In this article, we have briefly examined the entropy generation in magnetohydrodynamic (MHD) Eyring–Powell fluid over an unsteady oscillating porous stretching sheet. The impact of thermal radiation and heat source/sink are taken in this investigation. The impact of embedded parameters on velocity function, temperature function, entropy generation rate, and Bejan number are deliberated through graphs, and discussed as well. By studying the entropy generation in magnetohydrodynamic Eyring–Powell fluid over an unsteady oscillating porous stretching sheet, the entropy generation rate is reduced with escalation in porosity, thermal radiation, and magnetic parameters, while increased with the escalation in Reynolds number. Also, the Bejan number is increased with the escalation in porosity and magnetic parameter, while increased with the escalation in thermal radiation parameter. The impact of skin fraction coefficient and local Nusselt number are discussed through tables. The partial differential equations are converted to ordinary differential equation with the help of similarity variables. The homotopy analysis method (HAM) is used for the solution of the problem. The results of this investigation agree, satisfactorily, with past studies.

In this article, the non-Newtonian fluid model named Casson fluid is considered. The semi-infinite domain of disk is fitted out with magnetized Casson liquid. The role of both thermophoresis and Brownian motion is inspected by considering nanosized particles in a Casson liquid spaced above the rotating disk. The magnetized flow field is framed with Navier’s slip assumption. The Von Karman scheme is adopted to transform flow narrating equations in terms of reduced system. For better depiction a self-coded computational algorithm is executed rather than to move-on with build-in array. Numerical observations via magnetic, Lewis numbers, Casson, slip, Brownian motion, and thermophoresis parameters subject to radial, tangential velocities, temperature, and nanoparticles concentration are reported. The validation of numerical method being used is given through comparison with existing work. Comparative values of local Nusselt number and local Sherwood number are provided for involved flow controlling parameters.

The present investigation aims to examine the heat flux mechanism in the hagnetohydrodynamic (MHD) mixed convective flow of Williamson-type fluid across an exponential stretching porous curved surface. The significant role of thermal conductivity (variable), non-linear thermal radiation, unequal source-sink, and Joules heating is considered. The governing problems are obtained using the Navier–Stokes theory, and the appropriate similarity transformation is applied to write the partial differential equations in the form of single-variable differential equations. The solutions are obtained by using a MATLAB-based built-in bvp4c package. The vital aspect of this analysis is to observe the effects of the curvature parameter, magnetic number, suction/injection parameter, permeability parameter, Prandtl factor, Eckert factor, non-linear radiation parameter, buoyancy parameter, temperature ratio parameter, Williamson fluid parameter, and thermal conductivity (variable) parameter on the velocity field, thermal distribution, and pressure profile which are discussed in detail using a graphical approach. The correlation with the literature reveals a satisfactory improvement in the existing results on permeability factors in Williamson fluids.

In this paper, the heat transfer properties in the three-dimensional (3D) magnetized with the Darcy-Forchheimer flow over a shrinking surface of the $$Cu + Al_{2} O_{3} /$$C u + A l 2 O 3 / water hybrid nanofluid with radiation effect were studied. Valid linear similarity variables convert the partial differential equations (PDEs) into the ordinary differential equations (ODEs). With the help of the shootlib function in the Maple software, the generalized model in the form of ODEs is numerically solved by the shooting method. Shooting method can produce non-unique solutions when correct initial assumptions are suggested. The findings are found to have two solutions, thereby contributing to the introduction of a stability analysis that validates the attainability of first solution. Stability analysis is performed by employing if bvp4c method in MATLAB software. The results show limitless values of dual solutions at many calculated parameters allowing the turning points and essential values to not exist. Results reveal that the presence of dual solutions relies on the values of the porosity, coefficient of inertia, magnetic, and suction parameters for the specific values of the other applied parameters. Moreover, it has been noted that dual solutions exist in the ranges of $$F_{s} \le F_{sc}$$F s ≤ F sc , $$M \ge M_{C}$$M ≥ M C , $$S \ge S_{C} ,$$S ≥ S C , and $$K_{C} \le K$$K C ≤ K whereas no solution exists in the ranges of $$F_{s} > F_{sc}$$F s > F sc , $$M < M_{c}$$M < M c , $$S < S_{c}$$S < S c , and $$K_{C} > K$$K C > K . Further, a reduction in the rate of heat transfer is noticed with a rise in the parameter of the copper solid volume fraction.

Thermophoresis and Brownian motion simultaneously occur in numerous industrial processes and have significance from an engineering point of view. The modeling of such processes provides mathematical models. The solutions of these models are used to investigate the dynamics of thermophoresis and Brownian motion in the fluid subjected to Ohmic and viscous dissipation and thermal radiation in the presence of a magnetic field. The cross-rheological model is used for modelling. Numerical solutions to the problems are computed using the finite element method (FEM). Numerical modeling through FEM is easier for complex geometries and shapes. Further, adaptability, accuracy and convergence are its key features. The grid-independent analysis is performed, accuracy is ensured and convergence is studied. Thermophoresis effects have increasing effects on the concentration profile. The strongest thermophoresis effects in mono-nano-cross-fluid are found in comparison to hybrid and ternary nanofluids. The Brownian motion parameter has a decreasing impact on the concentration profile. The strongest impact of the Brownian motion parameter in the case of ternary nanofluid is noted. Moreover, Brownian motion plays a significant role in controlling the thickness of the concentration boundary layer. Heat generation causes an increase in the thermal boundary layer region. The ternary nanofluid generates the heat. Therefore, it is recommended that fluid should not be heat generative as it impacts the efficiencies of fluid adversely. Hence, for maximum transportation of heat, ternary nanofluid should not be heat generative. The destructive chemical reaction, an enhancement in wall mass flux is noted. However, wall mass flux decreases as the strength of the generative chemical reaction increases.

This article is devoted to study sustainability of entropy generation in an incompressible thermal flow of Newtonian fluids over a thin needle that is moving in a parallel stream. Two types of Newtonian fluids (water and air) are considered in this work. The energy dissipation term is included in the energy equation. Here, it is presumed that u_∞ (the free stream velocity) is in the positive axial direction (x-axis) and the motion of the thin needle is in the opposite or similar direction as the free stream velocity. The reduced self-similar governing equations are solved numerically with the aid of the shooting technique with the fourth-order-Runge-Kutta method. Using similarity transformations, it is possible to obtain the expression for dimensionless form of the volumetric entropy generation rate and the Bejan number. The effects of Prandtl number, Eckert number and dimensionless temperature parameter are discussed graphically in details for water and air taken as Newtonian fluids.