The motivation for this study is to include the effect of plasma flow in Alfven 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.
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