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This paper intends to build a mathematical model of a two-dimensional Williamson nanofluid flow due to the stretching of a sheet linearly. The flow geometry is influenced by the magnetic field that is applied externally to the system. The induced magnetic field is infinitesimal and hence neglected. However, the flow structure is incorporated with the Brownian motion and thermophoresis effects as it alter the physics of the flow. The equations that model the flow comprise the highly nonlinear coupled simultaneous system. Thus, a numerical technique namely finite difference accompanied with the Thomas algorithm is adopted to approach the flow system. The motion, temperature, and concentration of the Williamson nanofluid flow are studied for the different flow controlling parameters. The co-efficient of skin-friction, heat, and mass transfer rates is also computed. The streamlines, isotherms, and iso-concentration are plotted to picturise the flow phenomena in the complete domain. The study reveals that the temperature of the flow raises with the Brownian motion of the nanoparticles and the trend is opposite with the concentration. The strength of the streamlines, isotherms, and the concentration contour is identified to be high for the least magnitude of Weissenberg number. The Brownian motion raises the heat transfer rate and slows down the mass transfer rate.

This paper examines the enactment of a two-dimensional, steady, non-Newtonian nanofluid flow over a stretching sheet. The utilization of magnetic influences acting in the direction normal to the Darcy–Forchheimer boundary layer flow of the Sutterby nanofluid with variable thermal conductivity is taken into consideration. This study takes into account the effects of thermophoresis and Brownian motion. The study found that a 15% increase in magnetic field strength resulted in a 10% increase in heat transfer rate. Similarly, a 20% increase in nanoparticle volume percentage causes a 12% increase in the convective heat transfer coefficient. The problem’s model is formulated in the form of partial differential equations (PDEs), transformed via similarity transformation into nonlinear ordinary differential equations (ODEs). Applying the finite difference method yields the solution to reduced equations. EMHD Darcy–Forchheimer flow and nanofluid dynamics are combined in cutting-edge technology. This is important for many industrial and technical applications. Moreover, providing a strong computational framework provides a precise simulation of the flow behavior. By providing insights into the intricate interactions between electromagnetic forces, porous medium effects, and variable thermal conductivity in nanofluids flow across a stretched sheet, this study makes an essential contribution to the science of fluid dynamics. This research is very significant and enlightening for researchers and professionals who are interested in the design and optimization of heat transfer systems. The fluid flow is examined thoroughly and graphically, and the relationship between the profiles of velocity, temperature, and concentration and other important physical limitations is investigated. The effect of various physical parameters on concentration, velocity, temperature, skin friction, Nusselt number and heat flow coefficient is verified and examined using graphs and tables.

The main motivation of this study is to examine the effects and behavior of Casson nanofluid mainly in reference to oblique stagnation points across a stretching surface. Oblique stagnation point (OSP) motions have so many applications, like artificial fibers, sticky materials, drying paper, and freezing electrical equipment, and numerous applications for endothermic and exothermic processes exist, including in heat exchangers, cooking, and drying damp clothes. Because of these applications on various domains, the Casson nanofluid OSP motion via a stretching sheet is studied with endothermic/exothermic chemical processes and convective boundary conditions (CBC). Similarity transformations are employed to convert partial differential equations (PDEs) into a collection of ordinary differential equations (ODEs). Furthermore, some significant engineering coefficients are discussed and also evaluated the behaviors of several nondimensional factors using the Runge–Kutta–Fehlberg-45 numeric method with a shooting scheme and graphical representations. The outcomes signify that temperature and concentration both rise with a rise in the Casson parameter and activation energy (AE) respectively. A higher Biot value leads to a higher temperature profile. A temperature profile increases with an enhance in the Casson parameter. The addition of a solid fraction will enrich the mass transmission rate in combination with AE.

