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The purpose of this study is to present the significance of Joule heating, viscous dissipation, magnetic field and slip condition on the boundary layer flow of an electrically conducting Boussinesq couple-stress fluid induced by an exponentially stretching sheet embedded in a porous medium under the effect of the magnetic field of the variable kind. The heat transfer phenomenon is accounted for under thermal radiation, Joule and viscous dissipation effects.

The governing nonlinear partial differential equations are transformed to the nonlinear ordinary differential equations (ODEs) by using some appropriate dimensionless variables and then the consequential nonlinear ODEs are solved numerically by making the use of the well-known shooting iteration technique along with the standard fourth-order Runge–Kutta integration scheme. The impact of emerging flow parameters on velocity and temperature profiles, streamlines, local skin friction coefficient and Nusselt number are described comprehensively through graphs and tables.

Results reveal that the velocity profile is observed to diminish considerably within the boundary layer in the presence of a magnetic field and slip condition. The enhanced radiation parameter is to decline the temperature field. The slip effect is favorable for fluid flow.

Till now, slip effect on Boussinesq couple-stress fluid over an exponentially stretching sheet embedded in a porous medium has not been explored. The present results are validated with the previously published study and found to be highly satisfactory.

The purpose of this paper is to discuss the flow of Casson nanofluid past a nonlinear permeable stretching sheet in the presence of thermal radiation, chemical reaction, viscous dissipation, heat source, and magnetohydrodynamics.

Appropriate transformations are used to convert the boundary layer equations into nonlinear ODEs which are then solved numerically by using the Runge-Kutta-Fehlberg fourth-fifth order method along with shooting technique.

Solution of this systems is obtained for velocity, temperature, and concentration profiles. Graphical illustrations are added to discuss the effect of evolving parameters against above-mentioned distributions. Tabular values of local skin friction factor, local Nusselt number, and local Sherwood number are also added and studied accordingly.

A good agreement of the present results has been observed by comparing with the existing literature results. It is noted that skin friction coefficient, Nusselt number, and Sherwood number decrease with Casson parameter and increase with suction parameter.

In this communication, a theoretical simulation is aimed to characterize the Darcy–Forchheimer flow of a magneto-couple stress fluid over an inclined exponentially stretching sheet. Stokes’ couple stress model is deployed to simulate non-Newtonian microstructural characteristics. Two different kinds of thermal boundary conditions, namely, the prescribed exponential order surface temperature (PEST) and prescribed exponential order heat flux, are considered in the heat transfer analysis. Joule heating (Ohmic dissipation), viscous dissipation and heat source/sink impacts are also included in the energy equation because these phenomena arise frequently in magnetic materials processing.

The governing partial differential equations are transformed into nonlinear ordinary differential equations (ODEs) by adopting suitable similar transformations. The resulting system of nonlinear ODEs is tackled numerically by using the Runge–Kutta fourth (RK4)-order numerical integration scheme based on the shooting technique. The impacts of sundry parameters on stream function, velocity and temperature profiles are viewed with the help of graphical illustrations. For engineering interests, the physical implication of the said parameters on skin friction coefficient, Nussult number and surface temperature are discussed numerically through tables.

As a key outcome, it is noted that the augmented Chandrasekhar number, porosity parameter and Forchhemeir parameter diminish the stream function as well as the velocity profile. The behavior of the Darcian drag force is similar to the magnetic field on fluid flow. Temperature profiles are generally upsurged with the greater magnetic field, couple stress parameter and porosity parameter, and are consistently higher for the PEST case.

The findings obtained from this analysis can be applied in magnetic material processing, metallurgy, casting, filtration of liquid metals, gas-cleaning filtration, cooling of metallic sheets, petroleum industries, geothermal operations, boundary layer resistors in aerodynamics, etc.

From the literature review, it has been found that the Darcy–Forchheimer flow of a magneto-couple stress fluid over an inclined exponentially stretching surface with heat flux conditions is still scarce. The numerical data of the present results are validated with the already existing studies under limited cases and inferred to have good concord.

The purpose of this paper is to study the boundary layer flow and heat transfer of a power-law non-Newtonian nanofluid over a non-linearly stretching sheet.

The governing equations describing the problem are transformed into a nonlinear ordinary differential equations by suitable similarity transformations. The resulting equations for this investigation are solved numerically by using the variational finite element method.

It was found that the local Nusselt number increases by increasing the Prandtl number, stretching sheet parameter and decreases by increasing the power-law index, thermophoresis parameter and Lewis number. Increases in the stretching sheet parameter, Prandtl number and thermophoresis parameter decrease the local Sherwood number values. The effects of Brownian motion and Lewis number lead to increases in the local Sherwood number values.

The work is relatively original as very little work has been reported on non-Newtonian nanofluids.

THE modern aeroplane is constructed largely from sheet metal. As such, the most important production problems are those of sheet metal forming, and assembling. Production is here considered as not only the act of forming and assembling the required number of parts, but also the making of forming tools, and all processing of parts such as heat‐treating. Only that phase of the above concept of production which deals with the tooling for production and the forming and heat‐treating will be considered here. The design of the aircraft parts will also be discussed somewhat, for it is obvious that the design of the part (designed shape and materials used) frequently determines whether the part can or cannot be readily made.

