Advances in Chemical Physics: Proceedings of the Conference

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Extra info for Advances in Chemical Physics: Proceedings of the Conference on Instability and Dissipative Strutures in Hydrodynamics, Volume 32

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Further experiments by Hooper did not support the findings of Tritton and Zarraga: the dimensions of the cells were found to increase only slowly with R (see also de la Cruz Reyna, and McKenzie, Roberts and we is^^^). We should observe that Graham23has recently suggested an alternative method of obtaining a generalized thermodynamics for continua in motion, and has set up a functional equation o€ the Fokker-Planck type €or the probability of different states in BCnard convection at small R - R,.

These smooth Hermite spaces are discussed, for example, in Ref. 23. The case m = 1 (piecewise cubic) is of particular interest. Letting O = x,< x1< x1 * < x,, = 1, a basis for this space is given by (see Figs. 3b and 3c) ti = 1 x, < x (2- - 1) 2-2, 4-1 < xi+ xi-l < x < xi where t = ( x - xi)/Ai, z - = (x - xi- l)/Ai- I, and At = xi+ - xi. If we are considering, for example, a two-point boundary value problem with u(0) = u( 1)=0, then we represent our approximate u, by + n- I uZ"(x)= UiC#+(X)+ i: Ui'&(X) (9) SOME REMARKS ON VARIATIONAL METHODS 45 Our functional then becomes a function of the 2 n variables {ui}yii and { u , !

If u and K are complex, but % ( K ) > O , the solution approaches a steady oscillation with a frequency which differs from wc by order (R - Rc). If u is real and K is negative, A increases catastrophically; after a time (T) of order 1, (26) will fail to govern it. In this case subcritical instabilities occur, and the criterion R < R, of the linear theory is necessary but not sufficient for stability. If R < R,, small amplitude perturbations will decay, but those for which IA12>u(Rc- R ) / ( - K) will grow indefinitely.

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