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The present paper investigates the mixed convection in a two-sided and four-sided lid-driven square cavity in porous media. In the two-sided porous cavity, the left and right walls of the enclosure are maintained at constant but different temperatures, while the top and bottom walls are adiabatic. The top and the bottom walls of the enclosure move with a constant speed from left to right. In the four-sided porous cavity, the top and the bottom walls of the enclosure move from left to right and right to left, respectively, while the left and the right walls move from top to bottom and bottom to top, respectively, with a constant speed. The left and right walls of the enclosure are maintained at different heat fluxes, while the top and bottom walls are maintained at hot and cold temperatures, respectively. The governing equations are discretized by the fully implicit finite difference method, namely, Alternating-Direction-Implicit (ADI) method. The numerical results are analyzed for the effect of Darcy number (Da = 0.001, 0.01), Prandtl number (Pr = 7), Grashof number (Gr = 50,000), porosity (ε = 0.2) and viscosity ratio (Λ = 1, 3). The stability and convergence of the considered problem have been proved using the Matrix method.

The article delves into the realm of application of complex numbers and matrix transformations in pentagonal tiling. A basic understanding of the concepts in these two branches helps to plot the tiles in any computer algebra system (CAS). The application of the concepts has been demonstrated for the special case of Type 13 pentagonal tiling.

The present study investigated the characterization of heat and mass transfer in a two-sided lid-driven square cavity subjected to non-uniform sinusoidal heating on its horizontal walls. The amplitude ratio ( $$\varepsilon $$ ) and phase deviation ( $$\phi $$ ) of sinusoidal temperature at the bottom wall are assumed to be constant, while it behaves as a variable on the cavity’s top boundary. The cavity’s adiabatic and impermeable vertical walls are kept in motion at a uniform speed in the opposite directions, while the horizontal walls are stationary. An unsteady, laminar, and compressible fluid flow under Boussinesq approximation has admitted in the cavity. The considered problem is taken into account due to its great importance in various thermal engineering applications. The stream function–vorticity formulation with the alternating-direction-implicit method is used to solve the non-dimensional governing equations. Also, the stability and consistency of the numerical scheme have been proved using the matrix method. A MATLAB code is written, executed to do the numerical simulations for various non-dimensional parameters, such as Richardson number $$(0.01\le \hbox {Ri} \le 100)$$ , amplitude ratio ( $$\varepsilon =0$$ , 0.25, 0.5, 1), phase deviation ( $$\phi =0$$ , $$\pi /4$$ , $$\pi /2$$ , $$3\pi /4$$ , $$\pi $$ ), and buoyancy ratio $$(-5 \le N \le 5)$$ . The MATLAB code is validated with some of the available literature and found a good agreement with them. The streamline, isotherm, and concentration contours are examined graphically for the different values of the non-dimensional parameter. It is found that the distribution of the contours within the square cavity depends on the relative contributions of the parameters. The heat and mass transfer inside the square cavity are analyzed in terms of the average Nusselt number and the average Sherwood number. The rate of heat and mass transfer in the cavity is found almost constant in forced convection but increases significantly in natural convection. It is also found that the buoyancy ratio, amplitude ratio, and phase deviation contribute significantly only under natural convection.

This study presents the mixed convection inside a four-sided lid-driven square porous cavity whose right wall is maintained at a sinusoidal temperature condition, the left wall of the cavity is maintained at a cold temperature, while the top and the bottom walls are adiabatic. We have discussed two different cases depending upon the direction of the moving walls. Brinkmann-extended Darcy model is represented in terms of $$\psi $$ and $$\xi $$ using the stream function-vorticity formulation to simulate the momentum transfer in the porous medium. This formulation is used to solve the governing equations as a coupled system of equations which consists of the field variables, vorticity $$(\xi )$$ , stream function $$(\psi )$$ , and temperature (T). The velocity components (u, v) are derived from the stream function $$(\psi )$$ whereas the average Nusselt number is derived from temperature. The stability and consistency of the applied numerical scheme to the considered problem has been proven by matrix method. The numerical results are investigated by ranging the various dimensionless numbers such as Grashof number $$(10^3 \le { {Gr}} \le 10^5)$$ , Darcy number $$(10^{-1} \le { {Da}} \le 10^{-5})$$ , Reynolds number $$(10 \le { {Re}} \le 1000)$$ and keeping the Prandtl number $$({ {Pr}}=0.7)$$ fixed.

The current research paper delibrates the effect of variable viscosity of the nanofluid that saturates a porous media, on transient free convection. The viscosity term that appears in the equation of conservation of momentum depends on nanoparticle volume fraction. The effect of viscosity parameter on reduced Nusselt number and reduced Sherwood number for different combinations of Buoyancy ratio Nr, Brownian motion Nb and thermophoresis Nt is examined using the unconditionally stable Keller Box method. The conditions for existence of the numerical solution has been analytically derived. The study illustrates that the variation in viscosity has an insignificant effect on the reduced Nusselt number, but the reduced Sherwood number increases with an increase in the viscosity. Although the study can accommodate any arbitrary axisymmetric body, vertical flat plate, horizontal cylinder and sphere are three particular geometries that are considered in the paper.

The article explores the application of binary and ternary number systems to three classical mathematical puzzles–weight problem of Bachet de Meziriac, binary numbers magic trick, and the detection of a counterfeit coin. The article further elucidates the generalization of these puzzles using the balanced number system.

The intent of this paper is to present the numerical results for transient heat transfer across an axisymmetric non-isothermal body embedded in porous media saturated by nanofluid. A non-linear coupled PDE is reduced using dimensionless similarity variables and is solved using Keller Box method. Buoyancy ratio Nr, Brownian motion Nb and thermophoresis Nt are the parameters considered in this study. The effect of Lewis number Le and non-Newtonian parameter N on reduced Nusselt number and reduced Sherwood number is recorded in tabular form for isothermal as well as for non-isothermal bodies. The heat flux for different values of non-Newtonian parameter N is plotted in the case of sphere and cylinder at different time levels.

Similarity solutions for the problem of free convection flow over a non-isothermal horizontal plate embedded in porous media are investigated in the presence of internal heat generation. The porous medium is saturated with non-Newtonian power law fluid. Numerical results are obtained for the effect of power law temperature profile and fluid index on the heat transfer characteristics.

A similarity solution is proposed for the analysis of the steady free convection boundary layers over a non-isothermal axisymmetric body embedded in a fluid saturated porous medium. The effect of temperature dependent viscosity on heat transfer rates in the presence of internal heat generation is investigated. The linearized version of the Arrhenius law for the temperature dependent viscosity is considered. It is shown that the heat transferred is more for a less viscous fluid.

Similarity solution for the problem of free convection boundary layers over a non-isothermal two dimensional or axisymmetric body which is embedded in a porous medium with internal heat generation is presented. In each case, a corresponding class of wall temperature admitting similarity solution is found. The results are illustrared for a vertical flat plate, horizontal cylinder and a sphere