  The motivation for this study is to include the effect of plasma flow in
Alfv\'en wave (AW) damping via phase mixing and to explore the observational
implications. Our magnetohydrodynamic (MHD) simulations and analytical
calculations show that, when a background flow is present, mathematical
expressions for the AW damping via phase mixing are modified by the following
substitution: $C_A^\prime(x) \to C_A^\prime(x)+V_0^\prime(x)$, where $C_A$ and
$V_0$ are AW phase and the flow speeds, and the prime denotes a derivative in
the direction across the background magnetic field. In uniform magnetic fields
and over-dense plasma structures, where $C_A$ is smaller than in the
surrounding plasma, the flow, which is confined to the structure and going in
the same direction as the AW, reduces the effect of phase-mixing, because on
the edges of the structure $C_A^\prime$ and $V_0^\prime$ have opposite signs.
Thus, the wave damps by means of slower phase-mixing compared to the case
without the flow. This is the result of the co-directional flow that reduces
the wave front stretching in the transverse direction. We apply our findings to
addressing the question why over-dense solar coronal open magnetic field
structures (OMFS) are cooler than the background plasma. Observations show that
the over-dense OMFS (e.g. solar coronal polar plumes) are cooler than
surrounding plasma and that, in these structures, Doppler line-broadening is
consistent with bulk plasma motions, such as AW. If over-dense solar coronal
OMFS are heated by AW damping via phase-mixing, we show that, co-directional
with AW, plasma flow in them reduces the phase-mixing induced-heating, thus
providing an explanation of why they appear cooler than the background.
