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Thermocapillary and buoyant flows with low frequency jitter. I. Jitter confined to the plane

Grassia, P. and Homsy, G. M. (1998) Thermocapillary and buoyant flows with low frequency jitter. I. Jitter confined to the plane. Physics of Fluids, 10 (6). pp. 1273-1290. ISSN 1070-6631

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A temperature gradient is applied along a fluid filled slot with a flat upper interface, establishing flow via thermocapillarity and/or buoyancy. There is a known parallel flow along the slot, in which the fluid velocity varies vertically, and there is a known convected temperature profile. This parallel flow is then subjected to gravitational modulation or “jitter” which is applied at low frequency and in various directions. For gravity modulations in the plane of the basic flow, analytic solutions for velocity and temperature profiles are obtained for jitter of arbitrary amplitude. These solutions involve modifications to the earlier parallel flow solutions. Jitter in the vertical direction generates vorticity due to coupling with the applied horizontal temperature gradient. This alternately cooperates or competes with the steady basic flow over a cycle of the modulation, but does not qualitatively change the flow or temperature profiles. Jitter applied along the slot produces vorticity only when coupled to vertical convected temperature gradients and so is important when the basic flow is sufficiently strong (large Marangoni and/or Rayleigh number). Various cases are considered for the basic flow, which may be driven by thermocapillarity alone, by vertical gravity alone or by a mixture of thermocapillarity and vertical gravity. When strong streamwise jitter is added to any of these cases, the flow profile alternates during the modulation cycle between boundary layer structures and vertically stacked cells. The type of structure selected depends on the sense of the horizontal thermal stratification with respect to the jitter, and in that part of the cycle where this stratification is unstable, there are particular amplitudes of jitter which can give strong cellular motions or runaways. These runaways represent a resonant interaction with stationary Rayleigh-Bénard cells.