Abstract
A thermosyphon, in the engineering literature, is a device composed of a closed loop containing a fluid whose motion is driven by several actions, such as gravity and natural convection. In this work we prove some results about the asymptotic behaviour for solutions of a closed loop thermosyphon model with a viscoelastic fluid in the interior (Jiménez-Casas et al., Discrete Conti Dynam Syst (9th AIMS Conference Sool) 2013:375–384, 2013; Chaotic Model Simul 2:281–288, 2013). In this model a viscoelastic fluid described by the Maxwell constitutive equation is considered, this kind of fluids present elastic-like behavior and memory effects. Their dynamics are governed by a coupled differential nonlinear systems. In several previous works we have shown chaos in the fluid, even with this kind of viscoelastic fluid (Jiménez-Casas and Castro, Electron J Differ Equ (Conference 22), 53–61, 2015; Yasapan et al., Abstr Appl Anal 2013, Article ID: 748683, 2013; Discrete Conti Dynam Syst Ser B 20:3267–3299, 2015 among others). In this model, we consider a prescribed heat flux like Rodríguez-Bernal and Van Vleck (SIAM J Appl Math 58:1072–1093, 1998), Jiménez-Casas and Ovejero (Appl Math Comput 124:289–318, 2001) among others (all of them with Newtonian fluids). This work is, in some sense, a generalization of some previous results on standard (Newtonian) fluids obtained by Rodríguez-Bernal and Van Vleck (SIAM J Appl Math 58:1072–1093, 1998), when we consider a viscoelastic fluid.
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Acknowledgements
I want to thank M. Castro and J. Yasappan for the numerical experiments and the referee for their helpful comments. This research was partially supported by grants MTM2012-31298,MTM2016-75465-P and Project FIS2013-47949-C2-2 from Ministerio de Economia y Competitividad, Spain; by GR58/08 Grupo 920894 BSCH-UCM from Grupo de Investigación CADEDIF, Grupo de Dinámica No Lineal (U.P. Comillas)Spain and by the Project FIS2016-78883-C2-2-P (AEI/FEDER,UE).
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Jiménez-Casas, Á. (2019). Asymptotic Behaviour of Finite Length Solutions in a Thermosyphon Viscoelastic Model. In: García Guirao, J., Murillo Hernández, J., Periago Esparza, F. (eds) Recent Advances in Differential Equations and Applications. SEMA SIMAI Springer Series, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-030-00341-8_6
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