## 1 Introduction  
Various industrial applications depend on the properties and nature of porous media. Thermal insulation, grain stocking and drying processes, the petroleum reservoir, compacted beds for the chemical industry and geophysical systems are good examples of these applications. This importance of these materials makes researchers focus on understanding their nature and properties [1; 2; 3; 4; 5]. In fact, the first category of porous medium investigations is concerned with situations where the properties (porosity, permeability and thermal conductivity) of the medium are constants [6; 7; 8; 9; 10]. Moreover, the presence of nanoparticles inside a thermal application solvent is receiving increasing attention as it allows one to control the performance in terms of heat transfer and entropy [11; 12; 13]. Therefore, researchers are targeting the analysis of situations where the two (porous materials and nanofluids) interact. Recent trends of such studies can be found in [PERSON] [14] who discussed nanofluids motion in an inclined geometry filled with anisotropic porous elements. His outcomes mentioned that an increase in nanoparticles concentration by 4% results in an enhancement of the maximum temperature of the liquid by 12.27%. A porous medium with constant properties was considered by [PERSON] et al. [15] as a flow medium within a rectangular domain filled by nanofluids. The thermal dispersion impactsare of interest in this study. Their findings refer to a support in the values of the Nusselt (Nu) coefficient given as the Brownian parameter is growing. The two-energy model (LTNEM) in case of the constants porosity and permeability is used by [PERSON] (16) to survey non-Darcy flow within trapezoidal geometries. The flow is due to the presence of a triangular fin that is located at the bottom. The domain is divided into nanofluid layer and porous layer. The outcomes uncovered that variations of the Neild number from 1 to 1000 results in an enhancement in the Nu coefficient by 95.6% at the value of the thermal conductivity ratio \(Kr=0.1\). The second category of porous media studies is focused on the variable properties of the medium, i.e., variable porosity, variable permeability and variable thermal conductivity. Examples of these studies are those of [PERSON] et al. (17), [PERSON] et al. (18), [PERSON] et al. (19), [PERSON] et al. (20), [PERSON] et al. (21) and [PERSON] and [PERSON] (22). In all the mentioned studies, the porosity is varied exponentially with the smallest distance to the nearest wall and the permeability depends on the particle diameter. On the other side, the dynamical behaviors of nanofluids within irregular geometries (trapezoidal, triangle, wavy, etc.) have been examined by various investigators in recent years. Treatments of such domains may be made using the body-fitted method, i.e., converting the complex real physical model into a rectangular computational model. While cavities with regular shapes and fins have been studied extensively in many research works, such as (23; 24; 25), non-regular geometries have received less focus and the aforementioned method is effective in simulations of the transfer of heat problems concerning them. The real coordinates \((x,y)\) are expressed as functions in new coordinates \((\xi,\eta)\), and based on these functions, all the partial derivatives of the dependent variables are transformed into the new coordinates (26; 27). [PERSON] et al. (28) used a trapezoidal geometry as a flow domain to discuss the transfer of heat by Darcian nanofluid flow. They considered that the domain is inclined and divided into two layers (porous layer and nanofluid layer). They noted a remarkable enhancement in the rate of heat transfer in the case of silver-water mixture. [PERSON] et al. (29) discussed a trapezoidal domain that is divided into nanofluid porous layer and non-Newtonian fluid layer. Visualization of the heat lines within the system is of interest.