Free convection flow of non-Newtonian fluids over flat, heated surfaces is an important natural phenomenon that also occurs in human-made engineering processes under various physical and mechanical situations. In the current study, the free convection magnetohydrodynamic flow of Jeffrey fluid with heat and mass transfer over an infinite vertical plate is examined. Mathematical modeling is performed using Fourier’s and Fick’s laws, and heat and momentum equations have been obtained. The non-dimensional partial differential equations for energy, mass, and velocity fields are determined using the Laplace transform method in a symmetric manner. Later on, the Laplace transform method is employed to evaluate the results for the temperature, concentration, and velocity fields with the support of Mathcad software. The governing equations, as well as the initial and boundary conditions, satisfy these results. The impacts of fractional and physical characteristics have been shown by graphical illustrations. The obtained fractionalized results are generalized by a more decaying nature. By taking the fractional parameter $\beta ,\gamma \to 1$, the classical results with the ordinary derivatives are also recovered, making this a good direction for symmetry analysis. The present work also has applications with engineering relevance, such as heating and cooling processes in nuclear reactors, the petrochemical sector, and hydraulic apparatus where the heat transfers through a flat surface. Moreover, the magnetized fluid is also applicable for controlling flow velocity fluctuations.

The purpose of the current study is to investigate the non-Newtonian unsteady Williamson fluid on a stretching/shrinking surface along with thermophoresis and Brownian effects. Basically, the model consists of a time-dependent magnetic field. The fluid is considered to be electrically conducting due to the effect of the external magnetic field. The values of magnetic Reynolds number are so small that the induced magnetic field is assumed to be negligible. In the concentration equation, the effects of Brownian motion and thermophoresis are discussed. Employing the similarity transformations, the governing nonlinear Partial Differential Equations (PDEs) are converted into the Ordinary Differential Equations (ODEs). The resulting ODEs are solved with the combined effects of the Successive Over Relaxation (SOR) method and Finite Difference Method (FDM). The impact of all the including parameters such as suction parameter, injection parameter, stretching/shrinking parameter, the ratio of viscosity, local Weissenberg number, unsteadiness parameter, Eckert number, Prandtl number, Lewis number, Nusselt number, Brownian motion parameter, shear stress, heat transfer rate, and mass transfer rate are analyzed using graphs and tables. Results show that the values of fluid velocity are better for $S=8, -S=0, \lambda =0.3, {\beta }^{*}=0.9, Wi=0.3$, and $Aa=2.0$. It is also depicted from the results that the values of boundary layer thickness are better for $S=0, -S=-8, \lambda =0.3, {\beta }^{*}=0.1, Wi=1.5$, and $Aa=0.25$. From the above numeric results, it is concluded that the fluid velocity is reduced and the thermal boundary layer thickness is enhanced by the enhancement of the stretching parameter.

Heat transfer enhancement in heat exchangers results in thermal efficiency and energy saving. In double-pipe heat exchangers (DPHEs), extended or augmented fins in the annulus of the two concentric pipes, i.e., at the outer surface of the inner pipe, are used to extend the surface of contact for enhancing heat transfer. In this article, an innovative diamond-shaped design of extended fins is proposed for DPHEs. This type of fin is considered for the first time in the design of DPHEs. The triangular-shaped and rectangular-shaped fin designs of DPHE, available in the literature, can be recovered as special cases of the proposed design. An h-adaptive finite element method is employed for the solution of the governing equations. The results are computed for various performance measures against the emerging parameters. The results dictate that the optimal configurations of the diamond-shaped fins in the DPHE for an enhanced heat transfer are recommended as follows: If around 4–6, 8–12, or 16–32 fins are to be placed in the DPHE, then the height of the fins should be 20%, 80%, or 100%, respectively, of the annulus width. If frictional loss of heat is also to be considered, then for fin-heights of 20–80% and 100% of the annulus width, the placement of 4 and 8 diamond-shaped fins, respectively, is recommended for an enhanced heat transfer. These recommendations are for the radii ratio (i.e., the ratio of the inner pipe radius to that of the outer pipe) of 0.25. The recommendations are be modified if the radii ratio is altered.

