Abstract
The influence of boundaries on the dynamics of a compositional plume is studied using a simple model in which a column of buoyant fluid rises in a less buoyant fluid bounded by two vertical walls with a finite distance apart. The problem is governed by four dimensionless parameters: The Grashoff number, R, which is a measure of the difference in concentration of light material of the plume to its surrounding fluid, the Prandtl number, σ, which is the ratio of viscosity, ν, to thermal diffusivity, κ, the thickness of the plume, 2x0, and the distance, d, between the two vertical walls relative to the salt-finger length scale. The influence of the boundary on the fluxes of material, heat, and buoyancy is examined to find that the buoyancy flux possesses a local maximum for moderate to small thicknesses of the plume when they lie close to the wall. This has the effect of introducing a region of instability for thin plumes near the wall with an asymptotically larger growth rate. In addition, the presence of the boundary suppresses the three-dimensional instabilities present in the unbounded domain and allows only two-dimensional instabilities for moderate to small distances between the bounding walls.
Highlights
Studies on the dynamics of fluid alloys are relevant to industrial (e.g., Rees and Worster [1] and references therein), environmental (e.g., Wells et al [2] and references therein) and geophysical (e.g., Loper [3], Moffatt [4], Al-Lawatiaet al. [5]), applications
For wavenumbers for which the zeroth order solution is neutrally stable, Ω1 is purely imaginary and the non-homogeneity of the equations of problem 1 are all imaginary. It follows that the variables with subscript 1 are all imaginary. When we employ this result into the expression for Ω2, we find that Ω2 is real, and it will determine the stability of the plume outside the unstable regions of the zeroth order
We can observe that 1) when the plume is close to a sidewall, the preferred mode is two-dimensional, 2) for moderate to large values of a2, the preferred mode is of the modified sinuous (MS) type when the plume is thin but changes to modified varicose (MV) and back to MS as it approaches the wall, 3) in all cases the growth rate increases from its value for small thickness to a maximum before it decreases to a small value as the plume increases and approaches the sidewall
Summary
Studies on the dynamics of fluid alloys are relevant to industrial (e.g., Rees and Worster [1] and references therein), environmental (e.g., Wells et al [2] and references therein) and geophysical (e.g., Loper [3], Moffatt [4], Al-Lawatiaet al. [5]), applications. Theoretical studies of a compositional plume rising in a fluid of infinite extent have shown that the plume is unstable (see, e.g., Eltayeb and Loper [15]) even for small Grashoff numbers. This is found to be true even if the plume is subject to rotation or in the presence of a magnetic field even if another plume is present [5] [16]-[18]. The main purpose of this study is to examine the influence of boundaries on the dynamics of compositional plumes For this purpose, we introduce boundaries to the model discussed by Eltayeb and Loper [16].
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