Heat transfer in a symmetrical cavity with two semi-cylinders was explored in this study. Several parameters, such as (${10}^3\le Ra\le {10}^6$), (${10}^{-5}\le Da\le {10}^{-2}$), ($0.02\le \varphi \le 0.08$), ($0.2\le \epsilon \le 0.8$), and ($0\le Ha\le 100$) were selected and evaluated in this research. The outcome of the magnetic field and the temperature gradient on the nanofluid flow is considered. The geometric model is therefore described using a symmetry technique. The flow issue for the governing equations has been solved using the Galerkin finite element method (G-FEM), and these solutions are presented in dimensionless form. The equations for energy, motion, and continuity were solved using the application of the COMSOL Multiphysics^® software computer package. According to the results, there is a difference in the occurrence of the magnetic parameter and an increase in heat transmission when the right wall is recessed inward. The heat transmission is also significantly reduced when the right wall is exposed to the outside. The number of Nusselt grows directly proportional to the number of nanofluids in the environment. In contrast, all porous media with low Darcy and Hartmann numbers, high porosity, and low volume fraction have high Nusselt numbers. It is found that double streamlines for the hot side and single cooling for Darcy, Rayleigh, and Hartmann numbers. A cold isotherm at various physical parameters is needed in the top cavity. Rayleigh’s number and a solid volume fraction raise Darcy’s number, increasing heat transmission inside the cavity and thermal entropy determines entropy components.

The objective of this article is to model and analyze unsteady squeezing flow of fractional MHD Casson fluid through a porous channel. Casson fluid model is significant in understanding the properties of non-Newtonian fluids such as blood flows, printing inks, sauces and toothpaste etc. This study provides important results as unsteady flow of Casson fluid in fractional sense with aforementioned effects has not been captured in existing literature. After applying similarity transformations along with fractional calculus a highly non-linear fractional-order differential equation is obtained. Modeled equation is then solved along with no-slip boundary conditions through a hybrid of Laplace transform with homotopy perturbation algorithm. For validity purposes, solution and errors at various values in fractional domain are compared with existing results. LHPM results are better in terms of accuracy than other available results in literature. Effects of fractional parameter on the velocity profile, skin friction and behaviors of involved fluid parameters is the focal point of this study. Comprehensive, quantitative and graphical analysis is performed for investigating the effects of pertinent fluid parameters on the velocity profile and skin friction. Analysis revealed that fractional parameter depicts similar effect in case of positive and negative squeeze number. Also, skin friction decreases with an increasing fractional parameter. Moreover, in fractional environment Casson parameter has shown similar effect on the velocity profile in case of positive and negative squeeze number.

In this study, we have examined the stagnation point flow of MHD chemically reacting nanofluid along a stretching/shrinking sheet. The chemical reaction has vital applications in food preparation, the mechanism of water and oil emulsions, and chemical engineering. Therefore, this analysis is subjected to the effects of Joule heating, viscous dissipation, Brownian motion, chemical reaction, thermophoresis, and melting heat transfer. The governing transformed equations are solved numerically via shooting scheme together with Runge–Kutta–Fehlberg method. The impacts of various nanofluids and relevant regulating parameters such as chemical reaction parameter, magnetic parameter, radiation parameter, melting parameter, and Eckert number on dimensionless velocity, thermal, and concentration profiles as well as skin friction drag coefficient, Nusselt and Sherwood numbers are presented graphically. Although dual branch solutions are obtained in this investigation, only one solution is determined to be stable. It is seen that the coefficient of skin friction and mass transfer rate tend to be decreasing due to melting heat transfer. The results also revealed that higher estimations of magnetic parameter increase the heat transfer rate, but a reverse trend is noticed for surface drag coefficient and mass transfer rate in case of first solution.

In a cylindrical cavity, the convection and entropy of the hybrid nanofluid were studied. We have introduced a rectangular fin inside the cylinder; the fin temperature is at $T_h$. The right waving wall is cooled to $T_c$. The upper and lower walls are insulated. This study contains the induction of a constant magnetic field. The Galerkin finite element method (GFEM) is utilized to treat the controlling equations obtained by giving Rayleigh number values between $Ra$ (10³–10⁶) and Hartmann number ratio $Ha$ (0, 25, 50, 100) and Darcy ranging between $Da$ (10⁻²–10⁻⁵) and the porosity ratio is $\epsilon$ (0.2, 0.4, 0.6, 0.8), and the size of the nanoparticles is $\varphi$ (0.02, 0.04, 0.06, 0.08). The range is essential for controlling both fluid flow and the heat transport rate for normal convection. The outcomes show how Da affects entropy and leads to a decline in entropy development. The dynamic and Nusselt mean diverge in a straight line. The domain acts in opposition to the magnetic force while flowing. Highest entropy-forming situations were found in higher amounts of $Ra$, $Da$, and initial values of $Ha$. Parameters like additive nanoparticles ($\varphi$) and porosity ($\epsilon$) exert diagonal dominant trends with their improving values.

This paper is an attempt to study the nonlinear vibration of a pre-stressed single-walled carbon nanotube (SWCNT) with water-filled and simply supported ends. A new analytical formula is obtained for the nonlinear model based on the simplified Donnell’s shell theory. The effects of internal fluid on the coupling vibration of the SWCNT–water system are discussed in detail. Furthermore, the influence of the different nanotube thicknesses and radiuses on the nonlinear vibration frequencies is investigated according to the shell theory. Numerical calculations are done to show the effectiveness of the proposed schemes. The results show that the nonlinear frequency grew with the increasing nonlinear parameters (radius and thickness of nanotube). In addition, it is shown that the influence of the nonlinear parameters is greater at the lower mode in comparison with the higher mode for the same nanotube thickness and radius.