Convection with Local Thermal Non-Equilibrium and Microfluidic Effects

Abstract

In this chapter we describe work of [366] and of [393] on thermal convection in a Darcy porous material in a vertical layer subject to different temperatures on the vertical walls.For the case of a single temperature the problem where the porous medium occupies a vertical layer subject to differential heating from one side to the other has attracted much attention. This problem has much application to insulation, such as in the area of building design, and is of importance in window double glazing. For the single temperature situation the first proof that a vertical porous slab of Darcy type which is held at fixed but different temperatures on the vertical walls is stable to perturbations from the equilibrium state is due to [156]. His analysis is based on the linear theory and analyses the two-dimensional problem.[409] further analysed this problem but in three-dimensions and he treated the completely nonlinear situation by employing energy-like integral methods. In particular, [409] produced a threshold for the Rayleigh number which guarantees global nonlinear stability, i.e. regardless of how large the initial perturbation may be. He also employed a generalized energy method to demonstrate that the result of [156] is true in the fully three-dimensional case although the initial data must be restricted. Articles dealing with other interesting aspects of convection in a vertical porous slab or pipe involving effects like double diffusion, and LTNE, include the papers by [32, 33, 36, 46, 211, 220, 221, 329, 330, 362] and [460]. The methods of energy stability techniques applied specifically to convection in a vertical porous slab are addressed in [144, 229] and [356], and these are discussed in detail in the book by [414, pp. 126-134].The analogous problem in LTNE to that of [156] was first addressed by [366]. Specifically [366] dealt with the [156] problem of thermal convection in a vertical porous medium, but he employed the theory of local thermal non-equilibrium, allowing for different fluid and solid temperatures. The interesting work of [366] demonstrates that the vertical configuration is always stable according to linear theory, even with the much more complicated LTNE theory. [393] continued the problem of [366] but they analysed the complete nonlinear three-dimensional situation.

Bidispersive thermal convection

“…This result is optimal and shows that in this case the linear theory captures the physics of the onset of convection correctly. It is worth stressing that this not true in the equivalent bidispersive convection problem when there are different temperatures T f and T p for the macro and micropores phases, see Nield and Kuznetsov [1], Straughan [9,10], chapter 13.…”

“…Further work followed see e.g. Nield [3], Nield and Bejan [4], Nield and Kuznetsov [2,5,1,6,7,8], Straughan [9,10], chapter 13, with fundamental work on the thermal convection problem arising in Nield and Kuznetsov [1]. These works all employ different velocities U …”

“…Finally, we mention an application to the vital production of clean drinking water, see e.g. Ghasemizadeh et al [16], Zuber and Matyka [17], although many further applications are discussed in Straughan [10], chapter 13.…”

Thermal convection in a magnetized conducting fluid with the Cattaneo–Christov heat-flow model

“…For example, thermal waves are important in the study of thermal transport in nanomaterials and nanofluids [6,13], and thermal shocks in solids [14], and for heat transport in biological tissue and surgical operations [8,[15][16][17]. Similarly, thermal relaxation has been shown to impact on flow velocity profiles in Jeffrey fluids [18], and a number of thermal convection problems in fluids and porous media [19][20][21][22] (including thermo-haline convection [23,24]), while type-II flux laws analogous to equation (1.1) have found utility in related contexts involving advection-diffusion systems [25][26][27].…”

“…Straughan and colleagues [19][20][21][22][23]), recently we studied the impact of the Cattaneo-Christov heatflow model on the canonical Rayligh-Bénard problem of a Boussinesq fluid layer heated from below [28][29][30]. Our analysis in this earlier context showed that in addition to onset of instability by stationary convection (as predicted classically [29,30]), Cattaneo effects give rise to oscillatory convection as the preferred manner of instability whenever C exceeds some threshold value C T , where C T (P 1 ) is a function of the Prandtl number P 1 [28].…”