Azimuthal instability modes in a viscoelastic liquid layer flowing down a heated cylinder M. Moctezuma-Sánchez, L.A. Dávalos-Orozco ⇑ Instituto de Investigaciones en Materiales, Departamento de Polímeros, Universidad Nacional Autónoma de México, Ciudad Universitaria, Circuito Exterior S/N, Delegación Coyoacán, 04510 México D. F., México article info Article history: Received 5 March 2015 Received in revised form 11 June 2015 Accepted 11 June 2015 Keywords: Thin liquid film Thermocapillarity Marangoni convection Viscoelasticity Cylindrical layer Azimuthal modes abstract 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 grav- ity 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 azi- muthal 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. Ó 2015 Elsevier Ltd. All rights reserved. 1. Introduction The coating of surfaces by liquid films have important applica- tions in industry. The problems found when looking for the perfect finishing are due to hydrodynamic instabilities. In the absence of gravity a cause of instability is thermocapillarity. When the liquid layer is coating a flat wall Pearson [1] has shown that a liquid film is unstable to temperature gradients perpendicular to the layer. As a consequence convection cells appear which may have important consequences in the solidified film. Therefore, it is necessary to investigate this instability under different mechanical and thermal boundary conditions. When the free surface is deformable the problem is investigated first by Scriven and Sternling [2]. The restoring influence of gravity is taken into account by Takashima [3] in the stationary case and by Takashima [4] when the flow is time dependent. The double diffusive Marangoni convection is first investigated by Mctaggart [5]. Sometimes in applications the fluid has elastic properties due to the presence in solution of macro- molecules which change their form when shear stresses are applied to the liquid. These fluids are called viscoelastic (see for example Bird et al. [6]) and have been investigated widely in nat- ural convection phenomena (see a recent review paper by Dávalos-Orozco [7–9] by Pérez-Reyes and Dávalos-Orozco). Notice that one characteristic of the viscoelastic instabilities is that they can be time dependent, in contrast to Newtonian fluids con- vection. Yet it is shown [8] that these instabilities do not occur for any thermal boundary conditions. The thermal Marangoni instability has also been investigated for viscoelastic fluids by a number of authors. Getachew and Rosenblat [10] calculated the codimension-two points where sta- tionary and oscillatory convection compete to be the first unstable one when the Marangoni number increases. Wilson [11] investi- gates supercritical conditions of the thermocapillary instability of a viscoelastic fluid from the point of view of the growth rates. Siddheshwar et al. [12] investigate the instability of a Maxwell fluid under different thermal boundary conditions including the effect of viscosity variation with temperature. The thermocapillary instability of a Maxwell viscoelastic fluid is investigated by Herná ndez-Hernández and Dávalos-Orozco [13] assuming a flat free sur- face and presenting results for a wide range of wall thermal con- ductivities. The goal is to calculate the codimension-two points where the stationary and oscillatory Marangoni convection modes compete to be the first unstable one. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.06.035 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail address: ldavalos@unam.mx (L.A. Dávalos-Orozco). International Journal of Heat and Mass Transfer 90 (2015) 15–25 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt