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Distant side-walls cause slow amplitude modulation of cellular convection

  • Lee A. Segel (a1)


The effect of vertical boundaries on convection in a shallow layer of fluid heated from below is considered. By means of a multiple-scale perturbation analysis, results for horizontally unbounded layers are modified so that they ‘fit in’ to a rectangular region. The critical Rayleigh number and critical wave-number are determined. Motion is predicted to have the form of finite ‘rolls’ whose axes are parallel to the shorter sides of the dish. Aspects of the non-linear development and stability of this motion are studied. The general question of convective pattern selection in a bounded layer is discussed in the light of available theoretical and experimental results.



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Arscott, F. 1964 Periodic Differential Equations. New York: Pergamon.
Benney, D. & Newell, A. 1967 The propagation of nonlinear wave envelopes J. Math. Phys. 46, 133139.
Brindley, J. 1967 Thermal convection in horizontal fluid layers J. Inst. Math. Appl. 3, 313343.
Busse, F. 1967a On the stability of two-dimensional convection in a layer heated from below J. Math. Phys. 46, 140150.
Busse, F. 1967b The stability of finite amplitude cellular convection and its relation to an extremum principle J. Fluid Mech. 30, 625649.
Chen, M. & Whitehead, J. 1968 Evolution of two-dimensional periodic Rayleigh convection cells of arbitrary wave-numbers J. Fluid Mech. 31, 116.
Cowley, M. & Rosensweig, R. 1967 The interfacial stability of a ferromagnetic fluid J. Fluid Mech. 30, 671688.
Davis, S. 1967 Convection in a box: linear theory. J. Fluid Mech. 30, 465478.
Davis, S. 1968 Convection in a box: on the dependence of preferred wave-number upon the Rayleigh number at finite amplitude. J. Fluid Mech. 32, 619624.
Davis, S. & Segel, L. 1968 Effects of surface curvature and property variation on cellular convection Phys. Fluids 11, 470476.
Eckhaus, W. 1965 Studies in Nonlinear Stability Theory. New York: Springer.
Erdelyi, A. 1968 Two-variable expansions for singular perturbations J. Inst. Math. Appl. 4, 113119.
Görtler, H. & Velte, W. 1967 Recent mathematical treatments of laminar flow and transition problems Phys. Fluids Suppl. 10, S310.
Koschmieder, L. 1966 On convection on a uniformly heated plane Beit. z. Phys. Atmos. 39, 111.
Koschmieder, L. 1967 On convection under an air surface J. Fluid Mech. 30, 915.
Krishnamurti, R. 1968 Finite amplitude convection with changing mean temperature J. Fluid Mech. 33, 445463.
Milne-Thompson, L. M. 1950 Jacobian Elliptic Function Tables. New York: Dover.
Newell, A. & Whitehead, J. 1969 Finite bandwidth, finite amplitude convection. J. Fluid Mech. (To be published.)
Palm, E. 1960 On the tendency towards hexagonal cells in steady convection J. Fluid Mech. 8, 183192.
Sani, R. 1964 Note on flow instability in heated ducts ZAMP 15, 381387.
Scanlon, J. & Segel, L. 1967 Finite amplitude cellular convection induced by surface tension J. Fluid Mech. 30, 149162.
Schlüter, A., Lortz, D. & Busse, F. 1965 On the stability of finite amplitude convection J. Fluid Mech. 23, 129144.
Segel, L. 1965 The nonlinear interaction of a finite number of disturbances to a layer of fluid heated from below J. Fluid Mech. 21, 35984.
Segel, L. 1966 Nonlinear hydrodynamic stability theory and its applications to thermal convection and curved flows. Non-equilibrium Thermodynamics, Variational Techniques, and Stability (R. Donnelly, R. Herman, and I. Prigogine, eds.). University of Chicago Press, 165197.
Snyder, H. 1969 Wave-number selection at finite amplitude in rotating Couette flow. To appear in J. Fluid Mech.
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