**2. Mathematical model**

Buoyancy driven flows in porous enclosures have been simulated using different mathematical models and numerical techniques. Most commonly used mathematical model of governing momentum equation is the Darcy's law, which is valid for the laminar flow regime (Re < 10), where the velocities are low and the viscous forces are predominant over inertia forces [1]. Extensions of the governing momentum equations have been made by analogy with the Navier-Stokes equations with addition of Brinkman term in order to consider the viscous diffusion and Forchheimer term to study the inertia effects on the free convection [2]. Problems of natural convection in porous media were studied intensively in last few decades, mainly for the cases of two-dimensional geometries. Two types of geometries are commonly investigated: porous enclosures where temperature gradient is imposed horizontally [3–7] or vertically [8–11]. Studies considering three-dimensional geometries are rare and are usually confined on using a simplified mathematical model, e.g., Darcy model or to conditions of heating from below [12–18]. Researches considering three-dimensional cavities with the

Recently, several researchers have been investigated buoyant flow in porous media domains saturated with nanofluids [22–24]. Nanoscale particles are often added to working fluids in order to enhance heat transfer or cooling processes. A comprehensive review of the studies considering convection heat transfer in porous media saturated with nanofluid was published in [25].

The solutions of the problems of transport phenomena in porous media have been obtained using different numerical methods, where the most commonly used methods are the finite element method (FEM), the finite difference method (FDM), and the finite volume method (FVM). As an alternative to others, in engineering practice widely used methods, the BEM was developed mainly because it was very efficient for solving potential problems of fluid mechanics (inviscid fluid flow, heat conduction, etc.), where the mathematical transformation of the governing set of partial differential equation results in boundary integral equations only. To rewrite the partial differential equation into an equivalent integral representation, the known fundamental solutions of the differential operator [26] and the Green's theorem are used. The discretized system contains only a fully populated system of integrals over boundary elements, which represent the main advantage over the volume-based

When dealing with nonhomogenous and nonlinear problems, e.g., diffusion-convection problems, the domain integrals occur in the integral representation as well, which demands the extension of the classical BEM in order to additionally deal the problem within the domain. The main issue in this case is the evaluation of the domain matrices, which are full and unsymmetrical and require a lot of storage space. Several techniques have been developed in order to eliminate the domain integrals or transform them into the boundary integrals. One of the possibilities is the dual reciprocity boundary element method (DRBEM), which transforms domain integrals into a finite series of boundary integrals [27–29]. The nonhomogenous term is expanded in terms of radial basis functions. Since the discretization of the domain is represented only by grid points and the discretization of the geometry and fields on the boundary is piecewise polygonal, the DRBEM is still more flexible and efficient

condition of heating from the side were published in [19–21].

80 Numerical Simulations in Engineering and Science

methods.

against other numerical methods, e.g., FDM.
