  Unidirectional start-up flow of a viscoelastic fluid in a pipe with
fractional Maxwell's model is studied. The flow starting from rest is driven by
a constant pressure gradient in an infinite long straight pipe. By employing
the method of variable separations and Heaviside operational calculus, we
obtain the exact solution, from which the flow characteristics are
investigated. It is found that the start-up motion of fractional Maxwell's
fluid with parameters $\alpha$ and $\beta$, tends to be at rest as time goes to
infinity, except the case of $\beta=1$. This observation, which also can be
predicted from the mechanics analogue of fractional Maxwell's model, agrees
with the classical work of Friedrich and it indicates fractional Maxwell's
fluid presents solid-like behavior if $\be\neq 1$ and fluid-like behavior if
$\be=1$. For an arbitrary viscoelastic model, a conjecture is proposed to give
an intuitive way judging whether it presents fluid-like or solid-like behavior.
Also oscillations may occur before the fluid tends to the asymptotic behavior
stated above, which is a common phenomenon for viscoelastic fluids.
