The aim of this research is mainly concerned with the numerical examination of Darcy-Forchheimer relation in convective magnetohydrodynamic nanofluid flow bounded by non-linear stretching sheet. A visco-elastic and strictly incompressible liquid saturates the designated porous medium under the direct influence of the Darcy-Forchheimer model and convective boundary. The magnetic effect is taken uniformly normal to the flow direction. However, the model is bounded to a tiny magnetic Reynolds number for practical applications. Boundary layer formulations are taken into consideration. The so-formulated leading problems are converted into highly nonlinear ordinary problems using effectively modified transformations. The numerical scheme is applied to solve the governing problems. The outcomes stipulate that thermal layer receives significant modification in the incremental direction for augmented values of thermal radiation parameter R_d. Elevation in thermal Biot number γ₁ apparently results a significant rise in thermal layer and associated boundary layer thickness. The solute Biot number is found to be an enhancing factor the concentration profile. Besides the three main profiles, the contour and density graphs are sketched for both the linear and non-linear cases. Furthermore, skin friction jumps for larger porosity and larger Forchheimer number. Both the heat and mass flux numbers receive a reduction for augmented values of the Forchheimer number. Heat flux enhances, while mass flux reduces, the strong effect of thermal Biot number. The considered problem could be helpful in any several industrial and engineering procedures, such as rolling, polymeric extrusion, continuously stretching done in plastic thin films, crystal growth, fiber production, and metallic extrusion, etc.

This research work describes and investigates Williamson nanofluid flow over an exponentially stretching permeable vertical plate with temperature-dependent thermal conductivity and viscosity. The governing non-linear partial differential equations (PDEs) are metamorphosed into coupled non-linear ordinary differential equations (ODEs) by using similarity transformation. The succeeding equations were numerically solved using MATLAB function bvp4c for various values of parameters. For velocity, temperature, concentration, the skin friction coefficient, and the local Nusselt number, data are presented in the form of graphs and tables. It is noted that for increasing values of magnetic parameter $M$, Williamson parameter $\lambda ,$ and viscosity parameter $\alpha$, the boundary layer thickness of the velocity profile decreases, while it increases for the temperature profile. The findings of the present work are validated through the published results.

The co-infection of Human Immunodeficiency Virus (HIV) and Hepatitis B virus (HBV) poses a major threat to public health due to their combined negative impacts on health and increased risk of complications. A novel fractional mathematical model of the dynamics of co-infection between HBV and HIV for Taiwan is presented in this paper. Detailed analyses are conducted on the possible impact of HBV vaccination on the dynamics of HBV and HIV co-infection. The next-generation matrix technique is used to calculate the fundamental reproduction number R ₀ = max{R ₁, R ₂}, where R ₁ and R ₂ are the reproduction numbers for HBV and HIV, respectively. The disease-free and endemic equilibria of the co-infection model are calculated. An extensive investigation is carried out to determine the local and global stability of the disease-free equilibrium point through Rough Hurtwiz criteria and the construction of Lyapunov function, respectively. We demonstrate that when R ₁ < 1 < R ₂, HBV infection is eradicated, but HIV remains prevalent. If R ₂ < 1 < R ₁, the opposite outcome occurs. The real data from 2000-2023 for Taiwan is used to fit the model. The fitting results show how effectively our model handles the data. In addition, numerical simulations are run for different scenarios to observe how the vaccine and fractional parameters changed the model state variables, as well as how the solutions behaved and how quickly they reached the model’s equilibrium points. According to the model’s numerical analysis, greater vaccination efforts against HBV have a positive effect on the propagation of co-infection.

We investigated the dynamics of nanofluid and heat transfer in a three-dimensional circular annular using the $\kappa -\epsilon$ turbulence model and energy equations. The pipe contained two concentric and rotating cylinders with a constant speed in the tangential direction. A heat flux boundary condition was executed at the inner cylinder of the annular. The pipe was settled vertically, and water alumina nanofluid was allowed to enter, with the initial velocity depending on the Reynolds number, ranging from 30,000 to 60,000. The volume fraction of the solid particles was tested from 0.001 to 0.1. The speed of the rotation of the cylinders was tested in the range from 0.5 to 3.5. The simulations were developed using COMSOL Multiphysics 5.6, adopting the finite element procedure for governing equations. The results were validated using the mesh independent study and the average Nusselt number correlations. We found that the average Nusselt number in the middle of the channel decreases linearly with the increase in the volume fraction of the water alumina nanofluid. The novelty of the present work is that various correlations between the average Nusselt number and volume fraction were determined by fixing the Reynolds number and the rotation of the inner cylinder. We also found that fixing the Reynolds number and the volume fraction improves the average Nusselt number at the outlet linearly. In addition, it was stated that the increase in the total mass of the nanofluid would decrease the average temperature at the outer cylinder of the annular. Moreover, the maximum average improvement percentage in the average Nusselt number, which is about 21%, was observed when the inner cylinder rotation was changed from 1.5 to 2.5 m/s.

In this article, MHD flow of silver/water nanofluid past a stretched cylinder under the impact of thermal radiation with chemical reaction and slip condition is studied. The impact of Soret and Dufour effect is also analyzed during this flow. The uniqueness of the given problem is enlarged with the insertion of variable magnetic field, free stream velocity, thermal slip condition, and nonlinear thermal radiation. The PDEs are converted to ODEs by using suitable similarity transformation. The nonlinear system of ODEs is solved by applying convergent homotopy analysis method (HAM). The velocity, temperature, and concentration profiles for the free stream and at the plate are discussed through graphs and numerical tables. It is found that velocity field reduces, while the temperature profile rises for the increasing values of magnetic parameter. It is examined that effects of curvature on frication factor are increasing. Furthermore, temperature profile increases for greater Brownian motion and thermophoresis parameters. Transfer of heat enhances decreasing the radius of the cylinder also with heat generation parameter. The skin friction can be reduced by enhancing free stream and wall stretching velocities ratio. Velocity profile of the flow can be controlled by enhancing velocity slip and magnetic field.