Purpose: This research aims to investigate the flow and heat transfer characteristics of a hybrid nanofluid comprising water, single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs) over a stretching sheet under the influence of a magnetic field. Design/Methodology/Approach: The study employs a mathematical model that accounts for factors such as variable viscosity, thermal radiation, a porous medium and heat generation/absorption. The governing partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) using similarity transformations and then solved numerically using the bvp4c solver in MATLAB. Findings: The numerical results reveal that the velocity profile of the hybrid nanofluid is significantly enhanced by the presence of MWCNTs. Additionally, the temperature profile is influenced by parameters like the magnetic field, heat source/sink and Prandtl number. The Yamada–Ota (Y–O) model is found to have a more pronounced effect on heat transfer compared to the Xue model. Originality/Value: This study provides valuable insights into the behavior of hybrid nanofluids in complex flow scenarios. The findings can be applied to the design and optimization of various thermal systems, such as heat exchangers and cooling devices.

Magnetohydrodynamic flow of nanofluids and heat transfer between two horizontal plates in a rotating system have been examined numerically. In order to do this, the group method of data handling (GMDH)-type neural networks is used to calculate Nusselt number formulation. Results indicate that GMDH-type NN in comparison with fourth-order Runge–Kutta integration scheme provides an effective means of efficiently recognizing the patterns in data and accurately predicting a performance. Single-phase model is used in this study. Similar solution is used in order to obtain ordinary differential equation. The effects of nanoparticle volume fraction, magnetic parameter, wall injection/suction parameter and Reynolds number on Nusselt number are studied by sensitivity analyses. The results show that Nusselt number is an increasing function of Reynolds number and volume fraction of nanoparticles but it is a decreasing function of magnetic parameter. Also, it can be found that wall injection/suction parameter has no significant effect on rate of heat transfer.

MHD viscoelastic (Walters’-B) fluid flow close to the stagnation point region along an extending plate with the changeable fluid properties’ influences has been debated. Heat transfer’s features are scrutinized via Cattaneo–Christov (CC) theory. The mathematical model for the physical problem is tackled numerically via Chebyshev pseudospectral (CPS) technique. The existing outcomes are supported by recent research and have acquired a suitable agreement. The numerical outcomes reveal that temperature fields are more pronounced for Fourier’s law case. Further, the opposite behavior is noticed with the heat transfer rate. Higher values of the conjugate parameter result in an increment of the heat transfer rate and temperature field. Fluid flow’s features, as well as physical quantities, are substantially varied via variable fluid properties.

In this paper, a steady solution is presented for the equations that represent the MHD rarefied gas fluid flow and heat transfer due to a permeable stretching sheet with second-order velocity slip and thermal slip phenomenon. By using nondimensional transformations, the system of partial differential equations governing the problem is transformed into another system of nonlinear ordinary differential another equations. Novel solutions are investigated for the resulting ordinary differential equation which describe the momentum equation. The numerical results obtained agreed very well with previously reported cases available in the literature. Additionally, the effects of the magnetic parameter, first- and second-order velocity slip parameter, conductivity parameter, thermal slip parameter and the suction (injection) parameter on both the velocity and temperature profiles and on the local skin-friction coefficient are discussed and presented through tables and graphs.

Interaction between nonuniform heat source/sink, magnetic field and thermal radiation with heat flux through the flow of non-Newtonian power-law fluid due to a linearly stretching sheet was studied numerically using an implicit finite difference method (FDM). The heat flux is assumed to depend on both the thermal conductivity and the thermal radiation. Besides, the effects of all governing parameters, such as the magnetic parameter, thermal conductivity parameter, the power-law index parameter, Prandtl number, the space-dependent heat source/sink parameter, the temperature-dependent heat source/sink parameter, and the radiation parameter, on the profiles of velocities and temperature are studied and discussed. In particular, thermal radiation was found to play a key role in the heat transfer characteristics and in the formation of thermal boundary layer. Generally, our numerical results reveal that both the velocity and temperature distributions are marginally influenced by both the magnetic parameter and power-law index. A good agreement is observed between our results via finite difference method and the previously published numerical results for some special case.

