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In recent decades, developing energy transfer mechanisms has piqued the curiosity of scientists and researchers. Basic fluids such as motor oils, ethylene glycol, and other conventional fluids play an important role in many industries. On the other hand, these fluids have low thermal conductivity, which causes difficulties for some industries. Therefore, scientists were interested in developing nanofluids to produce fluids with high thermal conductivity. In this research, the flow and heat transfer of a hybrid fluid through a surface that expands and contracts under the magnetic effect and radiation field was studied. The fluid contains hybrid nanoparticles, which are copper and aluminum oxide with ethylene glycol (40%)+water (60%). An appropriate symmetric transformation was used to reduce the partial differential equation (PDEs) governing speed, angular velocity, and temperature into a system of ordinary differential equations (ODEs). A solution to ODEs was obtained using the modified Adomian decomposition method. The results showed that the presence of hybrid nanoparticles with other factors has an important role in more effective heat transfer, and the presence of a stagnation point led to adverse results with some parameters.

In this paper, we thoroughly examine the influences of slip effects and stagnation point flows in the context of an upper-convected non-Newtonian Maxwell nanofluid interacting with a stretching sheet. The existence of a heat generation, transverse magnetic field, and thermal radiation induces a flow resulting from a linearly stretched sheet. The application of the shooting method involves deriving nonlinear ordinary differential equations from the governing partial differential equations, followed by their solution. The effects of dimensionless governing parameters, including velocity ratio, Brownian motion parameter, thermophoresis parameter, velocity slip parameter, Lewis numbers, solutal slip parameter, Maxwell parameter, magnetic number, Eckert number, thermal slip parameter, chemical reactions parameter, and heat source parameter, are examined. The outcomes are illustrated and discussed through graphical representations, showcasing their impact on the velocity field, as well as heat and mass transfer characteristics. Tabular data are generated to display numerical values for physical parameters, including the skin-friction coefficient, local Sherwood number, and the reduced local Nusselt number. The findings suggest that an increase in the velocity slip parameter results in a reduction of both the local Sherwood number and the local Nusselt number. Furthermore, an increase in the strength of the magnetic field leads to a decrease in velocity profiles while simultaneously elevating temperature and concentration profiles.

In this article, we propose a high order method for solving steady and unsteady two-dimensional laminar boundary-layer equations. This method is convergent of sixth-order of accuracy. It is shown that this method is unconditionally stable. The unsteady separated stagnation point flow, the Falkner–Skan equation and Blasius equation are considered as special cases of these equations. Numerical experiments are given to illustrate our method and its convergence.

Carbon nanotubes (CNTs) influenced nanofluid is gaining popularity in the industry for solar energy and scratch heat exchanger applications. Consequently, this research focuses on evaluating the impact of nonlinear thermal radiation from a CNT-based nanofluid on an unsteady three-dimensional nonasymmetric Homann stagnant flux as a function of length and radius. CNTs have remarkable thermal physical properties that appear to be critical for nonlinear thermal transport. As a result, the nonlinear heat transfer properties of H₂O composed of single or multiple wall CNTs are studied. The nanomaterial has a length and radius of approximately $3\mathrm{\text{nm}}\le L\le 70\mathrm{\text{nm}}$ and $10\mathrm{\text{nm}}\le R\le 40\mathrm{\text{nm}}$. Partially differential equations with appropriate similarity transformations serve as a mathematical model for the process. The numerical solution of the simplified system of equations is achieved via the use of the well-known Runge–Kutta (RK) method in conjunction with the shooting approach. An effective way to show how a component affects velocity and temperature, skin frictions in both direction and Nusselt number are utilized in graphical representations. Increasing the unsteadiness parameter causes a reduction in the temperature profile and the velocity profile in both directions. As $\epsilon$ grows larger with $\varphi$, the skin friction in both directions decreases, while the Nusselt number profile grows larger. In addition, the variation in the Nusselt number is included in the tables, along with a comparison of the model without radiation to the model with radiation.

