6092) | |
| 15 | 323.9978 (6.3847) | 408.5190 (6.3847) | 493.0402 (6.3847) | 577.5614 (6.3847) | 662.0825 (6.3847) | |
| 20 | 331.3287 (6.2498) | 441.7716 (6.2498) | 552.2144 (6.2498) | 662.6573 (6.2498) | 773.1002 (6.2498) | |
| 10 | 1 | 494.4820 (8.9375) | 504.2737 (8.9375) | 514.0654 (8.9375) | 523.8572 (8.9375) | 533.6489 (8.9375) |
| 5 | 342.8150 (7.3292) | 375.4640 (7.3292) | 408.1130 (7.3292) | 440.7621 (7.3292) | 473.4111 (7.3292) | |
| 10 | 324.5245 (6.7147) | 383.5290 (6.7147) | 442.5334 (6.7147) | 501.5379 (6.7147) | 560.5423 (6.7147) | |
| 15 | 325.7133 (6.4415) | 410.6820 (6.4415) | 495.6507 (6.4415) | 580.6194 (6.4415) | 665.5881 (6.4415) | |
| 20 | 332.4015 (6.2855) | 443.2020 (6.2855) | 554.0025 (6.2855) | 664.8030 (6.2855) | 775.6035 (6.2855) | |
| 103 | 1 | 525.1702 (9.3988) | 535.5696 (9.3988) | 545.9690 (9.3988) | 556.3685 (9.3988) | 566.7679 (9.3988) |
| 5 | 345.3312 (7.4019) | 378.2199 (7.4019) | 411.1086 (7.4019) | 443.9972 (7.4019) | 476.8859 (7.4019) | |
| 10 | 325.3216 (6.7407) | 384.4710 (6.7407) | 443.6204 (6.7407) | 502.7697 (6.7407) | 561.9191 (6.7407) | |
| 15 | 326.1164 (6.4551) | 411.1902 (6.4551) | 496.2641 (6.4551) | 581.3379 (6.4551) | 666.4118 (6.4551) | |
| 20 | 332.6499 (6.2938) | 443.5332 (6.2938) | 554.4166 (6.2938) | 665.2999 (6.2938) | 776.1832 (6.2938) | |
| 105 | 1 | 525.5497 (9.4046) | 535.9567 (9.4046) | 546.3636 (9.4046) | 556.7705 (9.4046) | 567.1774 (9.4046) |
| 5 | 345.3597 (7.4027) | 378.2512 (7.4027) | 411.1426 (7.4027) | 444.0340 (7.4027) | 476.9254 (7.4027) | |
| 10 | 325.3305 (6.7410) | 384.4815 (6.7410) | 443.6325 (6.7410) | 502.7835 (6.7410) | 561.9345 (6.7410) | |
| 15 | 326.1209 (6.4552) | 411.1959 (6.4552) | 496.2709 (6.4552) | 581.3459 (6.4552) | 666.4209 (6.4552) | |
| 20 | 332.6527 (6.2939) | 443.5369 (6.2939) | 554.4212 (6.2939) | 665.3054 (6.2939) | 776.1896 (6.2939) | |
Comparison of Values from the Present Study with Existing Studies
| Free-Free | Rigid-Rigid | |||
|---|---|---|---|---|
| Present Study | Dhiman [10] | Present Study | Dhiman [10] | |
| M1 = 1 | 360.0088 (5.4644) | 342.52 (5.174) | 913.1268 (10.2291) | 889.15 (9.921) |
| M1 = 5 | 126.5677 (5.8260) | 116.49 (5.319) | 313.1211 (10.5816) | 299.56 (10.06) |
Variation of critical magnetic Rayleigh numbers Nc with h1, M3 and Das at fixed χ = 10 and δ = 0_03
| χ = 10 | Critical Magnetic Rayleigh Number | |||||
|---|---|---|---|---|---|---|
| M = 3 | h1 | Das = 0 | Das = 10 | Das = 102 | Das = 103 | Das = 105 |
| 3 | 1 | 1058.2 (7.1019) | 1713.5 (10.3132) | 2257.1 (11.8650) | 2371.8 (12.1002) | 2385.8 (12.1272) |
| 3 | 352.7195 (7.1019) | 571.1730 (10.3132) | 752.3706 (11.8650) | 790.5888 (12.1002) | 795.2814 (12.1272) | |
| 5 | 211.6317 (7.1019) | 342.7038 (10.3132) | 451.4224 (11.8650) | 474.3533 (12.1002) | 477.1689 (12.1272) | |
| 7 | 151.1655 (7.1019) | 244.7884 (10.3132) | 322.4445 (11.8650) | 338.8238 (12.1002) | 340.8349 (12.1272) | |
| 9 | 117.5732 (7.1019) | 190.3910 (10.3132) | 250.7902 (11.8650) | 263.5296 (12.1002) | 265.0938 (12.1272) | |
| 5 | 1 | 998.5998 (6.5835) | 1660.7 (9.6886) | 2210.0 (11.1916) | 2325.7 (11.4193) | 2339.9 (11.4454) |
| 3 | 332.8666 (6.5835) | 553.5690 (9.6886) | 736.6527 (11.1916) | 775.2371 (11.4193) | 779.9737 (11.4454) | |
| 5 | 199.7200 (6.5835) | 332.1414 (9.6886) | 441.9916 (11.1916) | 465.1422 (11.4193) | 467.9842 (11.4454) | |
| 7 | 142.6571 (6.5835) | 237.2439 (9.6886) | 315.7083 (11.1916) | 332.2445 (11.4193) | 334.2744 (11.4454) | |