This article investigates the pulsatile flow of viscous incompressible MHD nanofluid in a rectangular channel. At the upper and lower walls, the channel has symmetrical constrictions. The goal is to analyze the heat transfer features of the nanofluid flow under the effect of the magnetic field and thermal radiation. Five different nanofluids, formed with nanoparticles of copper (Cu), magnetite (Fe3O4), silver (Ag), titanium oxide (TiO2), and single wall carbon nanotube (SWCNT) in the base fluid of water, are considered in the study. The unsteady governing equations are transformed using the vorticity-stream function approach. The solution is obtained using the finite difference technique (FDM). The effect of various flow controlling parameters on velocity, temperature, Nusselt, and skin-friction profiles is inspected by using graphs. Across the channel, graphs of vorticity, streamlines, and temperature distribution are also displayed. The thickness of the thermal boundary layer upsurges with escalating values of the magnetic field, radiation parameters, and solid volume fraction, whereas it declines with escalating values of the Strouhal and Prandtl numbers. The profiles are usually found to have a more regular pattern upstream of the constriction than that downstream of the constriction. At the throat of the constriction, the carbon nanotube-based nanofluid attains higher temperatures than the other nanofluids downstream of the constriction. However, in the lee of the constriction, silver-based nanofluid attains higher temperatures than the other nanofluids downstream of the constriction. The behavior, in most cases, is opposite upstream of the constriction. The findings of the study can be utilized to cure stenosis in blood vessels, design biomechanical devices, and employ flow pulsation to control industrial operations.

In this study, heat transfer is analyzed for an incompressible steady laminar micropolar fluid flow among two symmetrical stretchable disks. The associated partial differential equations (PDEs) are converted into ordinary differential equations (ODEs) through appropriate similarity transformations and then solved numerically by applying the quasi-linearization method along with the finite difference method (FDM). This impact from certain inclusive parameters in the heat transfer and fluid flow is presented schematically. The results are tabulated and depict that the shear stress rises for the upper disk and the opposite trend for the lower disk. It is also noted that a couple stress and heat transfer rates were reduced across the disks. The slip factor enhances the tangential stress on the top disc and it reduces on the bottom disc. Twisting stress decreases where ϕ ′ η increases due to the effect of the slip factors.

The purpose of this study was to determine an exact solution for the fluid flow and heat transfer of laminar steady magnetohydrodynamics (MHD) nanofluid flow over a stretching/shrinking surface. Appropriate similarity transformations were used to transform the governing partial differential equations into coupled nonlinear ordinary differential equations. The current study showed good correspondence with previously published work. The solution was deduced from the solution of the flow field and temperature field. Furthermore, the dimensionless skin friction coefficient and Nusselt number were derived. The solution of the temperature field was deduced in terms of the generalized Laguerre polynomial. The value of the generalized Laguerre polynomial was calculated using the “LaguerreL” command in MuPAD. The impact of different physical parameters of the symmetry on the thermal performance, including the nanoparticle volume fraction parameter, magnetic parameter, mass suction/injection parameter and stretching/shrinking parameter, is discussed in detail for different nanoparticles. Furthermore, the effect of nanoparticle type on the fluid velocity component, temperature distribution, skin friction coefficient and Nusselt number was studied in detail. Four different nanoparticles were considered in this study. This work reveals that the nanoparticles within the base fluid have the potential to increase the heat transfer ability of many liquids. The results indicate that silver and titanium oxide nanoparticles had the largest and lowest skin friction coefficients, respectively, in the shrinking surface case, exhibiting opposite behavior in the stretching surface case among all the nanoparticles considered. The results also indicate that silver and titanium oxide nanoparticles had the largest and lowest Nusselt numbers, respectively, for both the stretching and the shrinking surface cases. It is suggested silver nanoparticles are not used for optimum heat transfer.