In this analysis, the stability of MHD stagnation point flow and heat transfer of an electrically conducting viscous incompressible fluid on a horizontally stretching/shrinking permeable cylinder with volumetric heat source are investigated. The cylindrical surface is subjected to a cross flow and a heat flux. This analysis finds applications in the areas of polymeric processing unit due to stretching/shrinking of surface and biofluid flows with therapeutic effects through the arteries, vein and digestive system that have a tube-like structure. Other aspects of the analysis are: magnetic field intensity whose strength can be remotely controlled and embodied generating/absorbing thermal power. The solution of the boundary value problems (BVP) is carried out with MATLAB’s inbuilt solver bvp4c. The most significant findings are recorded as: an increase in magnetic field strength increases the skin friction at the solid surface which is consistent with progressive thinning of boundary layer. This striking result is of interest in industrial applications because it is easy to regulate the magnetic field strength by electromagnetic devices such as controlling the voltage in the electric circuit. There is a point of neutrality of thermal energy distribution during the high fluctuation of temperature irrespective of the presence of heat generation or absorption. This can be attributed to the dominating effect of stretching/shrinking of the surface. The first solution provides stability of the flow with an increase in velocity in the presence of heat sink that is commensurate with the strength.

This paper examines nonlinear thermal radiative stagnation point flow of Walter-B nanofluid. The characteristics of nanofluid are explored using Brownian motion and thermophoresis effects. In the presence of uniform magnetic field, fluid is conducting electrically. Furthermore, phenomena of mass and heat transfer are studied by implementing the effects of chemical reaction, Joule heating and activation energy. Outcomes of distinct variables such as induced magnetic parameter, Eckert number, thermal radiation parameter, Weissenberg number, ratio of rate constant, heat capacity ratio, thermal Biot number, solutal Biot number, Prandtl number, heat generation parameter, Schmidt number on concentration, temperature and velocity distributions are explored. The numerical method is implemented to solve the governing flow expression. Further, Sherwood number, Nusselt number and skin friction coefficient are analyzed and discussed in tables. Weissenberg number have opposite behavior on velocity field while it increases for larger values of mixed convection parameter. Temperature of the fluid rises for higher values of thermal Biot number, thermophoresis diffusion coefficient, heat generation parameter and Eckert number Activation energy parameter and Weissenberg number have direct relation with concentration field.

For practical purposes, the study of ternary hybrid nanofluid flows near stretching/ shrinking surfaces, including heat generation/ absorption and velocity slip, has enormous value. It is crucial to understand how fluid mechanics deals with stagnation point flow, which is a common phenomenon in both engineering and scientific domains. In the evaporation process, the polymer enterprises, and the aircraft counter jet, the stagnation point flow may be found. An unsteady stagnation point flow is used to explore a ternary hybrid nanofluid (Cu–TiO₂–Al₂O₃/ polymer) in relation to a convectively heated stretching/ shrinking sheet. This research also considers the velocity slip condition in addition to the traditional surface under no-slip conditions. The differential equations and their partial derivatives are changed to ordinary differential equations by applying approved similarity transformations. The MATHEMATICA operating system employs the Shooting with Runge–Kutta-IV process to explain the reduced mathematical model. When preliminary assumptions are appropriate, the technique may provide solutions. According to the data, nanoparticle volume fraction has an effect on the skin friction coefficient and local Nusselt number. The coefficient of skin friction decreases when velocity slip occurs at the border, while the rate of heat transfer increases. According to the research, increasing the unstable parameter led to large increases in the coefficient of skin friction and heat transfer as opposed to just altering velocity slip. The outcomes show that ternary fluid has a greater skin fraction and heat transmission profile than hybrid and traditional nanofluids for all parameters. The recent evidence and published results for a particular case were contrasted to validate the findings, and good agreement was established.

