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# Convection due to an unstable density difference across a permeable membrane

Puthenveettil, Baburaj A and Arakeri, Jaywant H (2008) Convection due to an unstable density difference across a permeable membrane. In: Journal of Fluid Mechanics, 609 . pp. 139-170.

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## Abstract

We study natural convection driven by unstable concentration differences of NaCl across a horizontal permeable membrane at Rayleigh numbers (Ra) of $10^{10}$ to $10^{11}$; the Schmidt number $(Sc) = 600$. A layer of brine lies over a layer of distilled water, separated by the membrane, in square cross section tanks. The membrane is permeable enough to allow a small flow across it at higher driving potentials. Based on the predominant mode of transport across the membrane, three regimes of convection, viz. an advection regime, a diffusion regime and a combined regime, are identified. The near-membrane flow in all the regimes consists of sheet plumes formed from the unstable layers of fluid near the membrane. In the advection regime bserved at higher concentration differences $(\Delta C)$ across the membrane, there is a slow overturning through-flow across the membrane; the transport across the membrane occurs mostly by advection. This phenomenology explains the observed $Nu_b\sim Ra^2/Sc$ scaling of Nusselt number. The planforms of sheet plumes near the membrane show a dendritic structure due to the combined influence of the mean shear due to the large scale flow and the entrainment flow of the adjacent plumes. The near-membrane dynamics show initiation, elongation and merger of plumes. Increase in Ra results in more number of closely and regularly spaced sheet plumes. The mean plume spacing in the advection regime $\overline{\lambda}_b$, is larger than the mean plume spacing in Rayleigh - $B\'{e}nard$ Convection $(\overline{\lambda})$, and shows a different Ra dependence. The plume spacings in the advection regime $(\lambda_b)$ show a common log-normal probability density function at all Ra. We propose a phenomenology which predicts $\overline{\lambda}_b\sim \sqrt{Z_wZ_{V_i}}$, where $Z_w$ and $Z_{V_i}$ are respectively the near-wall length scales in Rayleigh - $B\'{e}nard$ convection (RBC) and due to the advection velocity. In the combined regime, which occur at intermediate values of $\Delta C$, the flux scales as $(\Delta C/2)^{4/3}$. At lower driving potentials, in the diffusion regime, the flux scaling is similar to that in turbulent RBC.

Item Type: Journal Article Publisher Copyright of this article belong to Baburaj A Puthenveettil and Jaywant H Arakeri. Division of Mechanical Sciences > Mechanical Engineering 22 Oct 2008 09:43 19 Sep 2010 04:51 http://eprints.iisc.ernet.in/id/eprint/16270

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