  Recent numerical studies revealed that transverse motions of coronal loops
can induce the Kelvin-Helmholtz Instability (KHI). This process could be
important in coronal heating because it leads to dissipation of energy at small
spatial-scale plasma interactions. Meanwhile, small amplitude decayless
oscillations in coronal loops have been discovered recently in observations of
SDO/AIA. We model such oscillations in coronal loops and study wave heating
effects, considering a kink and Alfv{\'e}n driver separately and a mixed driver
at the bottom of flux tubes. Both the transverse and Alfv{\'e}n oscillations
can lead to the KHI. Meanwhile, the Alfv{\'e}n oscillations established in
loops will experience phase mixing. Both processes will generate small
spatial-scale structures, which can help the dissipation of wave energy.
Indeed, we observe the increase of internal energy and temperature in loop
regions. The heating is more pronounced for the simulation containing the mixed
kink and Alfv{\'e}n driver. This means that the mixed wave modes can lead to a
more efficient energy dissipation in the turbulent state of the plasma and that
the KHI eddies act as an agent to dissipate energy in other wave modes.
Furthermore, we also obtained forward modelling results using the FoMo code. We
obtained forward models which are very similar to the observations of decayless
oscillations. Due to the limited resolution of instruments, neither Alfv{\'e}n
modes nor the fine structures are observable. Therefore, this numerical study
shows that Alfv{\'e}n modes probably can co-exist with kink modes, leading to
enhanced heating.
