Nonlinear dynamics of viscoelastic fluids in a closed loop thermosyphon

Resumen

Flow dynamics in a closed loop thermosyphon is a very complex and in- teresting phenomenon, as it incorporates several factors such as gravity, thermal conduction, natural convection and gradients due to a solute, all producing the emergence of complex dynamical behaviors inside the loop. The convection inside a closed loop thermosyphon is propelled and sustained by the buoyancy effect vari- ations in the density or is caused by the disccusion of solute in dissolution due to temperature gradients. The dynamics becomes even more complex when the uid inside the loop is viscoelastic, leading to various types of behavior such as chaotic, periodic, quasi-periodic and stable behavior. Although these kinds of systems have been widely studied in the literature for simple (Newtonian) fluids, the behavior of viscoelastic fluids has not been explored thus far. These kinds of fluids present elastic-like behaviors and memory effects. Various viscoelastic coefficients, thermal gradients and solute gradients produce diferent types of complex dynamical behaviors on the system. A theoretical study of the dynamics of Maxwell viscoelastic fluids in a closed loop thermosyphon is presented. For the first time, the mathematical derivations of the motion of a viscoelastic fluid in the interior of a closed loop thermosyphon under the effects of natural convection and a given external temperature gradient are derived. The asymptotic properties of the fluid inside the thermosyphon and the exact equations of motion in the inertial manifold that characterize the asymptotic behavior are studied. The dynamics of the system is characterized by observing the time series plots and the phase-diagrams of acceleration, velocity, temperature and in the case of binary fluids, also solute concentration of Maxwell viscoelastic fluids. A detailed analysis of the impact of viscoelasticity and its coexistence with the Soret effect has also been extensively done in this research. This thesis consists of the study of three related problems, all of them con- cerning the dynamics of viscoelastic uids in a closed loop thermosyphon. The first model is based on one component viscoelastic uids with Newton's linear cooling law. The second model is based on one component viscoelastic fluids with prescribed heat ux with dicusion. Finally, the third model is based on binary viscoelastic fluids with the Soret effect. In each case, we have approached the problem from a theoretical viewpoint followed by numerical experiments to unveil the behavior of the system in larger detail. The contribution of this research is the derivation of the novel system of equations to study the behavior of a viscoelastic material inside a thermosyphon. This model can be thought as a preliminary simplification of a more complex fully spatially extended system. The main result is to prove that the original system (which involves both ordinary and partial different equations) possesses an inertial manifold in which the dynamics can be accurately described by a low dimensional system of ODEs. By numerical integration of the reduced equations we have been able to better understand the role of viscoelasticity (as opposed to a simpler New- tonian fluid) through the parameter ". This parameter is an adimensional version of the so-called Maxwellian viscoelastic time which accounts for the characteris- tic timescale (or, alternatively, the typical timescale separating purely elastic from purely viscous behaviors).