Geometry and boundary conditions

Take the diameter of the sphere as the length scale,

, and introduce spherical polar coordinates relative to the positive

-axis:

$\displaystyle r$	$\textstyle \equiv$	$\displaystyle \left(x^2+y^2+z^2\right)^{1/2};$	(8.2)
$\displaystyle \theta$	$\textstyle \equiv$	$\displaystyle \arctan\frac{\left(x^2+y^2\right)^{1/2}}{z};$	(8.3)
$\displaystyle \phi$	$\textstyle \equiv$	$\displaystyle \arctan\frac{y}{x}.$	(8.4)

**Figure 8.1:** Cartesian axes for the spherical cavity subjected to a linear variation with of vapour mass fraction and temperature at the boundary.
$\begin{figure}\centering\epsfig{file=fig.sphere.geom.eps, width=92mm}\end{figure}$

Ostroumov's (1958) study is more general than mine in that it considers the finite conductivity of the surrounding solid. Since solids, in general, are much more conducting than gases, and since the primary purpose here is to illuminate confined convective flow, let us assume that the solid is infinitely conducting. If the temperature gradient in the solid far from the cavity is uniform and horizontal (parallel to the

-axis), the temperature at the boundary of the sphere is analogous to the flow potential on a solid sphere moving along the

-axis through a perfect fluid; i.e. it varies linearly with

(Lamb 1932, p. 123), though the gradient differs from that in the far solid. Explicitly, the temperature field

$\begin{displaymath} T = \frac{1}{3}\left(\frac{1}{8r^3}+2\right)r\sin\theta\cos\phi \end{displaymath}$

(8.5)

$\displaystyle \nabla^2 T = 0,\qquad (r>1/2)$			(8.6)
$\displaystyle \frac{\partial T}{\partial r} = 0, \qquad (r=1/2)$			(8.7)
$\displaystyle T \sim \frac{2}{3} x, \qquad (r\rightarrow\infty)$			(8.8)

**Figure 8.2:** Temperature in the highly conducting solid surrounding a spherical cavity. The field is axisymmetric everywhere, and linear in the far field.
$\begin{figure}\centering\begin{picture}(80,76)(-40,-36) \put(0,0){\makebox(0,0){... ...w$} $x$} % bbllx=54, bblly=134, bburx=532, bbury=373\} \end{picture}\end{figure}$

This point appears to have been missed by Lewis (1950) and Ostrach (1988), who thought that `this temperature corresponds to that which would occur in the solid without gas bubbles' (Ostrach 1988), which is true but irrelevant.

As in §7.5.2, it is assumed that the vapour mass fraction at the boundary is a linear function of temperature; thus,

$\begin{displaymath} m=T=x\qquad (r=\mbox{$\frac{1}{2}$}). \end{displaymath}$

(8.9)