The effects of Vadasz term, anisotropy and rotation on bi-disperse convection

https://doi.org/10.1016/j.ijnonlinmec.2021.103749Get rights and content

Highlights

  • A model describing the evolutionary behaviour of the thermal conduction solution in a rotating horizontal layer of anisotropic bi-disperse porous medium, taking into account the Vadasz term, is analysed.

  • It has been proved that rotation delays the onset of thermal convection.

  • Linear instability analysis and numerical simulations indicate that the Vadasz number does not affect the onset of stationary convection and has a destabilizing effect on the onset of oscillatory convection.

  • Numerical simulations show that when the anisotropic micropermeability parameter is much greater than the anisotropic macropermeability parameter, oscillatory convection cannot occur.

Abstract

The onset of thermal convection in a uniformly rotating and horizontally isotropic bi-disperse porous medium, taking into account the Vadasz term, is investigated. Via linear instability analysis, it has been proved that the Vadasz term allows the onset of convection via an oscillatory state but does not affect convection via a stationary motion.

Introduction

A bi-disperse porous medium (BDPM) is a dual porosity material characterized by a standard pore structure, but the solid skeleton has fractures or cracks in it. In particular, a BDPM is a compound of clusters of large particles that are themselves aggregations of smaller particles: the macropores between the clusters are referred to as f-phase (meaning fracture phase) and have porosity φ, while the remainder of the structure is referred to as p-phase (meaning porous phase) and the porosity of the micropores within the clusters is ϵ [1].

While thermal convection has been widely studied in both clear fluid and porous media by many authors (see for example [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14] and references therein), in the past as nowadays, double porosity materials have recently attracted many researchers due to their applications in engineering, medical field, chemistry (see [15], [16], [17], [18] and references therein) and a theoretical key development is attributable to Nield and Kuznetsov in [1], [19], [20]. Bi-disperse porous media may be pretty useful in a laboratory [1], but in particular anisotropic bi-disperse porous materials offer much more possibilities to design man-made materials for heat transfer or insulation problems [21], [22], [23].

The analysis of fluid motion in rotating porous media finds many applications in geophysics and in engineering, for example for rotating machinery, chemical process industry, centrifugal filtration processes (see [14] and references therein), hence the study of thermal convection in rotating BDPM may be necessary and useful as well (see [21], [24], [25]).

Regarding the Vadasz term effect, it has been largely analysed by many authors in single porosity media (see for instance [4], [5], [9], [13], [26], [27]) since the Vadasz term has a remarkable effect on the onset of convection in a rotating porous layer, in particular in [13] it has been proved that the Vadasz term leads to the onset of convection via an oscillatory state. On the other hand, the effect of Vadasz term on the onset of bi-disperse convection has been investigated by Straughan in [28], where he considered a fluid mixture saturating a BDPM, and by Capone and De Luca in [24], which deals with an isotropic and rotating BDPM.

The goal of the present paper is to analyse the combined effects of anisotropic permeabilities, uniform rotation about a vertical axis and inertia on the onset of thermal convection in an incompressible fluid saturating a single temperature bi-disperse porous medium. The paper is organized as follows. In Section 2 the mathematical model and the associated perturbation equations are introduced. In Section 3 we perform linear instability analysis of the thermal conduction solution, in particular, we find out that the Vadasz term allows the onset of thermal convection via an oscillatory state (named as oscillatory convection), but it does not affect the onset of thermal convection via a steady state (named as stationary convection). In Sections 3.1 Steady convection, 3.2 Oscillatory convection we determine the critical Rayleigh numbers for the onset of steady and oscillatory convection, respectively. In Section 4 we perform numerical simulations in order to analyse the behaviour of the instability thresholds with respect to fundamental parameters. The paper ends with a concluding section that recaps all the results.

Section snippets

Mathematical model

Let Oxyz be a reference frame with fundamental unit vectors i,j,k and let us assume that the plane layer L, of thickness d, of saturated bi-disperse porous medium is uniformly heated from below and rotates about the vertical axis z, let Ω=Ωk be the constant angular velocity of the layer. Furthermore, we consider a single temperature bi-disperse porous medium, i.e. Tf=Tp=T. We restrict our attention to the case in which the permeabilities of the saturated bi-disperse porous medium are

Onset of convection

In order to determine the linear instability threshold of the thermal conduction solution, let us linearize system (4), i.e. Juft(Kf)1ufγ(ufup)πf+RθkTk×uf=0,Kr(Kp)1upγ(upuf)πp+RθkηTk×up=0,uf=0,up=0,θt=wf+wp+Δθ.Being the system (6) autonomous, we seek solutions with time- dependence like eσt, i.e. us(t,x)=eσtus(x),θ(t,x)=eσtθ(x),πs(t,x)=eσtπs(x),with σ and s={f,p}. By virtue of (7), (6) becomes Jσuf(Kf)1ufγ(ufup)πf+RθkTk×uf=0,Kr(Kp)1upγ(upuf)πp+RθkηTk×up=0,

Numerical simulations

The aim of this section is to solve (18), (25) and numerically describe the asymptotic behaviour of the steady and oscillatory critical Rayleigh numbers with respect to T2, J, k, h, in order to describe the influence of rotation, Vadasz number, anisotropic macropermeability and anisotropic micropermeability on the onset of convection. Through numerical simulations, we have shown that the minimum of both (18), (25) with respect to n is attained at n=1, hence let us define fS(a2)=Λ1a2a1b1,and fO(a

Conclusions

The onset of convection in a rotating and anisotropic bi-disperse porous medium, taking into account the Vadasz term, has been studied via linear instability analysis. Let us remark that the Vadasz term allows the onset of oscillatory convection, which is not present when the inertia is neglected (see [21]). Moreover, if h=k=1, i.e. confining ourselves to the isotropic case, from (18), (25) we recover the stationary and oscillatory thresholds found in [24], respectively. Lastly, it has been

CRediT authorship contribution statement

F. Capone: Conceived and designed the analysis, Collected the data, Contributed data or analysis tools, Performed the analysis, Wrote the paper. G. Massa: Conceived and designed the analysis, Collected the data, Contributed data or analysis tools, Performed the analysis, Wrote the paper.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This paper has been performed under the auspices of the GNFM of INdAM.

G. Massa would like to thank Progetto Giovani GNFM 2020: “Problemi di convezione in nanofluidi e in mezzi porosi bidispersivi”.

The Authors should like to thank the anonymous referees for suggestions which have led to improvements in the manuscript.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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