Review Thermal instability in Brinkman porous media with Cattaneo–Christov heat flux S.A.M. Haddad ⇑ Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, UK article info Article history: Received 26 February 2013 Received in revised form 16 September 2013 Accepted 16 September 2013 Keywords: Porous convection Brinkman model Thermal instability Cattaneo–Christov law abstract We consider the thermal instability in a Brinkman porous media incorporating fluid inertia. Both free– free and fixed–fixed boundaries are investigated. We have incorporated the Cattaneo–Christov theory in the constitutive equation for heat flux. For fixed surfaces, the results are generated by using the D 2 Chebyshev tau method. The results reveals that employing the Cattaneo–Christov theory has a pro- nounced effects in determining the convection instability threshold. Ó 2013 Elsevier Ltd. All rights reserved. Contents 1. Introduction ......................................................................................................... 659 2. Basic equations ....................................................................................................... 660 3. Linear instability ..................................................................................................... 661 4. Free surfaces ......................................................................................................... 661 5. Oscillatory convection for two free surfaces ............................................................................... 662 6. Numerical method; fixed surfaces ....................................................................................... 664 7. Numerical results and conclusion ........................................................................................ 666 7.1. Free surfaces ................................................................................................... 666 7.2. Fixed surfaces .................................................................................................. 666 7.3. Conclusion ..................................................................................................... 667 Acknowledgements ................................................................................................... 668 References .......................................................................................................... 668 1. Introduction Thermal convection in porous media is a subject of important current interest and has many real world applications. For in- stance, in geophysics, in crystal growth, and many other areas, see e.g. the book by Straughan [20], and references therein. To investigate convection in a porous medium when the porosity or the viscosity of a porous medium becomes sufficiently large, the Brinkman model is employed as a balance of linear momentum. This model has been devised by Brinkman [1], who found there was a relationship between the permeability and the porosity of a porous medium. A number of writers have employed the Brink- man model to investigate convection in porous media, namely, Rudraiah et al. [17], Vasseur et al. [29], and Vasseur and Robillard [28]. Recently the Brinkman equation has been considered by Rees [16], Hill and Straughan [8], Malashetty et al. [13], Wang and Tan [30], Shivakumara et al. [18], Kelliher et al. [10], Dhananjay et al. [4], and the account in the book by Straughan [21]. The propagation of thermal waves is of particular importance to the field continuum mechanics, see e.g., Straughan [26, chap. 9]. The investigation of the effect of thermal wave on the onset of con- vection instability by generalising the Fourier law for heat conduc- tion was initiated by Straughan and Franchi [19]. Some other 0017-9310/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.09.039 ⇑ Tel.: +44 7403372028. E-mail address: s.a.m.haddad@durham.ac.uk International Journal of Heat and Mass Transfer 68 (2014) 659–668 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt