Linear onset of convective instability for Rayleigh-Bénard-Couette flows of viscoelastic fluids

Alves, Leonardo Santos de Brito; Hirata, Sílvia C.; Ouarzazi, M. N.

doi:10.1016/j.jnnfm.2016.03.007

Cited by 13 publications

(2 citation statements)

References 33 publications

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“…Furthermore, L α = L α ( α , β , ω , ∂ω/∂α ; R , ∂R/∂α ) and L β = L β ( α , β , ω , ∂ω/∂β ; R , ∂R/∂β ) are also linear and homogeneous operators, where q n (z), ω and R were allowed to depend on α and β. The latter dependence is included here to generalize this procedure to calculate critical points for the onset of convective instabilities (Alves et al 2016), even though this paper is focussed on absolute instabilities. Equations (2.8) and (2.9) are solved within the same shooting methodology used to solve for Eq.…”

Section: Sensitivity Analysismentioning

confidence: 99%

Identifying linear absolute instabilities from differential eigenvalue problems using sensitivity analysis

et al. 2019

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Identifying the convective/absolute instability nature of a local base flow requires an analysis of its linear impulse response. One must find the appropriate singularity in the eigenvalue problem with complex frequencies and wavenumbers and prove causality. One way to do so is to show that the appropriate integration contour of this response, a steepest decent path through the relevant singularity, exists. Due to the inherent difficulties of such a proof, one often verifies instead whether this singularity satisfies the collision criterion. In other words, one must show that the branches involved in the formation of this singularity come from distinct halves of the complex wavenumber plane. However, this graphical search is computationally intensive in a single plane and essentially prohibitive in two planes. A significant computational cost reduction can be achieved when root finding procedures are applied instead of graphical ones to search for singularities. They focus on locating these points, with causality being verified graphically a posteriori for a small parametric sample size. The use of root-finding procedures require auxiliary equations, often derived by applying the zero group velocity conditions to the dispersion relation. This relation, in turn, is derived by applying matrix forming to the differential eigenvalue problem and taking the determinant of the resulting system of algebraic equations. Taking the derivative of the dispersion relation with respect to the wavenumbers generates the auxiliary equations. If the algebraic system is decoupled, this derivation is straightforward. However, its computational cost is often prohibitive when the algebraic system is coupled. Other methods exist, but often they can also be too costly and/or not reliable for two wavenumber plane searches. This paper describes an alternative methodology based on sensitivity analysis and adjoints that allow the zero group velocity conditions to be applied directly to the differential eigenvalue problem. In doing so, the direct and auxiliary differential eigenvalue problems can be solved simultaneously using standard shooting methods to directly locate singularities. Auxiliary dispersion relations no longer have to be derived, although it is shown that they are the algebraic equivalent of the auxiliary differential eigenvalue problems obtained by this alternative methodology. Using the latter dramatically reduces computational costs. The search for arbitrary singularities is then not only accelerated in single wavenumber planes but it also becomes viable in two wavenumber planes. Finally, the new method also allows group velocity calculations, greatly facilitating the verification of causality. Several test cases are presented to illustrate the capabilities of this new method.

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Section: Sensitivity Analysismentioning

confidence: 99%

Identifying linear absolute instabilities from differential eigenvalue problems using sensitivity analysis

et al. 2019

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“…Recently, the linear stability analyses of non-isothermal shear flows have been extended to non-Newtonian fluids [12][13][14][15]. In particular, Hirata et al [14] investigated the linear stability of the PRB flow of an Oldroyd-B fluid.…”

Section: Introductionmentioning

confidence: 99%

Onset of Thermal Instabilities in the Plane Poiseuille Flow of Weakly Elastic Fluids: Viscous Dissipation Effects

Hirata

Ouarzazi

2021

Fluids

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The onset of thermal instabilities in the plane Poiseuille flow of weakly elastic fluids is examined through a linear stability analysis by taking into account the effects of viscous dissipation. The destabilizing thermal gradients may come from the different temperatures imposed on the external boundaries and/or from the volumetric heating induced by viscous dissipation. The rheological properties of the viscoelastic fluid are modeled using the Oldroyd-B constitutive equation. As in the Newtonian fluid case, the most unstable structures are found to be stationary longitudinal rolls (modes with axes aligned along the streamwise direction). For such structures, it is shown that the viscoelastic contribution to viscous dissipation may be reduced to one unique parameter: γ=λ1(1−Γ), where λ1 and Γ represent the relaxation time and the viscosity ratio of the viscoelastic fluid, respectively. It is found that the influence of the elasticity parameter γ on the linear stability characteristics is non-monotonic. The fluid elasticity stabilizes (destabilizes) the basic Poiseuille flow if γ<γ* (γ>γ*) where γ* is a particular value of γ that we have determined. It is also shown that when the temperature gradient imposed on the external boundaries is zero, the critical Reynolds number for the onset of such viscous dissipation/viscoelastic-induced instability may be well below the one needed to trigger the pure hydrodynamic instability in weakly elastic solutions.

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Convection in a Horizontal Porous Layer with Vertical Pressure Gradient Saturated by a Power-Law Fluid

et al. 2019

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Linear onset of convective instability for Rayleigh-Bénard-Couette flows of viscoelastic fluids

Cited by 13 publications

References 33 publications

Identifying linear absolute instabilities from differential eigenvalue problems using sensitivity analysis

Identifying linear absolute instabilities from differential eigenvalue problems using sensitivity analysis

Onset of Thermal Instabilities in the Plane Poiseuille Flow of Weakly Elastic Fluids: Viscous Dissipation Effects

Convection in a Horizontal Porous Layer with Vertical Pressure Gradient Saturated by a Power-Law Fluid

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