This paper investigates the flow and heat transfer of special third-grade fluid with a viscous dissipation effect over a stretching sheet. This model, adequate for many non-Newtonian fluids, is used to characterize the behavior of the fluids domain. The governing momentum and energy equation are reduced to ordinary nonlinear differential (self-similar) equations via the Lie group transformation method. The Homotopy Perturbation Method (HPM) is applied to solve these obtaining results. For validation, current results have been compared with the fourth-order Runga method (RK4) and shooting technique. The effects of physical parameters on fluid velocity and temperature profile were investigated with the aid of figures and tables by simply altering a single parameter while keeping the others constant. It is observed that both the non-Newtonian parameter and the Prandtl number have the effect of decreasing the temperature of the stretching surface, while the opposite behavior was found for the Eckert number.

General slip boundary condition is used to solve the viscous incompressible flows induced by a stretching sheet. These flow problems corresponds to the planar and axisymmetric stretching. A similarity solution is developed by shooting method using Runge–Kutta algorithm. The results are graphically displayed and discussed under the influence of slip parameter and critical shear rate. The comparison of stretching flow problem subject to Navier's boundary condition in the planar case is made with the available numerical results in the literature.

This paper depicts the fully developed natural convective flow on a conducting viscous fluid towards a nonlinearly stretching sheet. Furthermore, the porous dissipation, thermal radiation and heating parameter effects are implemented on both the vertical walls of the stretchy channel. To model the stretchy flow equations, the Cartesian coordinates’ system is utilized. Through the utilization of similarity variables, the nonlinear partial differential equations that describe the flow (mass, momentum and energy conservation) are converted into nonlinear ordinary differential equations. With the help of the MAPLE, a well-known fourth-order Runge–Kutta procedure is used to do a numerical evaluation of the stated nonlinear and non-dimensional set of equations. For each of the several nonlinear radiative parameters regulating the flow regime, the velocity and temperature distribution functions are determined, viz the nonlinear heating parameter ${\theta }_R$, Eckert number $Ec$, Prandtl number $Pr$, porosity variable $Pm$ and thermal radiation parameter $N_R$. Graphic representations are provided for every outcome. Furthermore, skin friction and Nusselt number are also computed to give an approximation of the surface shear stress and cooling rate, respectively. A remarkable compaction is obtained between computed numerical data and published results. It has been demonstrated that an increase in the value of the nonlinear parameter $Pm$ outcomes creates a reduction in the dimensionless translational velocity $g^{\prime }$ of both viscous and Newtonian fluids. Dimensionless temperature mostly upsurges with growth in nonlinear parameters $Ec$, $Pm$, ${\theta }_R$ and decreases with an intensification in convective parameters, $Pr$, $N_R$. There is a detailed discussion on the implications of all embedded stretching sheet variables on the flow. The flow regime is extremely useful in the technology of polymer processing as well as in the field of materials science.

In this paper, the impact of thermal slip and slip velocity on ZrO₂ (Zirconia)–water nanofluid flow through a vertically heated permeable stretching sheet surrounded by a porous medium has been deliberated. The concept of thermal radiation is assimilated into the study. Using similarity transformations, the concerned governing flow model is changed into nonlinear ODEs and gets a solution by the R-K method. The consequence of numerous pertinent parameters in the fluid flow arena is deliberated by employing graphical and tabular approaches. The current investigation will be supportive of understanding the thermal features of heat transfer nanofluids. Outcomes of the present effort are compared with that of previously published works and found a commendable level of settlement. Our analysis sightsees that Zirconia–water achieves high temperatures due to thermal radiation. Also, it is found that Zirconia–water nanofluid lessens the heat transfer rate related to their base fluid.

