Azimuthal instability modes in a viscoelastic liquid layer flowing down a heated cylinder

In this paper the non axisymmetric longwave instability of a thin viscoelastic liquid film flowing down a vertical heated cylinder is investigated. The stability of the film coating a cylinder in the absence of gravity is also investigated. In a previous paper it is found that viscoelasticity stimulates the appearance of azimuthal modes but the axial mode is the most unstable one. Other calculations in a former paper show that for flow outside a heated cylinder azimuthal modes can be the more unstable when the Marangoni number is large and, in particular, when the Reynolds number and wavenumber are small. Therefore, the small wavenumber and large cylinder radius approximation is assumed with the simultaneous action of viscoelasticity and thermocapillarity on the stability of azimuthal modes. In the presence and in the absence of gravity, it is found that, in comparison with the Newtonian case, it is easier to excite the azimuthal modes when viscoelasticity and thermocapillarity destabilize at the same time. Moreover, it is shown that, despite the axial mode is the most unstable one, there are wide wavenumber ranges where higher modes are the more unstable and they can show up by means of a periodic time dependent perturbation.