DISCRETE AND CONTINUOUS Website: www.aimSciences.org DYNAMICAL SYSTEMS Supplement 2013 pp. 375–384 FINITE-DIMENSIONAL BEHAVIOR IN A THERMOSYPHON WITH A VISCOELASTIC FLUID A. Jim´ enez-Casas Grupo de Din´ amica No Lineal Universidad Pontificia Comillas de Madrid C/Alberto Agulilera 23, 28015 Madrid, Spain Mario Castro Grupo Interdisciplinar de Sistemas Complejos (GISC) and Grupo de Dinmica No Lineal (DNL) Escuela Tcnica Superior de Ingeniera (ICAI) Universidad Pontificia Comillas, E28015, Madrid, Spain Justine Yassapan Grupo de Dinmica No Lineal (DNL) Departamento de Matemtica Aplicada y Computacin Escuela Tcnica Superior de Ingeniera (ICAI) Universidad Pontificia Comillas, E28015, Madrid, Spain Abstract. We analyse the motion of a viscoelastic fluid in the interior of a closed loop thermosyphon under the effects of natural convection. We consider a viscoelastic fluid described by the Maxwell constitutive equation. This fluid presents elastic-like behavior and memory effects. We study the asymptotic properties of the fluid inside the thermosyphon and derive the exact equations of motion in the inertial manifold that characterize the asymptotic behavior. Our work is a generalization of some previous results on standard (Newtonian) fluids. 1. Introduction. Chaos in fluids subject to temperature gradients has been a sub- ject of intense work for its applications in the field of engineering and atmospheric sciences. A thermosyphon is a device composed of a closed loop pipe containing a fluid whose motion is driven by the effect of several actions such as gravity and natural convection [7, 12, 11]. The flow inside the loop is driven by an energetic balance between thermal energy and mechanical energy. The interest on this system comes from engineering and as a toy model of natural convection (for instance, to understand the origin of chaos in atmospheric systems). As viscoelasticity is, in general, strongly dependent on the material composition and working regime, here we will approach this problem by studying the most essen- tial feature of viscoelastic fluids: memory effects. To this aim we restrict ourselves to the study of the so-called Maxwell model [8]. In this model, both Newton’s law of viscosity and Hooke’s law of elasticity are generalized and complemented through 2010 Mathematics Subject Classification. 35M33, 35B42, 76A10. Key words and phrases. Thermosyphon, viscoelastic fluid, heat flux, asymptotic behaviour. Partially supported by Projects FIS2009-12964-C05-03, and MTM2009-07540, GR58/08 Grupo 920894 BSCH-UCM, Spain. 375