  Phase-mixing of Alfv\'en waves in the solar corona has been identified as one
possible candidate to explain coronal heating. While this scenario is supported
by observations of ubiquitous oscillations in the corona carrying sufficient
wave energy and by theoretical models that have described the concentration of
energy in small scale structures, it is still unclear whether this wave energy
can maintain the million degree solar corona. The aim of this work is to assess
how much energy can be converted by a phase-mixing process triggered by the
propagation of Alfv\'enic waves in a cylindric coronal structure, such as a
coronal loop, and to estimate the impact on the coronal heating. To this end,
we run 3D MHD simulations of a magnetised cylinder where the Alfv\'en speed
varies through a boundary shell and a footpoint driver is set to trigger kink
modes which mode couple to torsional Alfv\'en modes in the boundary shell.
These Alfv\'en waves are expected to phase-mix and the system allows us to
study the subsequent thermal energy deposition. We run a reference simulation
to explain the main process and then we vary simulation parameters. When we
take into consideration high values of magnetic resistivity and strong
footpoint drivers, we find i) that phase-mixing leads to a temperature increase
of the order of $10^5$ K or less, depending on the structure of the boundary
shell, ii) that this energy is able to balance the radiative losses only in the
localised region involved in the heating, iii) and how the boundary layer and
the persistence of the driver influence the thermal structure of the system.
Our conclusion is that due to the extreme physical parameters we adopted and
the moderate impact on the heating of the system, it is unlikely that
phase-mixing can contribute on a global scale to the heating of the solar
corona.