In this study, the impact of activation energy and transiting parameters on the two-dimensional stagnation point flow of magnetized Sutterby nanofluid of nano-biofilm incorporating with microorganisms over the porous surface has been examined. Prior research suggests that both the fluid viscosity and thermal conductance are temperature-dependent. Also, the effects of solutal concentration on fluid viscosity, heat capacity and nanofluid properties have been elaborated. In recent years, numerous technical strategies comprising hydromagnetic fluxes and thermal intensification in porosity media, like molding, condenser heat exchanger, liquefied metal filtration, fusion control and nuclear reactor coolant, have been addressed. According to numerous empirical studies, the viscosity and thermal conductivity of the nanoparticles are largely dependent on the intensity of nanoparticles rather than the temperature. The classical RK-4 method with shooting technique has been used. The significance of involving parameters in the domains of heat, velocity, density and concentration has been illustrated. The effect of nondimensional parameters on the skin friction factor, Nusselt number and Sherwood number has been discussed. The nanoparticle density increases with the activating energy effects and thermophoresis factor and rapidly decreases for Lewis number and Brownian factor. The velocity, temperature and concentration profiles increase as the concentration-dependent properties do, but all physical quantities deteriorate for all concentration-varying factors.

Practical Applications: Numerous technical applications, including as polymer deposition, electrolysis control, medication delivery, spin-stabilized missile cooling and cooling of rotating machinery slices have sparked considerable interest in studying stagnation point flow. Nuclear power plants, photovoltaic panels and heat exchangers as well as microfluidic heating devices use them.

Purpose: To better understand the unsteady $(\mathrm{\text{Cu}}$–${\mathrm{\text{Fe}}}_3{\mathrm{\text{O}}}_4$–$\mathrm{\text{Si}}{\mathrm{\text{O}}}_2\mathrm{\text{/polymer}})$ ternary hybrid nanofluid stream at the stagnation zone with Joule heating, this research examines the unique prospective applicative properties.

Methodology: The flow equations will be modeled. By using similarity transformation, it is possible to transform nonlinear partial differential equations (PDEs) that are not precisely solvable into ordinary differential equations (ODEs) that can be numerically resolved. Runge–Kutta-IV and the shooting technique in MATHEMATICA have been demonstrated to have a significant effect on the predominance of heat exchange and the mobility features of ternary hybrid nanofluids.

Findings: Results show that the unsteadiness parameter influences the $x$-direction velocity and mono nanofluid has a larger velocity than other nanofluids, while the opposite is true for the $z$-direction velocity. Nanoparticle concentrations, magnetic and Eckert number characteristics increase the thermal distribution, whereas the unsteadiness and rotation parameter decreases it. Unsteadiness, rotation and magnetic factors all improve heat transfer, while the Eckert number parameter has the reverse effect. The ternary hybrid nanofluid also has a greater heat transfer rate than the hybrid and normal nanofluids.

Originality: Unsteady $(\mathrm{\text{Cu}}$–${\mathrm{\text{Fe}}}_3{\mathrm{\text{O}}}_4$–$\mathrm{\text{Si}}{\mathrm{\text{O}}}_2\mathrm{\text{/polymer}})$ ternary nanofluid stream generated by magneto hydrodynamic (MHD) in the stagnation zone was studied in detail in this study. To avoid any errors in heat transfer, it may assist other researchers in selecting critical parameters for modern industrial heat transfer and the right parameters for developing nonunique solutions.

The primary objective of this investigation is to explore the Cattaneo–Christov flux models impact on Williamson nanofluid over a stretching surface. Buongiorno’s model featuring diffusions (Brownian and thermophoretic) is opted for nonlinear analysis. Buoyancy-driven nonlinear convection flow in stagnation region is modeled. Surface is permeable and transpiration effects are considered. Energy expression captures heat source/sink aspects. The nondimensional differential systems are tackled analytically via homotopy analysis method (HAM). The profiles of dimensionless temperature, concentration and skin friction are examined graphically for the attributes of multiple physical parameters. It is revealed that the heat transfer elevates with the increment of thermophoresis, heat source and Brownian motion parameters while it dwindles with the improvement of thermal relaxation parameter. The mass transfer strengthens with the enlargement of thermophoresis parameter while diminishing with the enhancement of solutal relaxation and Brownian motion parameters. The skin friction is elevated for higher values of material variable against nonlinear mixed convection parameter.