This paper deals with the analysis of the flow and heat transfer performance of ternary nanofluid flowing past a stretching sheet under the influence of quadratic convection. The ternary nanofluid is formed by suspending three different nanoparticles, namely, titanium dioxide (TiO₂), silver (Ag) and graphene (Gr) in the base fluid water (H₂O). Thus, the ternary nanofluid obtained is Gr–Ag–TiO₂/H₂O where the hybrid nanofluid Ag–TiO₂/H₂O forms the base fluid for the resulting ternary nanofluid. The addition of TiO₂ nanoparticles enhances the photocatalytic nature of the base fluid and makes it useful in various applications concerning the medicinal field. The presence of Gr helps in intensifying the thermal conductivity of water while the suspension of silver nanoparticles ensures chemical stability. Meanwhile, the thermophysical properties of the ternary nanofluid are mathematically defined and the system of equations that describe the flow of a ternary nanofluid past a stretching sheet is framed using differential equations. The outcomes of this study are interpreted through graphs for velocity and temperature profiles of the ternary nanofluid. It was mainly observed that the thermal conductance of ternary nanofluid was higher than the monophase and hybrid nanofluid. Also, the presence of quadratic convection had a prominent impact on the ternary nanofluid flow. The Nusselt number was found to be greater for spherical nanoparticles and it was found to be least for blade-shaped nanoparticles.

Studying real-world problems with flow models of Newtonian and non-Newtonian fluids has gained particular attention because of its significance in engineering and other industries. According to trends in the field of research, interest in studying the characteristics of all such fluid flows is expanding. Due to the peculiar nature of the physical foundation of these non-Newtonian flows, no single constituent equation is available in the literature to explain all of their characteristics or rheological behavior. In the current investigation, the continuous 2D Casson fluid heat transfer flow is combined with the effects of radiation and an inclined magnetic field over a linear stretch surface. Newtonian condition is used to heat the sheet. The governing partial differential equations (PDEs) are transformed into nonlinear ordinary differential equations (ODEs) via the similarity transformation. The fourth-fifth-order Runge–Kutta Fehlberg (RKF45) method is then used to numerically solve the problem. The results for temperature distribution, and velocity field are computed and plotted graphically and discussed in detail. It is found that the magnetic parameter reduces fluid velocity and the Casson fluid parameter increases temperature distribution.

This study looks at the numerical results of magnetohydrodynamic liquid flowing across a stretching sheet at its stagnation point while also experiencing chemical reaction, viscous dissipation, thermal radiation and variable magnetic field. The fundamental partial differential equations that regulate physical phenomena are converted into nonlinear ordinary differential equations by using the appropriate similarity transformations. Later, resolved by numerically using Mathematica. The effects of various non-dimensional parameters like velocity ratio parameter ($\lambda$), porosity (k), magnetic parameter (M), radiation parameter (R), chemical reaction parameter ($\gamma$), etc. on the velocity, temperature and concentration profiles as well as on the local skin-friction and the local Nusselt number are discussed in detail and displayed through graphs. The numerical evaluation of the physical quantities was provided in tabular form for the various values of the relevant stream parameters. It is important to note that magnetic field interaction increases concentration distribution and fluid temperature while decreasing velocities at all domain flow locations.

In the present scenario, the production process of various electronic gadgets including industrial products needs a higher cooling system to get its better features. Therefore, this study reveals the radiating ternary nanofluid flow through a wedged stretching surface. The forced convection of an electrically conducting fluid in association with the particle concentration enriches the study. The novelty of the proposed phenomena is due to the consideration of various shapes such as spherical, cylindrical, and platelet shaped solid particles combined with various physical models of electrical and thermal conductivity, viscosity, specific heat, and heat capacitance. The performance of the ternary nanofluid is presented using a carbon nanotube, graphene, and aluminum oxide in the base liquid water. To achieve a system of non-dimensional form of the governing equations, suitable transformations are adopted and further, shooting-based Runge-Kutta fourth-order technique is employed for the solution of this set of equations. Further, the important outcomes are deployed as follows: the combined effect of the nanoparticles and the base liquid augments the fluid velocity as well as the temperature distributions and thermal radiation also overshoots the temperature profile.

This work is about the investigation of the flow of a micropolar nanoliquid over a stretching surface, taking into account the effects of thermal radiation, thermophoresis, and Brownian motion. The study focuses on the impact of these factors on heat and mass transfer rates, with the assumption that the Newtonian heat impact dominates. The homotropy approach is used to generate non-dimensional transformational parameters, which are then used to create a system of nonlinear differential equations. The study includes charts and tables that define transfer rates based on various parameters, and the results suggest that increased radiance levels and Nb parameters lead to improvements in heat and mass convection. The graphics used in the study are accurate and consistent with previous research in the field. Overall, this research provides insights into the complex dynamics of micropolar nanoliquid flow and the factors that impact heat and mass transfer in this system.

