Abstract
In chemical engineering applications, the operation of condensers and evaporators can be made more efficient by exploiting the transport properties of interfacial waves excited on the interface between a hot vapor overlying a colder liquid. Linear theory for the onset of instabilities due to heating a thin layer from above is computed for the Marangoni–Bénard problem. Symbolic computation in the long wave asymptotic limit shows three stationary, non-growing modes. Intersection of two decaying branches occurs at a crossover long wavelength; two other modes co-exist at the crossover point—propagating modes on nascent, shorter wavelength branches. The dispersion relation is then mapped numerically by Newton continuation methods. A neutral stability method is used to map the space of critical stability for a physically meaningful range of capillary, Prandtl, and Galileo numbers. The existence of a cut-off wavenumber for the long wave instability was verified. It was found that the effect of applying a no-slip lower boundary condition was to render all long waves stationary. This has the implication that any propagating modes, if they exist, must occur at finite wavelengths. The computation of 8000 different parameter sets shows that the group velocity always lies within 1 2 to 2 3 of the longwave phase velocity.
Highlights
Marangoni–Bénard convection still generates much interest in fluid and nonlinear dynamics due to its complexity
An analytical study based on linear theory has been presented that considers the development of instabilities that arise when a thin layer of a stably stratified fluid is heated from above for the Marangoni–Bénard problem
Particular focus was directed towards the value of the critical Marangoni number and to the influence of the lower boundary condition
Summary
Marangoni–Bénard convection still generates much interest in fluid and nonlinear dynamics due to its complexity. When the fluid layer is heated from below, convective instabilities can be driven by surface or buoyancy forces [1]. The role of surface-tension gradients in inducing convective instability through Marangoni stresses at the air–liquid interface in a thin layer initially at rest, heated from below, was characterized in the seminal works of Sternling and Scriven [2] and Smith [3] and the role of surface deformation and surface tension gradients in the onset of patterned convection and oscillatory instability is reviewed in [4]. When the layer is heated from above, only overstability can be excited at sufficiently high Marangoni or Rayleigh (buoyancy) numbers. There are many different configurations for condensers, but all of them would condense faster if interfacial waves are excited on the interface
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