Three-dimensional absolute and convective instabilities in mixed convection of a viscoelastic fluid through a porous medium

Hirata, Sílvia C.; Ouarzazi, M. N.

doi:10.1016/j.physleta.2010.04.039

Cited by 30 publications

(23 citation statements)

References 18 publications

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“…Assuming thermal equilibrium between the solid and the fluid phases and neglecting the viscous dissipation effects, energy and continuity equations can then be written as [1][2][3][4][5][6][7][8][9][10]:…”

Section: Mathematical Formulationmentioning

confidence: 99%

“…In rheology, one crucial problem is the formulation of the constitutive equations regarding viscoelastic fluid flows in porous media. Recently, a modified Darcy's law was employed to study the stability of a viscoelastic fluid in a horizontal porous layer using linear and nonlinear stability theory [1][2][3][4][5][6][7][8][9]. Kim et al [1] and Yoon et al [2] performed a linear stability analysis and showed that in viscoelastic fluids, such as polymeric liquids, a Hopf bifurcation, as well as a stationary bifurcation may occur depending on the magnitude of the viscoelastic parameter.…”

Section: Introductionmentioning

confidence: 99%

“…The question of whether standing or traveling waves are preferred at onset has been fully addressed by Hirata et al [4]. The three-dimensional convective and absolute instabilities of a viscoelastic fluid in the presence of a horizontal pressure gradient have been analyzed by Hirata and Ouarzazi [5]. Alves et al [6] studied the effect of viscous dissipation of viscoelastic fluids at the onset of convection.…”

Section: Introductionmentioning

confidence: 99%

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Onset of Primary and Secondary Instabilities of Viscoelastic Fluids Saturating a Porous Layer Heated from below by a Constant Flux

Gueye

Ouarzazi

Hirata

et al. 2017

Fluids

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Abstract:We analyze the thermal convection thresholds and linear characteristics of the primary and secondary instabilities for viscoelastic fluids saturating a porous horizontal layer heated from below by a constant flux. The Galerkin method is used to solve the eigenvalue problem by taking into account the elasticity of the fluid, the ratio between the viscosity of the solvent and the total viscosity of the fluid and the lateral confinement of the medium. For the primary instability, we found out that depending on the rheological parameters, two types of convective structures may appear when the basic conductive solution loses its stability: stationary long wavelength instability as for Newtonian fluids and oscillatory convection. The effect of the lateral confinement of the porous medium by adiabatic walls is to stabilize the oblique and longitudinal rolls and therefore selects transverse rolls at the onset of convection. In the range of the rheological parameters where stationary long wave instability develops first, we use a parallel flow approximation to determine analytically the velocity and temperature fields associated with the monocellular convective flow. The linear stability analysis of the monocellular flow is performed, and the critical conditions above which the flow becomes unstable are determined. The combined influence of the viscoelastic parameters and the lateral confinement on the characteristics of the secondary instability is quantified. The major new findings concerning the secondary instabilities may be summarized as follows: (i) For concentrated viscoelastic fluids, computations showed that the most amplified mode of convection corresponds to oscillatory transverse rolls, which appears via a Hopf bifurcation. This pattern selection is independent of both the fluid elasticity and the lateral confinement of the porous medium; (ii) For diluted viscoelastic fluids, the preferred mode of convection is found to be oscillatory transverse rolls for a very laterally-confined medium. Otherwise, stationary or oscillatory longitudinal rolls may develop depending on the fluid elasticity. Results also showed the destabilizing effect of the relaxation fluid elasticity and the stabilizing effect of the viscosity ratio for the onset of both primary and secondary instabilities.

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Section: Mathematical Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Onset of Primary and Secondary Instabilities of Viscoelastic Fluids Saturating a Porous Layer Heated from below by a Constant Flux

Gueye

Ouarzazi

Hirata

et al. 2017

Fluids

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“…The least unstable condition corresponds to a Maxwell fluid (Γ = 0) whereas the most unstable one corresponds to a Newtonian fluid (Γ = 1). It is important to note that an opposite trend has been observed in a porous medium by Hirata & Ouarzazi (2010). Such a qualitative difference in behaviour with respect to the viscosity ratio is likely to be due to the linear character of Darcy's law, used to model the momentum equations in the porous medium.…”

