1. (10 pts) Viscous effects alter a uniform flow
within a pipe roughly according to that shown
in Fig. 1, i.e. from uniform at the inlet ($x = 0$)
to a fully-developed profile downstream ($x > L$).
Suppose that, near the wall of the pipe, the ve-
locity component along the axis, i.e. in the $x$-
direction, evolves approximately as
$u = u_0 e^{-x/L}$,
where $u_0$ and $L$ are some constant velocity and
length references. Assuming here that other ve-
locity components are negligible, determine the
acceleration of a fluid element along this axis and
comment whether acceleration is positive or neg-
ative and where (i.e. value of $x$) its magnitude is
maximum.
