An invariance property

It should be expected on physical grounds that the system is invariant in some sense if the `cold' and `hot' walls are renamed; i.e. if one looks at the same cuboid from behind.

This is most easily seen if the governing equations are expressed in tensor form. For definiteness, consider uniform vapour mass fraction and temperature on the cold and hot walls and let the other four walls of the cuboid be perfectly conducting and impermeable. As usual, all walls are assumed to be non-slip. With the contravariant coordinates $(x^1,x^2,x^3)=(x,y,z)$, (2.52)-(2.55) can be rewritten:

$$u^i_{,i} = 0;$$ (2.94)

$$\mbox{\textit{Gr}}(1+N)\mbox{\textit{Sc}}\;u^im_{,i} = \gamma^{ij} m_{,ij}$$ (2.95)

$$\mbox{\textit{Gr}}(1+N)\;u^j u^i_{,j} +\mbox{$\mathcal A$}\gamma^{ij}p_{,j} - \frac{T+Nm}{1+N}\gamma^{ij}x^2_{,j} = \gamma^{jk}u^i_{,jk} ;$$ (2.96)

$$\mbox{\textit{Gr}}(1+N)\left[ \mbox{\textit{Pr}}_r + \mbox{\textit{Pr}}_I\left(1-\mathrm{e}^{-\varPhi }\right)m \right] u^i T_{,i}$$

$$- \frac{\mbox{\textit{Pr}}_I}{\mbox{\textit{Sc}}}\left(1-\mathrm{e}^{-\varPhi }\right) \gamma^{ij} T_{,i}m_{,j} = \gamma^{ij} T_{,ij};$$ (2.97)

where subscripts and superscripts refer to covariant and contravariant components, respectively, and subscripts following a comma indicate covariant differentiation. Summation over indices appearing in both upper and lower positions in a term is understood. (See, for example, Aris 1989, ch. 7, for a summary of tensor calculus).

The inhomogeneous boundary conditions are:

$$m=T = x^1$$ (2.98)

$$x^1_{,i}u^i = \frac{1-\mathrm{e}^{-\varPhi }} {\mbox{\textit{Gr}}(1+N)\mbox{\te... ...eft[\left(1-\mathrm{e}^{-\varPhi }\right)m-1\right]} \gamma^{ij}x^1_{,i} m_{,j}$$ (2.99)

at $x^1=0,1$ and

$$T = x^1$$ (2.100)

on the other four walls.

The components of the metric tensor, $\gamma_{ij}$, are unity on the diagonal and zero otherwise for this coordinate system; therefore, since $\gamma^{ij}$ denotes the $ij^{th}$ element of the inverse of the matrix of $\gamma_{ij}$,

$$\gamma^{11}=\gamma^{22}=\gamma^{33}=1$$ (2.101)

and

$$\gamma^{ij}=0\qquad (i\neq j).$$ (2.102)

The transformation discussed above is:

$$(\tilde{x}^1, \tilde{x}^2, \tilde{x}^3) = (1-x^1, x^2, -x^3),$$ (2.103)

which leaves the metric tensor unchanged, since it is only a rigid rotation and translation of the coordinate axes.

The solution to the new system in terms of the solution of the old system is

$$\tilde{u}^i = -u^i$$ (2.104)

$$\tilde{p} = x^2/\mbox{$\mathcal A$}- p$$ (2.105)

$$\tilde{T} = 1-T$$ (2.106)

$$\tilde{m} = 1-m$$ (2.107)

if

$$\tilde{\mbox{\textit{Gr}}} = -\mbox{\textit{Gr}}$$ (2.108)

$$\tilde{N} = N$$ (2.109)

$$\tilde{\mbox{\textit{Sc}}} = \mbox{\textit{Sc}}$$ (2.110)

$$\tilde{\varPhi } = -\varPhi$$ (2.111)

$$\tilde{\mbox{\textit{Pr}}_r} = \mbox{\textit{Pr}}_r + \mbox{\textit{Pr}}_I\left(1-\mathrm{e}^{-\varPhi }\right)$$ (2.112)

$$\tilde{\mbox{\textit{Pr}}_I} = -\mbox{\textit{Pr}}_I\frac{1-\mathrm{e}^{-\varPhi }}{1-\mathrm{e}^{\varPhi }}$$ (2.113)

$$\tilde{\varLambda } = -1-\varLambda ,$$ (2.114)

which agrees with the result that would be deduced by transforming the parameters according to their definitions in terms of the physical properties and boundary conditions.

The Sherwood and Nusselt numbers are given by

$$\mbox{\textit{Sh}} = -\frac{\mbox{\textit{Gr}}(1+N)}{\varPhi }x^1_{,j} u^j = \frac{1-\... ...left[1-m\left(1-\mathrm{e}^{-\varPhi }\right)\right]} \gamma^{ij}x^1_{,j}m_{,i}$$ (2.115)

$$\mbox{\textit{Nu}} = \frac{1-\mathrm{e}^{-\varPhi _T}}{\varPhi _T} \left[ \gamma^{ij}x^1_{,j} T_{,i} + (T+\varLambda ) \varPhi _T \mbox{\textit{Sh}}\right]$$ (2.116)

so that

$$\tilde{\mbox{\textit{Sh}}} = \mbox{\textit{Sh}}$$ (2.117)

$$\tilde{\mbox{\textit{Nu}}} = \mathrm{e}^{-\varPhi _T}\mbox{\textit{Nu}}.$$ (2.118)

The Nusselt number does not transform as simply as the Sherwood number as the driving force for energy transfer, unlike that for mass transfer, cannot be written in terms of a function of a single property at the boundaries; energy transfer is driven by both the temperature difference and the mass transfer. If $\Delta T_*$ had been taken as the driving force for energy, then the `Nusselt number'

$$\left.\frac{-\mbox{\boldmath$\hat{\imath}$}\cdot\mathbf{e}_*b}{\lambda\Delta T_*}\right\vert _{x^1=0,1}$$ (2.119)

would be an invariant of the transformation, like the Sherwood number. As will be seen in chapters 4-6, however, this would lead to the value of the Nusselt number depending more strongly on the thermal mass transfer rate factor, $\varPhi _T$.

When $\varPhi =0$, the transformation of the parameters simplifies to

$$\tilde{\varPhi } = -\varPhi = 0$$ (2.120)

$$\tilde{\mbox{\textit{Pr}}_r} = \mbox{\textit{Pr}}_r$$ (2.121)

and $\tilde{\mbox{\textit{Pr}}_I}=-\mbox{\textit{Pr}}_I$, although the interdiffusion Prandtl number is then irrelevant. In this case, both the Sherwood and Nusselt numbers are invariants of the transformation.

The significance of the present findings lies in what they reveal about the symmetry of the Sherwood and Nusselt numbers with respect to the parameters of the problem. This should be exploited when constructing the functional form of correlations, for example. An instance of this will be found in §8.2.10, when the behaviour of the system at low Grashof number is examined. Also, the simplicity of the symmetry of Sh with respect to $\varPhi$ is one advantage of using $\varPhi$ as the parameter representing finite mass transfer effects; this becomes important when considering a rational approximation for the system at low mass transfer rates (ch. 6, see especially §,6.2.2).