Multiple solutions, oscillons, and strange attractors in thermoviscoelastic Marangoni convection

Abstract

Through numerical solution of the governing time-dependent and non-linear Navier-Stokes equations cast in the framework of the Oldroyd-B model, the supercritical states of thermal Marangoni-Bénard convection in a viscoelastic fluid are investigated for increasing values of the relaxation time while keeping fixed other parameters (the total viscosity of the fluid, the Prandtl number and the intensity of the driving force, Ma=300). A kaleidoscope of patterns is obtained revealing the coexistence of different branches of steady and oscillatory states in the space of parameters in the form of multiple solutions. In particular, two main families of well-defined attractors are identified, i.e. multicellular steady states and oscillatory solutions. While the former are similar for appearance and dynamics to those typically produced by thermogravitational hydrodynamic disturbances in layers of liquid metals, the latter display waveforms ranging from pervasive standing waves to different types of spatially localised oscillatory structures (oscillons). On the one hand, these localised phenomena contribute to increase the multiplicity of solutions and, on the other hand, give rise to interesting events, including transition to chaos and phenomena of intermittency. In some intervals of the elasticity number, the interference among states corresponding to different branches produces strange attractors for which we estimate the correlation dimension by means of the algorithm originally proposed by Grassberger and Procaccia.

Rayleigh–Bénard convection in viscoelastic liquid bridges

“…Here, we limit ourselves to discussing briefly the main outcomes of the studies by Parmentier et al. (2000) and Lappa & Ferialdi (2018 a ), as relevant exemplars for the WER and SER, respectively. Most interestingly, the former authors could show that the so-called inverted ‘hexagonal cells’ (convection with fluid motion downward at the centre of the cells), which for Newtonian fluids are known to be possible only in liquid metals (or other fluids with small value of the Prandtl number) can become a stable mode of convection in the WER regime of MB flow.…”

“…Most interestingly, the former authors could show that the so-called inverted ‘hexagonal cells’ (convection with fluid motion downward at the centre of the cells), which for Newtonian fluids are known to be possible only in liquid metals (or other fluids with small value of the Prandtl number) can become a stable mode of convection in the WER regime of MB flow. Lappa & Ferialdi (2018 a ) concentrated on the SER illustrating that on increasing the level of elasticity of the considered fluid for a fixed value of the Marangoni number ( Ma ), classical MB convection characterized by polygonal cells all rigidly embedded in a crystalline texture (side-by-side arrangement) is taken over by an unsteady mode of convection where cells lose their original well-defined boundary and acquire a rounded shape. In such a new regime cells are separated by intermediate regions of more or less motionless fluid (referred to as ‘buffer’ areas in Lappa & Ferialdi 2018 a ), which allow them to move back and forth along uncorrelated horizontal directions.…”

“…For additional validation of the Oldroyd-B solver through comparison with the results by other authors for different types of convection (Lebon and co-workers), the reader is referred to Lappa & Ferialdi (2018 a ). The additional computations relying on the FENE-CR paradigm included in this section (tables 4 and 5) are used to demonstrate the overall consistency of the present numerical framework.…”

“…2000; Park & Ryu 2001 a , b , 2002; Li & Khayat 2005; Park et al. 2009; Park & Lim 2010; Hu, He & Chen 2016; Lappa & Ferialdi 2018 a ; Park 2018).…”

Thermosolutal Marangoni instability in a viscoelastic liquid film: effect of heating from the free surface

“…A review of the existing literature on Marangoni convection in polymeric films, however, suggests that, in the previously reported studies, this binary aspect of the fluid was either completely ignored (Getachew & Rosenblat 1985; Dauby et al. 1993; Parmentier, Lebon & Regnier 2000; Hu, He & Chen 2016; Lappa & Ferialdi 2018) or the process was analysed by separately considering the thermal and solutal effects (Doumenc et al. 2013; Yiantsios et al.…”