# Free Convection in a Multicomponent Fluid

• Jean Karl Platten
• Jean Claude Legros

## Abstract

The preceeding chapters were concerned with one component fluids, but multi-component fluids are the more likely to be encountered in nature. Therefore generally concentration gradients and concentration fluctuations are present and must be taken into account. Particularly, one should like to know how much concentration fluctuations affect the stability of a multicomponent system heated from below. Our interest in the two-component Bénard problem was stimulated by the suggestion that the critical Rayleigh number could be used in order to determine the value of the Soret coefficient (cf. chapter I, § 3). Indeed, the critical Rayleigh number can be written as
$${\rm{R}}{{\rm{a}}^{{\rm{Crit}}}} = {{{\rm{g}}[ - \rho _{\rm{m}}^{ - 1}(\partial \rho /\partial {\rm{T}}]\Delta {{\rm{T}}^{{\rm{Crit}}}}\;{{\rm{h}}^3})} \over {{{\cal Z}^\nu }}} = {{{\rm{g}}\rho _{\rm{m}}^{ - 1}\;\Delta _\rho ^{{\rm{Crit}}}\;{{\rm{h}}^3}} \over {{{\cal Z}^\nu }}}$$
(IX.1)
where $$\Delta _\rho ^{{\rm{Crit}}}$$ is the critical difference in density between top and bottom due to the thermal gradient. In a two-component system, this thermal gradient induces a concentration gradient and the local density is a function not only of the temperature, but also of the composition. Let us suppose the critical value of the Rayleigh number known and the second definition (IX.1) still valid even with a density function of temperature and composition. For a given fluid of known ℋ and ν in a particular apparatus of given depth h, one may evaluate $$\Delta _\rho ^{{\rm{Crit}}}$$, which is at the onset of convection
$$\Delta {\rho ^{{\rm{Crit}}}} = {({\textstyle{{\partial \rho } \over {\partial {\rm{T}}}}})_{\rm{N}}}\,\Delta {{\rm{T}}^{{\rm{Crit}}}} + {({\textstyle{{\partial \rho } \over {\partial {\rm{N}}}}})_{\rm{T}}}\,\Delta {{\rm{N}}^{{\rm{Crit}}}}$$
(IX.2)
Here, N is a variable describing the local composition of the fluid; it is convenient to take N as the mass fraction of one of the components. $$\rho _{\rm{m}}^{ - 1}{({\textstyle{{\partial \rho } \over {\partial {\rm{T}}}}})_{\rm{N}}}$$ as well as $$\rho _{\rm{m}}^{ - 1}{({\textstyle{{\partial \rho} \over {\partial {\rm{N}}}}})_{\rm{T}}}$$ are known properties of the fluid and ΔTCrit is measured.

## Keywords

Heat Flux Nusselt Number Rayleigh Number Critical Rayleigh Number Concentration Fluctuation
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