| 9 | 110.9555 (6.5835) | 184.5230 (9.6886) | 245.5509 (11.1916) | 258.4124 (11.4193) | 259.9912 (11.4454) | |
| 10 | 1 | 1010.8 (5.9857) | 1734.0 (9.0216) | 2332.5 (10.4899) | 2458.4 (10.7120) | 2473.8 (10.7375) |
| 3 | 336.9392 (5.9857) | 578.0082 (9.0216) | 777.4880 (10.4899) | 819.4643 (10.7120) | 824.6157 (10.7375) | |
| 5 | 202.1635 (5.9857) | 346.8049 (9.0216) | 466.4928 (10.4899) | 491.6786 (10.7120) | 494.7694 (10.7375) | |
| 7 | 144.4025 (5.9857) | 247.7178 (9.0216) | 333.2092 (10.4899) | 351.1990 (10.7120) | 353.4067 (10.7375) | |
| 9 | 112.3131 (5.9857) | 192.6694 (9.0216) | 259.1627 (10.4899) | 273.1548 (10.7120) | 274.8719 (10.7375) | |
| 15 | 1 | 1077.7 (5.7151) | 1875.3 (8.7376) | 2534.0 (10.1969) | 2672.5 (10.4175) | 2689.5 (10.4428) |
| 3 | 359.2290 (5.7151) | 625.0951 (8.7376) | 844.6717 (10.1969) | 890.8388 (10.4175) | 896.5038 (10.4428) | |
| 5 | 215.5374 (5.7151) | 375.0570 (8.7376) | 506.8030 (10.1969) | 534.5033 (10.4175) | 537.9023 (10.4428) | |
| 7 | 153.9553 (5.7151) | 267.8979 (8.7376) | 362.0022 (10.1969) | 381.7881 (10.4175) | 384.2159 (10.4428) | |
| 9 | 119.7430 (5.7151) | 208.3650 (8.7376) | 281.5572 (10.1969) | 296.9463 (10.4175) | 298.8346 (10.4428) | |
j_ama-2025-0040_utab_001
| Symbol | Explanation |
|---|---|
| d | Depth of the ferromagnetic fluid layer (m) |
| t | Time variable (s) |
| T | Temperature (K) |
| T0, T1 | Reference temperatures at z = 0 and z = d respectively (K) |
| r | Radius of a dust particle (m) |
| m | Mass of a dust particle (kg) |
| Nd | Number density of dust particles (particles/3) |
| N’ | Disturbance to number density of dust particles (particles/3) |
| Cpt | Heat capacity of dust particles (kJ/m3K) |
| CH,V | Specific heat at constant magnetic field and volume (kJ/m3K) |
| K1 | Thermal conductivity |
| K2 | Pyromagnetic coefficient = –∂M/∂T at (T0, H0) (A/mK) |
| K | Stokes drag coefficient (kg/s) |
| Das | Darcy number accounting for velocity slip |
| p | Fluid pressure (psi) |
| p′ | Pressure perturbation (psi) |
| q | Velocity of the ferrofluid (m/s) |
| q′ = (u′, v′, w′) | Velocity perturbation of the fluid (m/s) |
| qd | Dust particle velocity (m/s) |
| qd′ = (l′, r′, s′), | Dust velocity disturbance (m/s) |
| g | Gravitational acceleration, g = (0,0,–g) (m/s2) |
| D/Dt | Convective (material) derivative (s−1) |
| kx1, ky1 | Horizontal wave numbers in x and y directions (m−1) |
| k | Overall wave number, k = ✓(kx12 + ky12) (m−1) |
| Unit vector in the vertical (z) direction | |
| H | Magnetic field vector (A/m) |
| H, H0, Hext | Magnitude of field, reference field, and external magnetic field (A/m) |
| H′ | Magnetic field perturbation (A/m) |
| B | Magnetic flux density (T) |
| B | Magnitude of magnetic induction (T) |
| M | Magnetization (A/m) |
| M′ | Magnetization disturbance (A/m) |
| M0 | Magnetization at (H0, T0) (A/m) |
| b | Subscript indicating base (equilibrium) state |
| Greek Symbols | |
| α | Thermal expansion coefficient (K−1) |
| β | Constant thermal gradient |dT/dz| (K/m) |
| v | Kinematic viscosity (m2/s) |
| μ | Dynamic viscosity (kg/m·s |
| μ1 | Dynamic viscosity under ambient magnetic field (kg/m·s |
| μ0 | Magnetic permeability of vacuum (H/m) |
| ρ | Fluid density (kg/m3) |
| ρ0 | Reference density at T0 (kg/m3) |
| ρ′ | Density perturbation (kg/m3) |
| θ′ | Temperature fluctuation (K) |
| ω | Rate of growth of disturbances (s−1) |
| χ | Magnetic susceptibility = ∂M/∂H at (T0, H0) |
| ▽ | Gradient (del) operator (m−1) |
| ϕ′ | Perturbation in magnetic scalar potential (A) |