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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.

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.

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