This study aims to investigate the unsteady two-dimensional viscous flow and heat transfer over an unsteady permeable stretching/shrinking sheet (surface) with generalized slip velocity condition.

Similarity transformation is used to reduce the system of partial differential equations into a system of nonlinear ordinary differential equations. The resulting equations are then solved numerically using “bvp4c” function in MATLAB software.

Dual solutions are found for a certain range of the unsteady, suction and stretching/shrinking parameters. Stability analysis is performed, and it is revealed that the first (upper branch) solution is stable and physically realizable, whereas the second (lower branch) solution is unstable.

The results obtained can be used to explain the characteristics and applications of the generalized slip in boundary layer flow. Such condition is applied for particulate fluids such as foams, emulsions, polymer solutions and suspensions. Furthermore, the phenomenon of stretching/shrinking sheet can be found on the manufacturing of polymer sheets, rising and shrinking balloon or moving and shrinking polymer film.

The present numerical results are original and new for the study of unsteady flow and heat transfer over a permeable stretching/shrinking sheet with generalized slip velocity.

The purpose of this paper is to study the flow and heat transfer of a hybrid nanofluid, Cu–Al₂O₃/water, past a permeable stretching/shrinking sheet. The effects of Brownian motion and thermophoresis are considered here.

Similarity transformations are used to reduce the governing partial differential equations to a system of ordinary (similarity) differential equations. A MATLAB solver called the bvp4c is then used to compute the numerical solutions of equations (12) to (14) subject to the boundary conditions of equation (15). Then, the effects of various physical parameters on the flow and thermal fields of the hybrid nanofluid are analyzed.

Multiple (dual) solutions are found for the basic boundary layer equations. A stability analysis is performed to see which solutions are stable and, therefore, applicable in practice and which are not stable. Besides that, a comparison is made between the hybrid nanofluid and a traditional nanofluid, Cu/water. The skin friction coefficient and Nusselt number of the hybrid nanofluid are found to be greater than that of the other nanofluid. Thus, the hybrid nanofluid has a higher heat transfer rate than the other nanofluid. However, the increase in the shrinking parameter reduces the velocity of the hybrid nanofluid.

The present results are original and new for the study of the flow and heat transfer past a permeable stretching/shrinking sheet in Cu–Al₂O₃/water hybrid nanofluid.

The purpose of this paper is to consider steady two‐dimensional mixed convection flow along a vertical semi‐infinite power‐law stretching sheet. The velocity and temperature of the sheet are assumed to vary in a power‐law form.

The problem is formulated in terms of non‐similar equations. These equations are solved numerically by an efficient implicit, iterative, finite‐difference method in combination with a quasi‐linearization technique.

It was found that the skin‐friction coefficient increased with the ratio of free‐stream velocity to the composite reference velocity and the buoyancy parameter while it decreased with exponent parameter. The heat transfer rate increased with the Prandtl number, buoyancy parameter and the exponent parameter.

A very useful source of information for researchers on the subject of convective flow over stretching sheets.

This paper illustrates mixed convective flow over a power‐law stretched surface with variable wall temperature.

Visco‐elastic fluid flow and heat transfer in a porous medium over a non‐isothermal stretching sheet have been investigated. The flow is influenced by linearly stretching the sheet in the presence of suction, blowing and impermeability of the wall. Thermal conductivity is considered to vary linearly with temperature. The intricate non‐linear problem has been solved numerically by shooting technique with fourth order Runge‐Kutta algorithm after using perturbation method. The zeroth order solutions are obtained analytically in the form of Kummer’s function. An analysis has been carried out for two different cases, namely prescribed surface temperature (PST) and prescribed heat flux (PHF) to get the effect of porosity and visco‐elasticity at various physical situations. The important finding is that the effect of visco‐elasticity and porosity is to increase the wall temperature in case of blowing and to decrease in both the cases of suction and when the stretching sheet is impermeable.

This paper aims to study a non-Newtonian micropolar fluid flow over a bidirectional flexible surface for multiple exact solutions of momentum boundary layer and thermal transport phenomenon subject to wall mass flux, second-order slip and thermal jump conditions.

The coupled equations are transformed into ordinary differential equations using similarity variables. Analytical and numerical techniques are used to solve the coupled equations for single, dual or multiple solutions.

The results show that the stretching flow, shrinking flow, the wall drag, thermal profile and temperature gradient manifest large changes when treated for special effects of the standard parameters. The role of critical numbers is definitive in locating the domains for the existence of exact solutions. The nondimensional parameters, such as mass transfer parameter, bidirectional moving parameter, plate deformation strength parameter, velocity slips, material parameter, thermal jump and Prandtl number, are considered, and their physical effects are presented graphically. The presence of governing parameters exhibits special effects on the flow, microrotation and temperature distributions, and various exact solutions are obtained for the special parametric cases.

The originality and value of this work lie in its exploration of non-Newtonian micropolar fluid flow over a bidirectional flexible surface, highlighting the multiple exact solutions for momentum boundary layers and thermal transport under various physical conditions. The study provides insights into the effects of key parameters on flow and thermal behavior, contributing to the understanding of complex fluid dynamics.