The onset of Darcy– Benard penetrative convection in a liquid saturated porous layer of high permeability of practical importance is investigated by employing the Brinkman– Forchheimer– Lapwood extended Darcy flow model with fluid viscosity different from effective viscosity. The lower boundary is taken to be rigid and isothermal and the upper surface is free and subject to the general thermal condition. The critical eigen values are obtained numerically, in general, using Galerkin method. The stability of the system is found to be dependent on the dimensionless internal heat source strength Ns, permeability parameter  e and the ratio of effective viscosity to fluid viscosity  . It is observed that the increase in the value of permeability parameter is to delay while increase in the value of internal heat source strength is to hasten the onset of convection in a fluid saturated porous layer.

In this paper, we study the reaction-diffusion equation $\Delta u(x) = |u(x)|^{\gamma(x)-1} u(x) $ from a regularity point of view. This equation is used for modelling the distribution of a gas in an inhomogeneous porous catalyst. And the power $\gamma(x)$ can be a discontinuous function. In particular, we study the vanishing order of solution near the zero level set $\{u=0\}$.

An instability analysis in a horizontal porous layer is made for a fluid with an inverse density gradient. The governing equations resulted when instability analysis is applied to this problem, are non-linear and therefore mathematically complex. In published literature, the problem is solved with some simplifications such as ignoring certain terms in the governing equations, or finding algebraic approximations that are valid in some specific range of physical parameters. In this study, the problem is solved in its general form for two rigid, and one rigid and one free boundary conditions using variational methods. The resulting critical Rayleigh numbers and critical wave numbers vs. l2/K curves are compared to previous works.        

Introduction: Natural convection is an important phenomenon in porous media problems. It is encountered in a variety of applications, including in enhanced oil recovery systems and geothermal reservoirs. Physics-based numerical models are widely used to simulate natural convection in porous media. Although these models are usually effective, they commonly suffer from high computational costs. This is notably problematic in repetitive runs at large time and space scales, as in uncertainty analysis, data assimilation, and sensitivity analysis. In recent years, at least four different methods have been proposed to overcome this challenge, including optimizing the numerical solution algorithm, parallel computing, cloud computing, and data-driven methods. In most cases, while data-driven models are capable of handling low-dimensional problems, they have not been very successful in dealing with high-dimensional problems, both accurately and time efficient. To overcome these challenges, we propose using the encoder-decoder convolutional neural networks (ED-CNNs) for heterogeneous porous media. We apply the ED-CNN in the context of ‘image-to-image’ regression in the following two use cases in the context of natural convection simulations: (1) as a meta-model to estimate the heat map from the Rayleigh number distribution, and (2) as an optimizer to estimate the Rayleigh number distribution from the heat map. Methodology: The proposed ED-CNN is employed to model the hypothetical example of a square porous enclosure filled with a saturated porous medium. The boundaries are impermeable, and temperatures at two opposite side walls are different, resulting in the formation of natural convection. Heterogeneity in the Rayleigh number across the problem domain is applied through zonation.A numerical modeling tool is used to generate steady-state heat maps based on a number of randomly selected Rayleigh numbers. The numerical model input-outputs are transformed into square-shaped jpg images of 64 × 64 resolution. Two ED-CNNs are trained, one as a meta-model and the other as an optimizer. Different numbers of training input-output images (including 1000, 2000, 4000, and 5000) generated from the numerical model are employed to evaluate the performance of proposed networks. Two evaluation criteria are used to assess the performance of the developed ED-CNN models: (1) the root mean squared error (RMSE), and (2) the coefficient of determination (R^2-score). The ED-CNNs have been developed using Keras and Tensorflow python libraries. Results and discussion: Results show that the ED-CNN accuracy, both as a meta-model and as an optimizer, is satisfactory. For the meta-model case (i.e. prediction of the temperature distribution from the Rayleigh map), the RMSE is mostly smaller than 0.15, and the R^2-score is around 0.92. In the case of ED-CNN as optimizer (i.e. estimation of the Rayleigh distribution from the heat map), RMSE is mostly in the interval [0.017-0.034], while the R^2-score is around 0.89. Acceptable results can be obtained using 2000 input-output image pairs and 150 epochs for the meta-model case, and 4000 image pairs and 200 epochs for the optimizer case. Analysis of the spatial distribution of errors shows that maximum errors occur in the middle of the problem domain where the heat map is least sensitive to the Rayleigh number. The ED-CNN model is also evaluated as an uncertainty analysis tool by comparing maps of mean and standard deviation based on the numerical model and ED-CNN predictions, showing a significant agreement with estimation error between them.Conclusion: In this paper, we examine the performance of ED-CNNs, as a specialized architecture of deep neural networks, to solve the forward and inverse problems of natural convection in porous media. For this purpose, we frame the problem as one of image-to-image regression and show that the developed model is able to provide high accuracy approximations with limited training samples, effectively solving the curse of dimensionality problem associated with heterogeneous domains. In practice, the proposed methodology can be applied to image datasets obtained from not only numerical modeling, but also high-resolution imaging and non-destructive scanning techniques, to either estimate the temperature distribution due to natural convection, or to characterize the porous media based on the temperature distribution.