This thermal case pronounced the stability framework for stagnation point flow of magnetized alumina and copper nanoparticles with due exponentially shrinking permeable surface. The thermal stability and enhancement of water base liquid had been taken into account with uniform impulsion of hybrid nanomaterials. The induced flow results via exponentially shrinking permeable surface. The similarity transformation simplifies the mathematical model where governing formulated system for hybrid nanofluid is altered into the nondimensional form. A numerical solver called bvp4c is employed in MATLAB software to aid in the problem-solving process, and dual branches have been found. The significance of pertaining parameters associated to the flow model is inspected in view of thermal properties. The findings show that there are two branches for suction strength $(S>S_c)$ and magnetic strength $(M>M_c)$. The bifurcation values $S_c$ and $M_c$ reduce for the occurrence of dual branches as the solid volume percentages of copper increase. Furthermore, for the upper branch solutions, the skin friction and heat transfer rate rise as ${\varnothing }_{\mathrm{\text{Cu}}}$ increases. The temporal stability analysis determines the stability of the dual branches, and it is discovered that only one of them is stable and physically applicable. The presence of suction parameter effectively controls the thermal transportation phenomenon.

Aluminum alloys are used to make wheels that are suitable for aeroplanes and automobiles, as well as all types of ground vehicles and watercraft. Aluminum alloys are made through melting, sintering (assembly of formed parts utilizing metal particles that melt together at intense temperatures), or mechanical braiding. Aluminum alloys have had a major impact on aeroplane manufacturing. Aluminum alloys like AA7075 and AA7072 are especially useful in transportation applications including maritime, aviation, and automotive, and are also utilized in the construction of bicycles, glider rock climbing equipment, and planes. This attempt sheds light on the magnetically influenced methanol-based micropolar nanofluid containing aluminum alloy nanoparticles (AA7075) over a variable thickened stretching sheet. A variable magnetic field is applied normal to the flow direction. The flow is considered at a stagnation point. Also, the Joule heating impact is considered in this analysis. The similarity transformations are used for the transformation of partial differential equations into ordinary differential equation. The authors have chosen to solve the proposed model with the help of NDSolve technique which can handle a wide range of ordinary and partial differential equations (ODEs and PDEs). The results showed that, as the volume fraction of AA7075 nanoparticles grows the velocity profile of the AA7075–methanol nanofluid decreases, while the microrotation and temperature profiles of the AA7075–methanol nanofluid increases. The velocity profile of the AA7075–methanol nanofluid reduces, while the microrotation and temperature profiles of the AA7075–methanol nanofluid increase with the increasing magnetic parameter. The growing micropolar parameter augments the velocity and temperature profiles of the AA7075–methanol nanofluid, whereas a dual impact of the micropolar parameter is found against the microrotation profile of the AA7075–methanol nanofluid. The growing variable wall thickness factor augments the velocity, microrotation and temperature profiles of the AA7075–methanol nanofluid. It is found that the embedded factors highly affected the AA7075–methanol nanofluid as compared to methanol fluid.

