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Outstanding features such as thermal conductivity and superior electrical conductivity of nanofluids unfold a new window in the context of their extensive applications in engineering and industrial domains. The purpose of this study to simulate numerically the magneto-nanofluid flow and heat transfer over a curved stretching surface. Heat transport is explored in the presence of viscous dissipation. At the curved surface, the convective boundary condition is adopted. Three different nanoparticles, namely, copper, aluminium oxide and titanium dioxide are taken into consideration because of easily available in nature.

The basic flow equations are framed in terms of curvilinear coordinates. The modelled partial differential equations are transformed into a system of non-linear ordinary differential equations by means of appropriate similarity transformation. The subsequent non-linear system of equations is then solved numerically by using the Runge–Kutta–Felhberg method with the shooting scheme via bvp4c MATLAB built-in function. Impacts of various physical parameters on velocity, pressure and temperature distributions, local skin-friction coefficient, local Nusselt number and wall temperature are portrayed through graphs and tables followed by a comprehensive debate and physical interpretation.

Graphical results divulge that augmenting values of the magnetic parameter cause a decline in velocity profiles and stream function inside the boundary layer. The magnitude of the pressure function inside the boundary layer reduces for higher estimation of curvature parameter, and it is also zero when the curvature parameter goes to infinity. Furthermore, the temperature is observed in a rising trend with growing values of the magnetic parameter and Biot number.

This research study is very pertinent to the expulsion of polymer sheet and photographic films, metallurgical industry, electrically-conducting polymer dynamics, magnetic material processing, rubber and polymer sheet processing, continuous casting of metals, fibre spinning, glass blowing and fibre, wire and fibre covering and sustenance stuff preparing, etc.

Despite the huge amount of literature available, but still, very little attention is given to simulate the flow configuration due to the curved stretching surface with the convective boundary condition. Very few papers have been examined on this topic and found that its essence inside the boundary layer is not any more insignificant than on account of a stretching sheet. A numerical comparison with the published works is conducted to verify the accuracy of the present study.

The purpose of this paper is to theoretically study the problem of the unsteady boundary layer flow past a permeable curved stretching/shrinking surface in the presence of a uniform magnetic field. The governing nonlinear partial differential equations are converted into ordinary differential equations by similarity transformation, which are then solved numerically.

The transformed system of ordinary differential equations was solved using a fourth-order Runge-Kutta integration scheme. Results for the reduced skin friction coefficient and velocity profiles are presented through graphs and tables for several sets of values of the governing parameters. The effects of these parameters on the flow characteristics are thoroughly examined.

Results show that for the both cases of stretching and shrinking surfaces, multiple solutions exist for a certain range of the curvature, mass suction, unsteadiness, stretching/shrinking parameters and magnetic field parameter.

The paper describes how multiple (dual) solutions for the flow reversals are obtained. It is shown that the solutions exist up to a critical value of the shrinking parameter, beyond which the boundary layer separates from the surface and the solution based upon the boundary layer approximations is not possible.

The purpose of this article is to analyze the magnetohydrodynamic viscous liquid flow with heat absorption/generation. Flow is induced by an unsteady stretching curved surface. A time-dependent magnetic field is utilized.

The resulting nonlinear system is solved through shooting method.

An increment in the values of curvature A and unsteadiness parameters correspond to higher velocity, temperature and concentration fields. There is a reduction in the temperature and related layer via Prandtl number. Skin friction is increasing factor of magnetic number. Local Nusselt and Sherwood numbers are lower for greater magnetic number.

To the best of author's knowledge, no such consideration has been given in the literature yet.

This paper aims to discuss the salient aspects of the Darcy–Forchheimer flow of viscous liquid in carbon nanotubes (CNTs). CNTs are considered as nanofluid, and water is taken as the continuous phase liquid. The flow features are discussed via curved surface. Water is taken as the base liquid. Flow is generated via nonlinear stretching. Energy expression is modeled subject to heat generation/absorption. Furthermore, convective conditions are considered at the boundary. The Xue model is used in the mathematical modeling which describes the features of nanomaterials. Both types of CNTs are considered, i.e. single-walled CNTs and multi-walled CNTs.

Appropriate transformations are used to convert the flow expressions into dimensionless differential equations. The bvp4c method is used for solution development.

Velocity enhances via higher estimations of nanoparticles volume fraction while decays for higher Forchheimer number, curvature parameter, behavior index and porosity parameter. Furthermore, thermal field is an increasing function of nanoparticle volume fraction, behavior index, Forchheimer number and porosity parameter.

Here, the authors have discussed two-dimensional CNTs-based nanomaterial Darcy–Forchheimer flow of viscous fluid over a curved surface. The authors believe that all the outcomes and numerical techniques are original and have not been published elsewhere.

The purpose of present communication is to analyze Darcy–Forchheimer flow of viscous nanofluid by curved stretchable surface. Flow in porous medium is characterized by Darcy–Forchheimer relation. Brownian diffusion and thermophoresis are considered. Convective heat and mass boundary conditions are also used at the curved stretchable surface.

The resulting nonlinear system is solved through shooting technique.

Skin friction coefficient is enhanced for larger porosity parameter and inertia coefficient while reverse trend is noticed for curvature parameter. Local Nusselt number is enhanced for higher Prandtl number and thermal Biot number, whereas the opposite trend is seen via curvature parameter, porosity parameter, inertia coefficient, thermophoresis parameter and Brownian motion parameter. Local Sherwood number is enhanced for Schmidt number, Brownian motion parameter and concentration Biot number, while reverse trend is noticed for curvature parameter, porosity parameter, inertia coefficient and thermophoresis parameter.

