The nonexistence of hydrostatic solutions

One of the most interesting problems in natural convection is whether or not buoyancy forces lead to flow. In addition to much work on the analogous single fluid heat transfer problem (see, for example, Busse 1978; Chandrasekhar 1981, ch. 2), there have been studies on isothermal (Sparrow et al. 1985; Suehrcke & Harris 1995; Suehrcke et al. 1996) and heated (Olson & Rosenberger 1979; Abernathy & Rosenberger 1981) gas-vapour mixtures. A still greater variety of phenomena is displayed by liquid mixtures, for which the Schmidt number is typically much larger than the Prandtl number (Turner 1973, pp. 251-9; 1974, 1985; Griffiths, R. W. 1979).

In contrast to these studies, where the imposed density gradient is typically parallel or antiparallel to the gravitational field, the above (§2.5) compositional and thermal boundary conditions mean that they are perpendicular. This always leads to flow; i.e. $\mathbf{u}\not\equiv\mathbf{0}$.

Taking the curl of the equation of motion (2.54) and setting the velocity to zero gives:

$$\mathbf{0} = -\mbox{\boldmath$\hat{\jmath}$}\times\frac{\mbox{\boldmath$\nabla$}(T+Nm)}{1+N},$$ (2.93)

which implies that $(T+Nm)$ depends only on $y$ (Gershuni & Zhukhovitskii 1976, p. 6). This is only consistent with the boundary conditions described in §2.5 if $N=-1$ and $T\equiv m$. The scheme of nondimensionalization (2.27) breaks down in this case, as $\mathbf{u}$ must be unbounded if $\mathbf{u}_*$ is nonzero. The possibility of hydrostatic solutions is a mathematically correct deduction from $N=-1$ and the equation of motion, but not a physical possibility. Flows at $N=-1$ were numerically calculated by Mahajan and Angirasa (1993) for free convection from a semi-infinite vertical plate.

To see how there can be a flow at $N=-1$, consider the vapour transport analog of Patterson and Imberger's (1980) cavity suddenly heated from the side. The fluid is initially isothermal and uniform, and then the temperature and vapour mass fraction at one vertical wall are suddenly raised and lowered, respectively. The different diffusivities of temperature and species, inversely proportional to the Prandtl and Schmidt numbers, respectively, mean that the thermal and solutal disturbances propagate at different rates (conduction and diffusion must precede convection). Assume, for definiteness, properties appropriate for air-water vapour: $\mbox{\textit{Pr}}>\mbox{\textit{Sc}}$ and $\zeta>0$. After a short time, the layer of desiccated air will be thicker than the layer of heated air, and the concentration gradients less than the temperature gradients. The outer isothermal desiccated air will be heavier than the still uniform air, further out, and so it will begin to fall. The inner heated desiccated air will be somewhat buoyed up by the thermal expansion and so fall less slowly (or even rise if the Prandtl number is sufficiently greater than the Schmidt number). There is, therefore, a nonzero flow, even though $N=-1$. The establishment of a nonzero steady flow from this initial condition is probably not impossible. Numerical solutions of this problem were obtained by Lin et al. (1990; reviewed in §3.3.11), but not for $N=-1$. The stability of the hydrostatic solution at $N=-1$ was considered by Gobin and Bennacer (1994) for zero mass transfer rate factor; though their problem seems rather artificial, since surely non-Boussinesq effects (property variations) or imperfections in the boundary conditions would prevent the required perfect balancing of the opposing buoyancy forces.

Hydrostatic solutions can be completely ruled out when the mass transfer rate factor, $\varPhi$, is finite. The transpiration boundary condition (2.59) would require the normal derivative of $m$ to vanish at the mass transfer interfaces. If the domain's boundary consists of mass transfer interfaces and impermeable surfaces, then a hydrostatic solution would require that $\mbox{\boldmath$\hat{n}$}\cdot\mbox{\boldmath$\nabla$}m=0$ everywhere on the boundary. Further, when $\mathbf{u\equiv 0}$, the species equation (2.53) reduces to Laplace's equation. It is well known that the only harmonic functions whose normal derivatives vanish over the entire boundary of a domain are constants (Lamb 1932, p. 41). This is clearly inconsistent with the imposed mass fraction difference, $\Delta m_*\neq 0$, so that hydrostatic solutions are indeed impossible.

This is to be contrasted with the situation for infinite horizontal layers (Pellew & Southwell 1940), or closed cylinders (Charlson & Sani 1970, 1971) filled with single fluids, or impermeable enclosures filled with gas-vapour mixtures (Olson & Rosenberger 1979; Abernathy & Rosenberger 1981) where there do exist hydrostatic solutions for all values of $\mbox{\textit{Gr}}(1+N)$, if the imposed temperature gradient is vertical.