Natural flow

Solution of (7.20) requires a particular integral. Two rational choices for this integral can be obtained by considering the asymptotic solutions for large and small $\mathcal S$.

The solution valid in the limit $\mbox{$\mathcal S$}\rightarrow\infty$ is (cf. equation 4.36):

$$v_n^{\Vert}=\frac{X(1-4X^2)}{24}.$$ (7.24)

This result, due to Jones and Furry (1946), and used by Batchelor (1954) and many others (Vest & Arpaci 1969; Hart 1971; Gershuni & Zhukhovitskii 1976, ch. 10; Nagata & Busse 1983; Daniels 1985; Chait & Korpela 1989) in the analysis of narrow cavities, describes the flow between infinite parallel plane vertical walls. It satisfies (7.21) at $X=\pm 1/2$ but not at $Z=\pm1/2$. The combination $v_f^{\Vert}+v_n^{\Vert}$ was noted by Aung (1972) as the solution for the transversely heated duct problem with $\mbox{$\mathcal S$}\rightarrow\infty$.

The solution of equation (7.20) matching the boundary conditions (7.21) then follows readily from Fourier's method:

$$v_n = v_n^{\Vert} + \frac{1}{4\upi ^3}\sum_{k=1}^\infty \f... ...\upi \mbox{$\mathcal S$}Z)}{\cosh(k\upi \mbox{$\mathcal S$})}.$$ (7.25)

The difference $v_n-v_n^{\Vert}$ quantifies the effect of the end-walls.

For small $\mathcal S$, the geometry approaches that of the flow in the experiments of Hele-Shaw (1898, 1899)--the dominant effect on the flow being from the viscous damping of the end-walls. These experiments were analysed by Stokes (1899) who showed that the general form for the vertical velocity component is:

$$v_n(X,Z) = \mbox{$\mathcal S$}^2v_n(X,0)\frac{1-4Z^2}{8}$$ (7.26)

The function of this form satisfying (7.20) is

$$v_n^{=} = \mbox{$\mathcal S$}^{2}\frac{X(1-4Z^2)}{8}.$$ (7.27)

This function satisfies the boundary conditions (7.21) at $Z=\pm1/2$, but not those at $X=\pm 1/2$. The corresponding full solution is:

$$v_n = v_n^{=} - \frac{2S^2}{\upi ^3}\sum_{k=0}^{\infty} \... ...{\sinh[(2k+1)\upi /2\mbox{$\mathcal S$}]} \cos[(2k+1)\upi Z],$$ (7.28)

which clearly tends to (7.27) as $\mbox{$\mathcal S$}\rightarrow 0$, for fixed $X$, $\vert X\vert<1/2$. The difference $v_n-v_n^{=}$ quantifies the effect of the hot and cold walls on the Hele-Shaw flow.

Both (7.25) and (7.28) are exact solutions of (7.20) and (7.21), and also; when combined with (7.16), (7.13) and (7.22); the full equation of motion (2.54), for all values of $\mbox{$\mathcal S$}\in(0,\infty)$.