Sketch of the annular cylindrical configuration.
(Color online) Various simple mode patterns. The shading describes the vertical velocity at z = 0.5; lighter shades indicate larger, positive velocities, while darker shades indicate larger, negative velocities. The arrows are the radial and azimuthal velocities at z = 1 seen from above. (a) The mode (1,0) produced using Rw = 0.01 and R = 0.6. (b) The mode (0,1) produced using Rw = 0.1 and R = 1.2. (c) The mode (1,1) produced using Rw = 0.2 and R = 1.6. (d) The mode (2,1) produced using Rw = 0.5 and R = 2.
(Color online) Contour plot of the critical Rayleigh number, Ra c , as a function of Rw and R when the sidewalls are insulated.
(Color online) Contour plot of the critical Rayleigh number, Ra c , for conducting sidewalls.
(Color online) Depiction of the mode maps for insulated sidewalls. (a) The values of m and (b) the values of p.
Comparison between the mode map of Bau and Torrance, and that of the present analysis for insulating sidewalls. (a) The mode map made by Bau and Torrance and is a copy of Figure 3 in Ref. 14. (b) Our version of the Bau and Torrance mode map. Reprinted with permission from H. H. Bau and K. E. Torrance, Phys. Fluids 24(3), 382 (1981). Copyright 1981, American Institute of Physics.
(Color online) Depiction of the mode maps for conducting sidewalls. (a) The values of m and (b) the values of p.
Variation of Ra c with R for (i) circular cylinder with an insulated sidewall (Ref. 11, lowest curve), (ii) circular cylinder with a perfectly conducting sidewall (Ref. 13, middle curve), (iii) annular cylinder with perfectly conducting sidewalls and with Rw = 10− 4. (The figure is adapted from Figure 1 in Ref. 13.)
The variation of the amplitude of convection with Rayleigh number for modes (1,0) and (2,0) for Rw = 0.1 and R = 0.7 and with insulated sidewalls. The dashed/dotted line is the amplitude of the basic mode (1,0). The continuous line is the amplitude of the stable part of the secondary mode (2,0) branch, while the dashed line is the unstable part.
(Color online) A stable mixed mode convection pattern corresponding to modes (5,0) and (2,1). The sidewalls are insulated while Rw = 0.7, R = 2.1, and Ra = 40. The pattern displays m = 2 characteristics near the inner radius and m = 5 characteristics near the outer radius. The perceived value of p is either 0 or 1 depending on the chosen azimuthal position.
The variation of the amplitude of convection with Ra for the case, Rw = 0.7 and R = 2.1, with insulated sidewalls. The continuous line corresponds to the stable part of basic mode (5,0) branch, which has the critical Rayleigh number 39.51, while the dashed line is the unstable region of this mode. The dashed/dotted line is the stable mixed mode branch.
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