Local differences in fluid density have important role in contaminant transport. Study of variable-density flow and solute transport in fractured porous media is necessary to figure out phenomena like contaminant transport of high density. In this research, the effect of permeability and molecular diffusion, respectively as a characteristic of porous medium and solute, on convective flow in fractured porous media has been studied using numerical modeling by FRAC3DVS/Hydro Geosphere (HGS) model. Conceptual model has been considered as a porous medium contained regular vertical and horizontal fractures. Four scenarios of different porous matrix permeability and free-solution diffusion coefficients have been taken account into the modelling process. The results indicated various patterns of solute transport in fractured porous medium in four different scenarios. So that, in the first one with high molecular diffusion coefficient, the solute is diffused into porous matrix from fractures symmetrically. While in the second scenario, with low molecular diffusion coefficient, the solute is transported in deeper depth and is diffused into porous matrix from fractures in an uprising way, in opposite direction of the contaminant entrance direction. In the next scenario, the more porous matrix permeability, the more convective flow velocity and solute transport in porous matrix. Finally, less differences between the matrix and fracture permeability in the last scenario leads to decrease of the fracture effects on the convective flow pattern, so that flow pattern in the fractured porous medium becomes similar to flow pattern in the homogenous porous medium.

In present paper, a numerical analysis for a rectangular cavity filled with a anisotrop porous media has been studied. It is assumed that the horizontal walls are adiabatic and impermeable, while the side walls of the cavity are maintained at constant temperatures and concentrations. The buoyancy force that induced the fluid motion are assumed to be cooperative. In the two extreme cases of heat-driven (N£1) and solute-driven (N³1) natural convection, scale analysis is applied to predict the order of magnitudes involved in the boundary layer regime. Especially, the effects of anisotropic properties on heat and mass transfer have been considered. The variation of Nusselt and Sherwood numbers for values of permeability ratio for a wide range of thermal Rayleigh number, buoyancy ratio, and Lewis number are presented. It is demonstrated that the anisotropic properties of the porous medium considerably modify the heat and mass transfer rates from that expected under isotropic conditions.

Objective: People who have higher social health can more successfully deal with the challenges and ups and downs of playing key social roles and participate more in collective activities and prevent social deviations and anomalies. In this regard, foresight and the development of a forward-looking strategy model by the media can play an important role in promoting the social health of their audience (individuals).Method: This research has been done qualitatively and quantitatively. In the qualitative phase, content analysis and Q methods were used, and in the quantitative phase, factor analysis was used to analyze the Q method data.Results: The content analysis of the media in this study showed that the most media production in the field of social deviations with 576 cases and the lowest media production in the field of violence with 237 cases. The field of social quality with 485 and the field of social issues with 312 cases are in the second and third ranks. A small questionnaire was made and based on it, the desired model was designed using PLS software. Conclusion: This model showed that the function of media is effective in promoting social health of society and can have the greatest impact on improving quality of life, reducing violence, reducing social deviations and social issues.

In this paper, we have investigated the onset of double diffusive convection (DDC) in a couple stress fluid saturated rotating anisotropic porous layer in the presence of Soret and Dufour effects using linear stability analyses which is based on the usual normal mode technique. The onset criteria for both stationary and oscillatory modes obtained analytically. The effects of the Taylor number, mechanical anisotropy parameter, Darcy Prandtl number, solute Rayleigh number, normalized porosity parameter, Soret and Dufour parameters on the stationary and oscillatory convections shown graphically. The effects of couple stresses are quite significant for large values of the non-dimensional parameter and delay the onset of convection. Taylor number has stabilizing effect on double diffusive convection, Dufour number has stabilizing effect in stationary mode while destabilizing in oscillatory mode. The negative Soret parameter stabilizes the system and positive Soret parameter destabilizes the system in the stationary convection, while in the oscillatory convection the negative Soret coefficient destabilize the system and positive Soret coefficient stabilizes the system.

In the current study, we considered a 2D enclosed porous media in three cases: rectangular, rectangular with chamfer corners, and pentagon with variable hollows. The impact of three cavity shapes, including triangle, rectangle, and lozenge shapes, on the Nusselt number at the local state, vertical velocity, total entropy, horizontal velocity, and isotherm of the porous enclosure is analyzed. The finite element method (FEM) is used to solve governing equations for fluid flow and heat transfer in a saturated non-Darcy porous medium with dissipation influences and radiation, subject to corresponding boundary conditions. In this work, we presented the pentagonal porous media shape (Case 3) as a novelty. The impact of the dimensionless parameters, such as Eckert number, on velocity profiles and temperature distribution is discussed for nine modes of different porous media geometries with varied hollows. The essential dimensionless parameters have been simulated by employing the following ranges: radiation parameter (0.5

In this paper, the phenomenon of liquid drop penetration inside the porous medium is studied by the two-phase fluid volume method. Considering the importance of two-phase and multi-phase flows and achieving the least error in the simulation of this phenomenon, in this research, a model for simulating the two-phase flow of droplet penetration in porous media is proposed, the proposed model has more agreement with experimental results compared with other similar models available in literature, the improvement is the novelty of this paper. The behavior of drop penetration has been predicted by the fluid volume method. The effect of changes in surface tension, viscosity, contact angle, viscosity and permeability and drop spreading level in porous media have been investigated. When the contact angle of the drop is 60 and 20 degrees, the changes of the expansion level are almost the same. The speed of spreading and penetration of water drop without the presence of gravity with surface tension of 0.02 is less than the other two surface tensions of 0.001 and 0.0072, while with the presence of gravity, water droplet with surface tension of 0.001 spreads and penetrates more than the other two cases. The results showed that the fluid volume method used in this research was 9% more accurate than the network Boltzmann method based on the Chan and Shen method with the physics of the same problem compared to the experimental results.