This paper considers two-dimensional electrically conducting and incompressible ternary hybrid nanofluid flow on a stretching sheet with the convective boundary condition and heat source effect. Relevant similarity formulas are effectuated in converting the governing equations into a system of ordinary differential equations (ODEs) and are further treated numerically using the spectral quasilinearization method (SQLM), with error analysis. The prominent dimensionless parameters controlling the flow, and heat transfer characteristics are discussed. The results of this study show that Eckert number, heat source parameter, and magnetic effect boost the temperature profile. This work expected significant information for the future applications of innovative heat transfer devices, as well as a valuable reference for researchers to study flow behavior under various assumptions.

This novel study unfolds the heat and mass transfer investigation of Williamson nanofluid (WNF) through a porous medium past a stretching plate, along with considering heat generation/absorption. Nanoparticles hold significant significance in thermal engineering, industrial operations, and biomedical advancements, contributing to enhanced heat transfer, cooling mechanisms, thermal extrusion processes, and applications in cancer treatment, particularly in addressing brain tumors. The coupled ordinary differential equations (ODEs) are gained from governing partial differential equations (PDEs) by applying sufficient transformations. Then the ODEs of a nonlinear nature, along with boundary conditions, are solved through bvp4c, a built-in MATLAB program. A good agreement has been found in comparing the present study with already published papers. Numerical values for skin friction, mass transfer, and heat transfer are shown through a table against involved physical parameters, especially for both injection $(S<0)$ and suction $(S>0)$ cases, by varying the values of unsteadiness parameter A and Weissenberg number $W_e$. The effects of the physical parameters on velocity, temperature, and mass profile are graphically depicted and illustrated minutely. It is noted that both the magnetic and Williamson parameters cause the thickness of the boundary layer to be reduced. It can be deduced from these findings that the rate of heat transfer over the surface of the plate decreases as the unsteadiness parameter increases. Furthermore, it is observed that an increase in the parameters of thermophoresis and Brownian motion leads to a higher temperature of the nanofluid.

The study investigates the behavior of a Maxwellian nanofluid flowing steadily over an extending sheet in a permeable medium with a magnetic field. The energy equation includes the radiation, and the conservation of momentum equation considers the magnetic field and porous medium. The novelty of this study is in its examination of complex phenomena involving magnetic fields, radiation, Maxwell fluids, and nanoparticle suspensions in a porous medium via an exponential stretching sheet. The study of Maxwell fluids has been enriched by the substantial contributions made by researchers, delving into their rheological behavior, modelling, and applications. Their work has provided valuable insights into the complexities of these materials and has advanced the knowledge in this specialized area of fluid mechanics. The governing partial differential equations (PDEs) are converted into ordinary differential equations (ODEs) utilizing an appropriate similarity transformation. These resultant ODEs are then resolved by employing a numerical approach, specifically the finite element method. The numerical simulations provide valuable information regarding the distributions of velocity, temperature, and nanoparticle concentration. Furthermore, the presented tabulated outcomes display alterations in skin friction, mass transfer rate, heat transmission coefficients, and their reliance on different emerging parameters. The velocity diminishes as the magnetic field, suction, Maxwell fluid, and medium factors increase, while it enhances with the stretching sheet limitation. The temperature diminishes with the Prandtl number but upsurges with higher radiation, thermophoresis, and Brownian motion. As the thermophoresis limit increases, concentration rises, but it decreases with an upsurge in the Lewis and Brownian motion. The heat transfer rate rises with radiation, thermophoresis, and Brownian motion, but the mass transfer rate decreases with Lewes and Brownian motion. The outcomes of the study could have implications for various engineering and industrial applications that control and manipulate fluid flows and heat transfer.