Flute oscillations of cooling coronal loops with variable cross-section

We consider fluting oscillations in a thin straight expanding magnetic flux tube the presence of background flow. The is divided into core region that wrapped transitional region, where damping takes place. method multiple scales used for derivation system governing equations. This applicable to study both standing and propagating waves. Furthermore, equations obtained tubes with sharp boundary. An adiabatic invariant derived using Wentzel-Kramer-Brillouin slowly varying density, theoretical results are then investigate effect cooling on flute curved semi-circlular shape. have analysed numerically dependencies dimensionless amplitude range values expansion factor ratio internal external plasma densities at an initial time. find increases due higher factor. Higher wave number lead localisation oscillation closer Finally, we show value densities, amplification cooling. Therefore, conclude number, density ratio, variation all relevant parameters process oscillating tube.