Long wavelength instabilities of square patterns. (English) Zbl 0787.76024
Summary: The long wavelength instabilities of square and rectangular planforms are studied analytically and numerically, using amplitude equations which describe the general interaction of two orthogonal coupled roll patterns. The zigzag and two-dimensional Eckhaus instabilities are found, and in addition it is discovered that the three-dimensional equivalent of the Eckhaus instability splits into two variants. The square Eckhaus instability is the direct equivalent of the two-dimensional case, whereas the rectangular Eckhaus instability is truly three-dimensional in character. In the case of square patterns, nonlinear phase diffusion equations are derived close to the onset of the instabilities. A short wavelength cross square mode is also discussed briefly.
MSC:
| 76E15 | Absolute and convective instability and stability in hydrodynamic stability |
Keywords:
zigzag instability; amplitude equations; Eckhaus instability; nonlinear phase diffusion equationsReferences:
| [1] | Busse, F. H.; Riahi, N., J. Fluid Mech., 96, 243 (1980) · Zbl 0429.76055 |
| [2] | Proctor, M. R.E., J. Fluid Mech., 113, 469 (1981) · Zbl 0484.76088 |
| [3] | Le Gal, P.; Croquette, V., Phys. Fluids, 31, 3440 (1988) |
| [4] | Jenkins, D. R.; Proctor, M. R.E., J. Fluid Mech., 139, 461 (1984) · Zbl 0554.76041 |
| [5] | Riahi, N., J. Fluid Mech., 152, 113 (1985) · Zbl 0585.76134 |
| [6] | Oliver, D. S.; Booker, J. R., Geophys. Astrophys. Fluid Dyn., 27, 73 (1983) |
| [7] | White, D. B., J. Fluid Mech., 191, 247 (1988) |
| [8] | Busse, F. H.; Frick, H., J. Fluid Mech., 150, 451 (1985) · Zbl 0588.76073 |
| [9] | Jenkins, D. R., J. Fluid Mech., 178, 491 (1987) · Zbl 0634.76086 |
| [10] | Moses, E.; Steinberg, V., Phys. Rev. Lett., 57, 2018 (1986) |
| [11] | Bigazzi, P.; Ciliberto, S.; Croquette, V., J. Phys. (Paris), 51, 611 (1990) |
| [12] | Moses, E.; Steinberg, V., Phys. Rev. A, 43, 707 (1991) |
| [13] | Müller, H. W.; Lücke, M., Phys. Rev. A, 38, 2965 (1988) |
| [14] | Knobloch, E., Phys. Rev. A, 40, 1549 (1989) |
| [15] | Clune, T.; Knobloch, E., Phys. Rev. A, 44, 8084 (1991) |
| [16] | Pieranski, P.; Dubois-Violette, E.; Guyon, E., Phys. Rev. Lett., 30, 736 (1973) |
| [17] | Salan, J.; Guyon, E., J. Fluid Mech., 126, 13 (1983) |
| [18] | Feng, Q.; Decker, W.; Pesch, W.; Kramer, L., J. Phys. II France, 2, 1303 (1992) |
| [19] | Newell, A. C.; Whitehead, J. A., J. Fluid Mech., 38, 279 (1969) · Zbl 0187.25102 |
| [20] | Segel, L. A., J. Fluid Mech., 38, 203 (1969) · Zbl 0179.57501 |
| [21] | Busse, F. H.; Clever, R. M., J. Fluid Mech., 91, 319 (1979) |
| [22] | Clever, R. M.; Busse, F. H., J. Fluid Mech., 65, 625 (1974) · Zbl 0291.76019 |
| [23] | Eckhaus, W., Studies in nonlinear stability theory (1965), Springer: Springer Berlin · Zbl 0125.33101 |
| [24] | Zaleski, S., (Thèse de 3ème cycle (1980), Université de Paris VI) |
| [25] | Milner, S. T., J. Fluid Mech., 225, 81 (1991) · Zbl 0850.76758 |
| [26] | Malomed, B. A.; Tribel’skiĭ, M. I., Sov. Phys. JETP, 65, 305 (1987) |
| [27] | Cross, M. C., Phys. Fluids, 23, 1727 (1980) · Zbl 0444.76066 |
| [28] | Pomeau, Y.; Manneville, P., J. Physique Lett., 40, L609 (1979) |
| [29] | Kuramoto, Y., Prog. Theor. Phys., 71, 1182 (1984) · Zbl 1046.76502 |
| [30] | Busse, F. H., Rep. Prog. Phys., 41, 1929 (1978) |
| [31] | Greenside, H. S.; Coughran, W. M., Phys. Rev. A, 30, 398 (1984) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