To the best of author’s knowledge, no such consideration has been given in the literature yet.

The study aims to determine an efficiency of external magnetic field on the bacteria surrounded by thousands of magnetic magnetite nanoparticles. The interstitial nanoliquid in which an artificial bacteria swims in biological cell is utilized with variable thermal conductivity. Two dimensions unsteady motion of second grade fluid are considered. The stretching wall is taken as a curved surface pattern.

The mathematical results have been obtained by Chebyshev pseudospectral method.

The impact of the various governing parameters is described by numerical tables and diagrams. It is proven that the pure blood velocity curves are higher when compared with the magnetite/blood. It is demonstrated from clinical disease that dangerous tumors show diminished blood flow. This study concludes that the blood velocity profile increases by increasing the values of fluid parameters. This implies that the medication conveyance therapy lessens the tumor volume and helps in annihilating malignancy cells. The blood temperature distribution raises as the magnetite nanoparticles concentration increases. Consequently, the physical properties of the blood can be enhanced by immersing the magnetite nanoparticles. Further, the present outcomes cleared the thermal conductivity as, a variable function of the temperature, has an important role to enhance the heat transfer rate.

To the best of authors’ knowledge, this study is reported for the first time.

This paper aims to explore on stagnation point flow of Ag-CuO/water over a horizontal stretching/shrinking cylinder by adding the effect of chemical reaction, B together with the magnetic field, M.

A set of reduced ordinary differential equations from the governing equations of partial differential equations is obtained through similarities requirements. The resulting equations are solved using bvp4c in MATLAB2019a. The impact of various physical parameters such as curvature parameter, ϒ, chemical reaction rate, B, magnetic field, M and Schmidt numbers, Sc on shear stress, f′′0 local heat flux, -θ′(0) and mass transfer, -∅′(0) also for velocity, f′(η), temperature, θ(η) and concentration, ∅(η) profiles have been plotted and briefly discussed. In this work, some vital characteristics such as local skin friction, Cf, local Nusselt number, Nux and local Sherwood number, Shx are chosen for physical and numerical analysis.

The findings expose that the duality of solutions appears in a shrinking region ( ε < 0). The value of skin friction, heat transfer rate and mass transfer rate reduction for existing of M, but in contrary result obtain for larger ϒ, B and Sc. Furthermore, the hybrid nanofluid demonstrates better heat transfer compared to nanofluid.

The hybrid nanofluid has widened its applications such as in electronic cooling, manufacturing, automotive, heat exchanger, solar energy, heat pipes and biomedical, as their efficiency in the heat transfer field is better compared to nanofluid.

The findings on stagnation point flow of Ag-CuO/water over a horizontal stretching/shrinking cylinder with the effect of chemical reaction, B and magnetic field, M is new and the originality is preserved for the benefits of future researchers.

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 current determination is committed to characterize the boundary layer flow of Williamson nanofluid prompted by nonlinear strained superficial under heat and mass transport mechanisms. Buongiorno model is presented to view the influence of nanoparticles in fluid flow. Scrutiny has been conceded under the action of the transversely smeared magnetic field. Heat and mass relocation exploration are conducted in the companionship of radiation effects and actinic compensation.

Similarity variable is designated to transmute nonlinear partial differential equations of conservation laws of mass, momentum, energy and species into ordinary dimensional expressions. These constitutive and complicated ordinary differential expressions assessing the flow situation are handled efficaciously by manipulating Runge–Kutta–Fehlberg procedure (RK-5) with shooting routine.

The graphical demonstration is deliberated to scrutinize the variation in velocity, temperature and concentration profiles with respect to flow regulating parameters. Numerical data are displayed through tables in order to surmise variation in skin friction coefficient and Nusselt number. The augmenting values of fluid parameter and magnetic parameter reduces the horizontal fluid velocity, whereas normal velocity upsurges for mounting values of stretching ratio parameter. Moreover, mounting values of radiation parameter and thermophoresis parameter upsurges the temperature profile, whereas, growing values of Prandtl number lessen the temperature field.

The current exploration is used in many industrial and engineering applications in order to discuss the transport phenomenon.

Flow over a nonlinear stretched surface has numerous applications in the industry. The present attempt examines the combined influence of various physical characteristics for the flow of Williamson fluid and no such attempt exist in the available literature.

The purpose of this paper is to review previous research studies on mathematical models for entropy generation in the magnetohydrodynamics (MHD) flow of nanofluids. In addition, the influence of various parameters on the velocity profiles, temperature profiles and entropy generation was studied. Furthermore, the numerical methods used to solve the model equations were summarized. The underlying purpose was to understand the research gap and develop a research agenda.

This paper reviews 141 journal articles published between 2010 and 2022 on topics related to mathematical models used to assess the impacts of various parameters on the entropy generation, heat transfer and velocity of the MHD flow of nanofluids.

This review clarifies the application of entropy generation mathematical models, identifies areas for future research and provides necessary information for future research in the development of efficient thermodynamic systems. It is hoped that this review paper can provide a basis for further research on the irreversibility of nanofluids flowing through different channels in the development of efficient thermodynamic systems.

Entropy generation analysis and minimization constitute effective approaches for improving the performance of thermodynamic systems. A comprehensive review of the effects of various parameters on entropy generation was performed in this study.