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.

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.

A DG-FEM based numerical investigation has been performed to explore the influence of the various geometric configurations on the thermal performance of the conjugate heat transfer analysis in the triangular finned double pipe heat exchanger. The computed results dictate that Nusselt number in general rises with values of the conductivity ratio of solid and fluid, for the specific configuration parameters considered here. However, the performance of these parameters shows strong influence on the conductivity ratio. Consequently, these parameters must be selected in consideration of the thermal resistance, for better design of heat exchanger.

An efficient parallel scheme based on the nodal discontinuous Galerkin finite element method (nodal-DGFEM) for the numerical solution of the partial differential equations governing fluid flow phenomena is discussed. The flow solver is demonstrated to perform numerical simulation of two-dimensional flow regimes on unstructured triangular grids. The parallel implementation serves to fulfill the requisition of the numerical method regarding high-performance computing resources. The distributed memory programming model with the domain decomposition approach is adopted. The message passing interface library is used for communication among the parallel processes, which are assigned domain-decomposed subproblems. The presented parallelization strategy accurately and efficiently tackles the communication of multi-node data on the element edges between the neighboring parallel processes. The efficacy and efficiency of the parallel solver are demonstrated through solving the well-known problem of non-viscous isentropic convecting vortex flow on parallel systems. The parallelization would extend the scope of the DGFEM by producing solutions in reasonable time frames.