Multicellular states of viscoelastic thermovibrational convection in a square cavity

Abstract

The problem of pure thermovibrational flow in a two-dimensional square cavity containing a viscoelastic liquid is investigated in the framework of a numerical approach based on the governing balance equations for mass, momentum, and energy in their complete and non-linear time-dependent form. For problem closure, these equations are complemented with the transport equation for the elastic stress formulated using the finitely extensible nonlinear elastic Chilcott–Rallison (FENE–CR) constitutive model. A complete parametric study is carried out to highlight the different path of evolution taken by the considered viscoelastic fluid with respect to the corresponding Newtonian counterpart when the Gershuni number is increased. Attention is paid to the patterning scenario in terms of time-averaged flow and related multicellular structures. It is shown that the triadic relationship among the typical characteristic time scales involved in these phenomena, namely, the thermally diffusive time, the fluid relaxation time, and the period of vibrations, can lead to a kaleidoscope of states, which differ in regard to the prevailing symmetry and the related spatiotemporal behaviors. Moreover, the complex interaction between the external vibrations and the elastic property of the polymer molecules, mediated by viscous effects, can produce an interesting “intermittent response.”

Competition of overstability and stabilizing effects in viscoelastic thermovibrational flow

“…The scope of the validation process is to verify the ability of the numerical procedure to capture and predict the onset of relevant fluid dynamic instabilities. An extensive discussion of the several tests used to validate the solver described in the earlier section is available in [27,68]. Good agreement with the existing literature was found for both isothermal-fluid benchmarks (where the emerging instabilities are entirely elastic in nature) and non-isothermal fluid conditions (Rayleigh-Bénard convection).…”

“…Even in this case it is straightforward to verify the good agreement between our calculations and the data available in the literature. Along the same lines, we wish to recall that in [68], we found Ra c ≈ 472 for ϑ = 0.06, the corresponding value determined by [67] being Ra c ≈ 470. For what concerns the validation of the Newtonian thermovibrational solver, the reader will find further details in [66].…”

“…that of the considered external forcing: the non-dimensional frequency Ω can also be seen as a ratio between period of the oscillations T * ω = 2π/ω and t α up to a multiplicative constant 2π. Given these arguments and following the same rationale of [68], therefore, we also introduce Σ, i.e. the ratio of the relaxation time λ and T * ω :…”

“…Following the same approach already undertaken in our preceding analysis [68], here we report the nondimensional amplitude A of the unsteady convective state for different values of Ra and the value of Ra c obtained through (quadratic) extrapolations of A to 0 (Table I). Even in this case it is straightforward to verify the good agreement between our calculations and the data available in the literature.…”

“…Although different interesting new physical mechanisms were identified, the root dynamics driving these types of solutions are still largely unknown. In particular, no studies have appeared where the "pure" thermovibrational flow in viscoelastic fluids has been examined (with the only exception of that by Boaro and Lappa [68] where however, the focus was on the situation with vibrations perpendicular to the temperature difference for which overstability is not a relevant mechanism).…”