Viscoelastic convection: Few-modes model and numerical simulations of field equations for Maxwellian fluids

Abstract

The structure formation of convection rolls in Maxwellian fluids that are heated from below in a Rayleigh-Bénard setup is investigated close to onset with a simple few-modes ansatz and by solving the hydrodynamic field equations with a finite-difference method. Depending on the magnitude of the viscoelastic relaxation time one can have besides stationary convection also oscillatory patterns in the form of standing or traveling waves. The existence and stability regions of these convection structures are determined. The convection behavior of the model is compared with the results of full numerical simulations. Furthermore, the effect of modulating the heating periodically in time on the stability of the quiescent conductive state of the fluid and on its convection behavior is investigated as a function of the fluid's viscoelasticity.

Thermal convection of viscoelastic shear-thinning fluids

“…Both works of Parmentier et al (2000) and Li and Khayat (2005) suggest the possibility of oscillatory behaviour for thermal convection of viscoelastic fluids, a fact that is supported by experiments of Kolodner (1998). Recently, Salm and Lücke (2012) considered the same modes as those taken in Khayat (1994) and studied the oscillatory flow planforms for the thermal convection of Maxwellian fluids, with emphasis on the wavenumber selection for oscillatory rolls. As will be shown below, extra modes are required to capture the viscoelastic nature of thermal convection; a gap that is considered in this work.…”

Stabilizing effect of elasticity on the motion of viscoelastic/elastic fluids

“…He found that a finite elastic stress in the undisturbed state is necessary for elasticity to affect stability. Since Herbert's pioneering work, many physicists have continued to develop the linear theory and nonlinear numerical method in the studies of thermal instability in viscoelastic fluids, see [11,14,47,40,59,61,65,66] and the references cited therein. Moreover, it has also been widely investigated how thermal convection in viscoelastic fluids evolves under the effects of other physical factors, such that rotation [13,42,67], magnetic fields [2,3,54], the porous media [77,62] and so on.…”

On thermodynamics of incompressible viscoelastic rate type fluids with temperature dependent material coefficients

“…However, the frequently adopted approach to the temperature equation, especially amongst practitioners focused on numerical simulations, is far less subtle, see for example Harder (1991), Li and Khayat (2005), Del Negro et al (2009), Choudhary et al (2010), Salm and Lücke (2012), Thielmann et al (2015) and Cao et al (2016). The temperature equation is usually written in the form…”