### Convection with Local Thermal Non-Equilibrium and Microfluidic Effects

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We endorse the statement by Lai that despite the increased volume of research in this field, experimental results are still very few. In particular experimental data on thermal dispersion are very scarce and this is hindering the study of the functional relationship between effective thermal conductivity and thermal dispersion.

In this chapter we turn our attention to processes of combined simultaneous heat and mass transfer that are driven by buoyancy. The density gradients that provide the driving buoyancy force are induced by the combined effects of temperature and species concentration nonuniformities present in the fluid-saturated medium.

## Convection in Porous Media on Apple Books

The present chapter is guided by the review of Trevisan and Bejan , which began by showing that the conservation statements for mass, momentum, energy, and chemical species are the equations that have been presented here in Chaps. In particular the material in Sect. The new feature is that beginning with Eq. Useful review articles on double-diffusive convection include those by Mojtabi and Charrier-Mojtabi , , Mamou b , Diersch and Kolditz , and Mojtabi et al.

In the examples of forced and natural convection discussed until now, the fluid that flowed through the pores did not experience a change of phase, no matter how intense the heating or cooling effect. In this chapter, we turn our attention to situations in which a change of phase occurs, for example, melting or evaporation upon heating and solidification or condensation upon cooling. These convection problems constitute a relatively new and active area in the field of convection in porous media. Most of the studies of convection in porous media published before were motivated by geophysical applications and many published since have geophysical ramifications; see, for example, the reviews by Cheng a, b, b.

On the other hand, geothermal reservoir modeling involves several features that are outside the scope of this book, and it usually involves specialist computer packages and techniques. An important book dealing with geological fluid dynamics is that by Phillips In this book, the emphasis is on flow patterns and specifically geological processes, involving dissolution, chemical reaction, and deposition. Some examples are discussed below in Sect. Another important book is that by Woods Titel Convection in Porous Media. Verlag Springer International Publishing.

Print ISBN Figure 2 displayed the profile of basic temperature functions along with the fluid layer. To check the system stability we introduced the following perturbed quantities:. Now, the following non-linear system of perturbation equations after dropping prime is obtained:. In order to analyze the stability of the above system, we first reduce the complexity of the system by using the following scaling parameter,. For a stability analysis, there are mainly two techniques that may be employed, one is the method of linearized instability and another is the energy method.

Infinitesimal small perturbations are assumed to the steady state, in linear analysis and so the impact of nonlinear terms becomes negligible and hence discarded from the set of partial differential equations. The linear theory only provides boundaries for instability; thus it is highly desirable to develop the nonlinear energy stability theory. For obtaining optimal nonlinear thresholds the idea of coupling parameter Joseph can be employed for constructing energy functional.

It will provide sharpen nonlinear thresholds as compared to the classical approach. So, we have the following linearized system:. Boundary conditions corresponding to the above system are given below [ Straughan ]:.

## Unknown error

For numerical analysis, Chebyshev-pseudospectral method is used. It an efficient numerical technique to find out eigenvalues and eigenfunctions in the system of continuum mechanics. The key point of this method is, one can calculate as many eigenvalues as required, not just the dominant one. After transformation the system of linearized equations is:. Next, discretizing equation 27 — 29 following the Chebyshev scheme and hence can be written in form of a system of the matrix as:.

Results obtained by numerical simulation for the linear instability are presented in section 4. Adopting the standard nonlinear energy approach in the stability measure, we first multiply eqs. Using equations 32 — 34 in 36 , we get. To obtain nonlinear stability bounds, the variational problem 40 must be solved. So, from equations 45 , 46 and 47 we have the following system of equations:. The onset of convection in a Darcy-Brinkman porous medium is investigated for a local thermal non-equilibrium model in the presence of internal heat source.

The various heat generating functions are taken into consideration which induces instability within the fluid layer when the heat is extracted through the boundary at a constant rate. Stability analysis performed by normal mode technique and energy method and Chebyshev-pseudospectral scheme used to solve generalized eigenvalue problem.

A comparison of critical values of Darcy-Rayleigh number and wave number with respect to various parameters is performed for all heat generating functions and corresponding results depicted numerically and. For the validation of the results, the model is reduced to the LTE limits and in the absence of heat extraction through the lower boundary the results are compared with the existing results and are presented in Table 1.

Increase in value of heat parameter adds on the stability of the system. The heat extraction through the lower boundary has a counter effect to the internal heat generation, which slows down the heat transfer and results in a delay in convection. Again, from tables, It is also observed that increase in the value of H , the value of the critical Rayleigh number increases and the onset of convection delays. Cell size gets. The neutral stability curves for Rayleigh number and wave number for distinct parameters are shown in Figures 3 — 6. A neutral stability curve, in the expression of critical Darcy-Rayleigh number, provides the stability criterion.

These curves show that for a range of wave number the linear and nonlinear thresholds are so close that they are indistinguishable graphically, signifying the suitability of linear theory to predict the nature of the onset of convection. But after a certain range of wave number, nonlinear threshold leave a region of potential subcritical instability.

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Further, from figure 3 it is observed that as the value of Darcy-Brinkman number increases, quantitative change occurs in the value of the critical Rayleigh number but the qualitative behavior remains the same. The increase in the values of the inter-phase heat transfer parameter has a significant impact on the critical Rayleigh number as shown in Figure 5. For small values of H , no significant heat transfer takes place between the solid and fluid phase and hence the properties of the solid phase does not affect the convection and system is in a strong LTNE situation.

Next, as H increases, we found a noticeable change in the critical values of the Rayleigh number because the solid properties affect the onset of convcetion. Figure 6 represents that as the value of increases, the value of critical Rayleigh number decreases; this is because of the increase in the value of leads to substantial transfer of heat through the solid and fluid phases. For very small values of H , the critical Rayleigh number is independent of the values of and approaches the LTE results.

On the other hand, for comparatively large values of H for graphical results the value of H is taken equal to 5 , a significant change in the values of critical Rayleigh number can be observed with the increasing values of.