The prime objective of binary chemical reaction (BCR) is concentrated on the study and optimization of chemical reaction to accomplish finest reactor design and performance, which elaborated the interfaces of flow phenomena, reaction kinetics and heat and mass transport. The reactor performance is likely to be linked to the reaction operating constraints and feed composition through the aforementioned factors. The applications of BCR are generally in the petroleum and petrochemical regions, but with the help of chemical engineering and reaction chemistry concepts, it could be used in different areas, like waste treatment, chemical pharmaceuticals, nanoparticles in advanced materials, microelectronics, enzyme technology, biochemical engineering, living systems, renewable energy systems, sustainable development, environment/pollution prevention, as well as to optimize a different reaction framework via simulation and modeling methodology. Owing such physical applications in mind, this research deals with the binary chemical reactive flow of non-Newtonian fluid (Walter’s B) subject to activation energy. Stagnation point is accounted. Radiative flux and ohmic heating effects are considered in the development of energy expression. Concentration and microorganism equations are considered. The governing system is altered to ordinary one through the important similarity variables. Results are obtained through bvp4c technique. All results are discussed graphically. Furthermore, surface drag force (skin friction) and heat and mass transfer (Nusselt and Sherwood) rates are calculated and displayed graphically. Significant results are listed in conclusion.

Steady flow of incompressible Maxwell–Sutterby fluid at stretching sheet is discussed in the presence of stagnation point region. The magnetic Reynolds number is considered very high and induced magnetic and electric fields are applied to the fluid flow. Temperature-dependent properties with radiation influence are considered in this analysis. The heat source or sink and melting impact are also debated in this analysis. A differential model of mathematics is developed by employing a governing constitutive equation. The differential equations’ model is condensed and becomes ordinary differential equations by implementing the appropriate transformations. Furthermore, these equations are elucidated by the numerical scheme. The physical influence of physical parameters is exhibited in the graphs and tabular form. The escalating values of M cause a shear thinning attitude in the fluid; as a result, devaluation in the velocity is detected in the case of $a/c=0.5$, while it depicted the counter behavior for $a/c=1.5$. The skin friction is heightened with improving values of $\mathrm{\text{Re}}$ because $Re$ boosted which improved the viscosity of liquid as well as heightened the friction at sheet. The enhancing values of $M_p$ cause a decrement in skin friction. The melting parameter is enriched which reduces the viscosity of fluid due to temperature boosting as well as friction reduction. The diminution in skin friction is found for enhancing the values of $\beta$.

This paper investigates the stagnation point flow of a second-grade nanomaterial, considering nanofluid effects via thermophoresis and Brownian motion. The study addresses convective heat and mass transfer conditions within the flow induced by a stretching cylinder. The transformed systems are solved using the optimal homotopy solutions (OHAM) method, revealing the impact of various parameters on the quantities of interest. The results indicate that an increase in curvature, viscoelasticity, velocity ratio and injection parameters leads to an enhancement in velocity, while the opposite trend is observed due to suction. The viscoelasticity and curvature parameters also increase skin friction. Additionally, thermophoresis enhances the temperature and concentration fields, while Brownian motion reduces mass diffusion.

This study investigates the behavior of a mixed convective boundary layer flowing around a rigid spherical shape with non-isothermal wall temperature. The fluid assumed in this examination exhibits both incompressibility and viscosity. The major purpose is to examine the impact of the non-uniform surface temperature on the flow patterns, including velocity outlines and temperature outlines. An examination is conducted on the modified conservation equations of the boundary layer flow utilizing the local non-similarity method. The MATLAB built-in code bvp4c is used to find computational solutions. The outcomes are then shown for both air and water, specifically at a temperature of 21°C. The influence of several factors, such as the mixed convective variable $\lambda$ and the exponent $\gamma$ in the wall temperature function $G(x$), is shown in velocity and temperature profiles. In addition, the study computes the local frictional force factor and local wall heat transport factor and compares these values with results from earlier research. Significantly, the study expands upon data made around the lower stagnating point to include other sites within the sphere, thereby offering a thorough comprehension of the whole flowing field.

An efficient analytical technique is used to obtain approximate analytical series solutions of Brinkman equation for the two-dimensional and axi-symmetric stagnation point flows in a porous medium. Analytical approximate solutions of the classical two-dimensional and axi-symmetric stagnation point flows are also obtained as limiting cases. The obtained zeroth order series solutions agree well with the computed numerical solutions.