Section: Results For Weakly Elastic Fluidsmentioning

confidence: 95%

Convective and absolute instabilities in Rayleigh–Bénard–Poiseuille mixed convection for viscoelastic fluids

et al. 2015

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The convective and absolute nature of instabilities in Rayleigh-Bénard-Poiseuille (RBP) mixed convection for viscoelastic fluids is examined numerically with a shooting method as well as analytically with a one-mode Galerkin expansion. The viscoelastic fluid is modelled by means of a general constitutive equation that encompasses the Maxwell model and the Oldroyd-B model. In comparison to Newtonian fluids, two more dimensionless parameters are introduced, namely the elasticity number λ 1 and the ratio Γ between retardation and relaxation times. Temporal stability analysis of the basic state showed that the three-dimensional thermoconvective problem can be Squire-transformed. Therefore, one must distinguish mainly between two principal roll orientations: transverse rolls TRs (rolls with axes perpendicular to the Poiseuille flow direction) and longitudinal rolls LRs (rolls with axes parallel to the Poiseuille flow direction). The critical Rayleigh number for the appearance of LRs is found to be independent of the Reynolds number (Re). Depending on λ 1 and Γ , two different regimes can be distinguished. In the weakly elastic regime, the emerging LRs are stationary, while they are oscillatory in the strongly elastic regime. For TRs, it is found that in the weakly elastic regime, the stabilization effect of Re is more important than in Newtonian fluids. Moreover, for sufficiently elastic fluids a jump is observed in the oscillation frequencies and wavenumbers for moderate Re. In the strongly elastic regime, the effect of the imposed throughflow is to promote the appearance of the upstream moving TRs for low values of Re, which are replaced by downstream moving TRs for higher values of Re. Moreover, the results proved that, contrary to the case where Re = 0, the elasticity number λ 1 (the ratio Γ ) has a strongly stabilizing (destabilizing) effect when the throughflow is added. The influence of the rheological parameters on the transition curves from convective to absolute instability in the Reynolds-Rayleigh number plane is also determined. We show that the viscoelastic character of the fluid hastens the transition to absolute instability and even may suppress the convective/absolute transition. Throughout this paper, similarities and differences with the corresponding problem for Newtonian fluids are highlighted.

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“…When the thermal convective instability in porous media is considered, a modified Darcy model which extended the Darcy model to include a time derivative term is usually adopted to study the oscillatory convection as well as stationary convection [22,40,59]. Besides, by analogy with the constitutive equation of Oldroyd-B fluid, some researchers proposed a Darcy-Oldroyd model for modeling flows of viscoelastic fluids in porous media, which introduces a relaxation parameter characterizing the elasticity of viscoelastic fluid and a retardation parameter characterizing the viscous damping of viscoelastic fluid [13,60,61]. In this work, the Darcy model is extended with the addition of the Coriolis term in rotating reference frame and is applied to characterize the flow in porous media [1].…”

Section: Mathematical Formulationmentioning

confidence: 99%

Heat Transfer and Flows of Thermal Convection in a Fluid-Saturated Rotating Porous Medium

Kang

Xia

Liu

2015

Mathematical Problems in Engineering

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Thermal convection at the steady state for high Rayleigh number in a rotating porous half space is investigated. Taking into account the effect of rotation, Darcy equation is extended to incorporate the Coriolis force term in a rotating reference frame. The velocity and temperature fields of thermal convection are obtained by using the homotopy analysis method. The influences of Taylor number and Rayleigh number on the Nusselt number, velocity profile, and temperature distribution are discussed in detail. It is found that the Nusselt number decreases rapidly with the increase of Taylor number but tends to have an asymptotic value. Besides, the rotation can give rise to downward flow in contrast with the upward thermal convection.

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Three-dimensional absolute and convective instabilities in mixed convection of a viscoelastic fluid through a porous medium

Cited by 30 publications

References 18 publications

Onset of Primary and Secondary Instabilities of Viscoelastic Fluids Saturating a Porous Layer Heated from below by a Constant Flux

Onset of Primary and Secondary Instabilities of Viscoelastic Fluids Saturating a Porous Layer Heated from below by a Constant Flux

Convective and absolute instabilities in Rayleigh–Bénard–Poiseuille mixed convection for viscoelastic fluids

Heat Transfer and Flows of Thermal Convection in a Fluid-Saturated Rotating Porous Medium

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