The equation of motion

The equation of motion used here is not that derived by Bird et al. (1960, pp. 320, 563) for free convection. Neglecting the variation of pressure due to motion leads to an overconstrained system so that the equation for continuity of the mixture, (2.1) or (2.6), could not be enforced (see Gresho 1988).

The basic steady-state Navier-Stokes equation (Bird et al. 1960, p. 80) is:

$$\rho\mathbf{u}_*\cdot\mbox{\boldmath$\nabla$}_* \mathbf{u}_*... ...+\rho\mathbf{g} +\mu\mbox{\boldmath$\nabla$}_* ^2\mathbf{u}_*$$ (2.8)

where $p$ and $\mu$ are the pressure and viscosity, and $\mathbf{g}$ is the body force per unit mass, assumed hereafter to be uniform and downward. The density and viscosity are assumed uniform in (2.8), except in the body force term, $\rho\mathbf{g}$, where $\rho$ is replaced by its linear Taylor series expansion as a function of the temperature and vapour mass fraction about some reference state (Bird et al. 1960, p. 563):

$$\rho \left[1 - \beta(T_*-T_{*r})-\zeta(m_*-m_{*r})\right],$$ (2.9)

where $\beta$ and $\zeta$ are the thermal and vapour mass fraction coefficients of volumetric expansion. This Boussinesq approximation forms the basis of the vast majority of studies in natural convection. Its consistency and limitations are discussed by Spiegel and Veronis (1960), Ostrach (1964), Chenoweth and Paolucci (1986), Gebhart et al. (1988, ch. 2) and Perez-Cordon and Mengual (1997), amongst others. Few of the works reviewed in §3.3 did not use the approximation, and, as noted there, no qualitatively different features were found nor were any quantitative effects on the overall vapour or energy transport rates reported. Particular attention is drawn to my earlier non-Boussinesq numerical simulations, which gave results quite consistent with Boussinesq models (McBain 1995, 1997b). Boyadjiev and Halatchev (1998) came to the same conclusion for vapour transport from a vertical semi-infinite plate. Apart from the obvious simplification of the governing equations, the principal advantage in using the approximation is the increase in generality of the results, that is, the differences between various species are reduced to their essentials.

On rearranging the simple potential part of the body force and the pressure term,

$$\mathbf{u}_*\cdot\mbox{\boldmath$\nabla$}_* \mathbf{u}_* = ... ...$\hat{\jmath}$} +\nu\mbox{\boldmath$\nabla$}_*^2\mathbf{u}_*,$$ (2.10)

where $y_*$ and $\hat{\jmath}$ are the vertical coordinate and unit vector, and $\nu=\mu/\rho$ is the kinematic viscosity.

It is obvious from (2.10) that the temperature field must be known for the distribution of velocity, and therefore vapour, to be calculable. To this end, the equation of (thermal) energy is obtained in §2.1.3.