The low Grashof number expansion

It may be worthwhile at the outset to note that although low Reynolds number expansions for unbounded domains, e.g. viscous flow past a solid sphere (Lamb 1932, p. 609; Van Dyke 1964, ch. 8), are singular perturbation problems, this is not the case for bounded domains (Munson & Joseph 1971). This makes sense: the region of nonuniformity in the former class of problems, where the neglected inertial terms are not negligible in comparison to the retained viscous terms, is typically a neighbourhood of the point at infinity.

Assume asymptotic expansions for the vapour mass fraction, velocity, pressure and temperature of the form:

$\displaystyle m$	$\textstyle \sim$	$\displaystyle m_0 + [\mbox{\textit{Gr}}(1+N)] m_1 + [\mbox{\textit{Gr}}(1+N)]^2 m_2 + \cdots$	(8.10)
$\displaystyle \mathbf{u}$	$\textstyle \sim$	$\displaystyle \mathbf{u}_0 + [\mbox{\textit{Gr}}(1+N)] \mathbf{u}_1 + [\mbox{\textit{Gr}}(1+N)]^2 \mathbf{u}_2 +\cdots$	(8.11)
$\displaystyle p$	$\textstyle \sim$	$\displaystyle p_0 + [\mbox{\textit{Gr}}(1+N)] p_1 + [\mbox{\textit{Gr}}(1+N)]^2 p_2 + \cdots$	(8.12)
$\displaystyle T$	$\textstyle \sim$	$\displaystyle T_0 + [\mbox{\textit{Gr}}(1+N)] T_1 + [\mbox{\textit{Gr}}(1+N)]^2 T_2 + \cdots$	(8.13)

Substituting these in to the low mass transfer rate equations, (6.16)-(6.19), and taking the limit $\mbox{\textit{Gr}}\rightarrow 0$ leads to a hierarchy of problems:

$\displaystyle A_{mi}$	$\textstyle =$	$\displaystyle \nabla^2 m_i;$	(8.14)
$\displaystyle A_{Ti}$	$\textstyle =$	$\displaystyle \nabla^2 T_i;$	(8.15)
$\displaystyle m_i = T_i$	$\textstyle =$	$\displaystyle \left\{ \begin{array}{lr} x, & (i=0) \\ 0, & (i=1,2,\ldots) \end{array} \right.,\quad (r=\mbox{$\frac{1}{2}$}),$	(8.16)

$\displaystyle A_{mi}$	$\textstyle \equiv$	$\displaystyle \mbox{\textit{Sc}}\sum\limits_{k=0}^{i-1} \mathbf{u}_{i-1-k}\cdot\mbox{\boldmath$\nabla$}m_{k}$	(8.17)
$\displaystyle A_{Ti}$	$\textstyle \equiv$	$\displaystyle \mbox{\textit{Pr}}\sum\limits_{k=0}^{i-1} \mathbf{u}_{i-1-k}\cdot\mbox{\boldmath$\nabla$}T_{k}$	(8.18)

$\displaystyle \mbox{\boldmath$\nabla$}\cdot \mathbf{u}_i$	$\textstyle =$	$\displaystyle 0$	(8.19)
$\displaystyle \mathbf{I}_i - \mathbf{B}_i$	$\textstyle =$	$\displaystyle -\mbox{\boldmath$\nabla$}p_i +\nabla^2 \mathbf{u}_i$	(8.20)
$\displaystyle \mathbf{u}_i$	$\textstyle =$	$\displaystyle \mathbf{0},\quad(r=\mbox{$\frac{1}{2}$})$	(8.21)

$\displaystyle \mathbf{I}_i$	$\textstyle \equiv$	$\displaystyle \sum\limits_{k=0}^{i-1} \mathbf{u}_{i-1-k}\cdot\mbox{\boldmath$\nabla$}\mathbf{u}_{k}$	(8.22)
$\displaystyle \mathbf{B}_i$	$\textstyle \equiv$	$\displaystyle \frac{T_i+Nm_i}{1+N}\mbox{\boldmath$\hat{\jmath}$}.$	(8.23)

The series for $A_{mi}$ , $A_{Ti}$ and $\mathbf{I}_{i}$ are to be taken as zero if the lower index (zero) exceeds the upper; i.e. when

. Each

and

satisfies a Laplace or Poisson equation, with source given in terms of previously calculated quantities. These are, in principle, all soluble by expanding the source term and the independent variable in spherical harmonics. For the equations of motion, however, the presence of the unknown pressure terms and the continuity constraints means that at each order a Stokes problem with known `body force' must be solved. The Stokes problem in a sphere can be reduced to an uncoupled unconstrained set of scalar partial differential equations (Poisson and inhomogeneous biharmonic equations) by decomposing the velocity term into its poloidal and toroidal parts. This technique is summarized in appendix B.

The evaluation of $\mathbf{I}_i$ may be tedious, but can be simplified by noting that except for

the terms of the series occur in pairs:

$\begin{displaymath} \mathbf{u}_{i-1-k}\cdot\mbox{\boldmath$\nabla$}\mathbf{u}_k +\mathbf{u}_k\cdot\mbox{\boldmath$\nabla$}\mathbf{u}_{i-1-k}, \end{displaymath}$

(8.24)

$\begin{displaymath} \mathbf{u}\cdot\mbox{\boldmath$\nabla$}\mathbf{v}+\mathbf{v}... ... + (\mbox{\boldmath$\nabla$}\times\mathbf{u})\times\mathbf{v}. \end{displaymath}$

(8.25)

$\displaystyle \mathbf{u}\cdot\mbox{\boldmath$\nabla$}\mathbf{u}$	$\textstyle =$	$\displaystyle \left(u_r\frac{\partial u_r}{\partial r} + \frac{u_{\theta}}{r}\f... ...ial \phi} - \frac{u_{\theta}^2+u_{\phi}^2}{r}\right)\;\mbox{\boldmath$\hat{r}$}$	(8.26)
		$\displaystyle + \left(u_r\frac{\partial u_{\theta}}{\partial r} + \frac{u_{\the... ...u_r u_{\theta}-u_{\phi}^2\cot\theta}{r}\right) \;\mbox{\boldmath$\hat{\theta}$}$
		$\displaystyle + \left(u_r\frac{\partial u_{\phi}}{\partial r} + \frac{u_{\theta... ..._{\phi}+u_{\theta}u_{\phi}\cot\theta}{r}\right) \;\mbox{\boldmath$\hat{\phi}$}.$