In chemical engineering applications, the operation of condensers and evaporators can be\nmade more efficient by exploiting the transport properties of interfacial waves excited on the interface\nbetween a hot vapor overlying a colder liquid. Linear theory for the onset of instabilities due to heating\na thin layer from above is computed for the Marangoniââ?¬â??BÃ?©nard problem. Symbolic computation\nin the long wave asymptotic limit shows three stationary, non-growing modes. Intersection of two\ndecaying branches occurs at a crossover long wavelength; two other modes co-exist at the crossover\npointââ?¬â?propagating modes on nascent, shorter wavelength branches. The dispersion relation is then\nmapped numerically by Newton continuation methods. A neutral stability method is used to map the\nspace of critical stability for a physically meaningful range of capillary, Prandtl, and Galileo numbers.\nThe existence of a cut-off wavenumber for the long wave instability was verified. It was found that\nthe effect of applying a no-slip lower boundary condition was to render all long waves stationary.\nThis has the implication that any propagating modes, if they exist, must occur at finite wavelengths.\nThe computation of 8000 different parameter sets shows that the group velocity always lies within 12\nto 23\nof the longwave phase velocity.
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