  We investigate the shallow flow of viscous fluid into and out of a channel
whose gap width increases as a power-law ($x^n$), where $x$ is the downstream
axis. The fluid flows slowly, while injected at a rate in the form of
$t^\alpha$, where $t$ is time and $\alpha$ is a constant. The invading fluid
has higher viscosity than the ambient fluid, thus avoiding Saffman-Taylor
instability. Similarity solutions of the first kind for the outflow problem are
found using approximations of lubrication theory. Zheng et al [2014] studied
the deep-channel case and found divergent behaviour of the similarity variable
as $n\rightarrow1$ and $n\rightarrow3$, when fluid flows into and out of the
channel respectively. No divergence is found in the shallow case presented
here. The characteristic equilibration time for the numerically simulated
constant-volume flow to converge to the similarity solution is calculated
assuming inverse dependence on the ratio disagreement between the current front
using the method of lines (MOL). The inverse power dependence between
equilibration time and ratio disagreement is found for channels of different
powers. A similarity solution of the second kind for the inflow problem is
found using the phase plane formalism and the bisection method. An exponential
decay relationship is found between $n$ and the degree $\delta$ of the
similarity variable $xt^{-\delta}$, which does not show any divergent behaviour
for large $n$. An asymptotic behaviour is found for $\delta$ that approaches
$1/2$ as $n\rightarrow\infty$.
