Forced flow

The equation (7.19) for $v_f$ is simply that for fully developed laminar flow in a duct of rectangular section, the solution for which is well known (Dryden, Murnaghan & Bateman 1956, p. 197):

$$\frac{v_f}{-\,\mathrm{d}p/\,\mathrm{d}Y} = \frac{v_f^{\parallel}}{-\,\mathrm{d}p/\,\mathrm{d}Y}$$ (7.22)

$$- 4\sum_{k=0}^{\infty}\frac{(-1)^k}{[(2k+1)\upi ]^3} \cos[(2k+1)\up... ...osh[(2k+1)\upi \mbox{$\mathcal S$}Z]}{\cosh[(2k+1)\upi \mbox{$\mathcal S$}/2]},$$

where $v_f^{\parallel}$ is the solution valid for $\mbox{$\mathcal S$}\rightarrow\infty$ (Lamb 1932, p. 582):

$$\frac{v_f^{\parallel}}{-\,\mathrm{d}p/\,\mathrm{d}Y} = \frac{1-4X^2}{8}.$$ (7.23)