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The hybrid nanofluid is extensively used in manufacturing for industrial uses because of its exceptional property of enhancing the heat transfer process. The purpose of the present work is to find novel explanations for the behavior of thermal radiation and inclined magnetohydrodynamics effects on the convective viscoelastic flow of water Al2O3–Cu hybrid nanofluids over an accelerating permeable surface with mass transpiration. The viscoelastic liquid concept is postulated with the benefit of hybrid nanofluids employing conventional flow patterns that are impacted by the magnetic field. Thermophysical properties of Al2O3–Cu and water are employed. Nonlinear PDE for momentum, temperature, and concentration are converted into non-dimensional ODE by employing the proper similarity transformations. The current study is reported to be in very good accordance with earlier research. The velocity field and energy distributions were depicted graphically to show the influence and typical behaviors of physical factors such as the viscoelastic parameter, the Richardson number, the radiation number, etc. In industrial applications, the temperature distribution influenced by radiation is quite important, specifically in accelerated plates where cooling the liquid is necessary to achieve the desired outcome. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive exploration of the transient evolution of double-diffusive salt fingers within a two-layer thermohaline system, employing an advanced high-resolution numerical model. The investigation spans a wide range of Rayleigh numbers (7 × 109–7 × 103) and characterized by a density stability ratio near one, indicative of neutral buoyancy conditions. The study reveals complex finger structures in both concentration and buoyancy fields. As the Rayleigh number varies, the system undergoes a transition from being convection-dominated at high values to a diffusion-dominated state at lower Rayleigh numbers. The paper investigates the time series evolution of convective flux ratios and buoyancy, shedding light on the dynamic evolution of these phenomena. The paper further contributes an energy budget analysis specifically tailored for salt-fingering phenomena in the two-layer system. This analysis further examines the temporal evolution of potential energy, background potential energy attributed to temperature and salinity, and overall kinetic energy. A particular focus is placed on investigating the temporal evolution of excess density across the interface, offering nuanced insights into the system's behaviour. Additionally, the joint probability density function (PDF) of vertical velocity and salinity anomaly is analysed across varying Rayleigh numbers. A generalised evolution pattern of PDF is also presented to understand system's dynamical evolution behaviour. [ABSTRACT FROM AUTHOR]

Ternary nanofluids have been an interesting field for academics and researchers in the modern technological era because of their advanced thermophysical properties and the desire to increase heat transfer rates. Furthermore, the innovative, sophisticated artificial neural network strategy with the Levenberg–Marquardt backpropagation technique (LMBPT) is proposed for research on heat and mass transport over non-Newtonian ternary Casson fluid on a radially extending surface with magnetic field and convective boundary conditions. The main objective of the current research is to conduct a comparative study of numerical solutions of the ternary nanofluid model of heat/mass transport utilizing the artificial neural network (ANN) together with the (LMBPT). To accurately represent complex patterns, neural networks modify their parameters flexibly, resulting in more accurate predictions and greater generalization with numerical outcomes. The model equations were reduced from partial to ODEs through applying appropriate similarity variables. The shooting technique and the byp-4c algorithm were then used to analyze the numerical data. The current study reveals that a rise in the Casson parameter diminishes the fluid velocity but an opposite nature is seen in thermal distribution for rising behavior of heat source/sink and Biot number, and the concentration profile tends to deteriorate when the mass transfer is elevated. Furthermore, the resulting values of the significant engineering coefficients are numerically analyzed and tabulated. [ABSTRACT FROM AUTHOR]

This article studies the flow of Ostwald–de Waele ternary nanofluid over a permeable shrinking sheet. The governing partial differential equations are converted into ordinary differential equations using similarity variables and solved analytically. The article includes closed-form algebraic solutions and graphical flow dynamics analysis, dependent on Darcy number, volume fraction, and mass transpiration. The study shows that the presence of porous media significantly upsurges the mass transpiration and magnitude of skin friction. The calculation uses a combination of Titanium dioxide (TiO2), Cobalt ferrite (CoFe ), and Magnesium oxide (MgO) nanoparticles in pure water, and the ternary nanofluid performs better than the conventional fluid in terms of thermal conductivity. This is important in the fields of manufacturing, machine operations, and engineering, where improving heat transfer is crucial. [ABSTRACT FROM AUTHOR]

In this study, carbon nanotubes, which serve as nanoparticles, are added to the basic fluid. By using a similarity transformation, the governing equations are converted into a set of ordinary differential equations (ODEs), which are then solved analytically. In order to simulate the flow and heat transfer behavior of carbon nanotubes, the Prandtl numbers for water and kerosene are 6.72 and 21, respectively. The precision of the analytical solution found in this study for the nonlinear flow of fluid containing carbon nanotubes is what makes it so beautiful. With the available experimental data, the proposed model is reliable. The main physical parameters for the Jeffrey fluid flow on the stretching/shrinking surface using carbon nanotubes are shown in tables and graphs and described in detail for the thermal and boundary layers. Carbon nanotubes enhance the heat more than the nanofluid; for this purpose, the work on carbon nanotubes flow through stretching/shrinking surfaces has many applications in biomedical, solar energy, generator cooling, nuclear system cooling, etc. Therefore, it is quite significant to assimilate the analytical extension of heat transfer fluid in the presence of magnetohydrodynamics under the influence of slip velocity. Further, carbon nanotubes can effectively elucidate the base materials' thermal performance and mechanical properties. Here, we assimilated the extension of heat transfer fluid in the incidence of magnetohydrodynamics under the influence of slip velocity analytically. Further, carbon nanotubes can successfully elucidate the base materials' thermal performance and mechanical properties. [ABSTRACT FROM AUTHOR]

The Darcy–Forchheimer model is a commonly used and accurate method for simulating flow in porous media, proving beneficial for fluid separation, heat exchange, subsurface fluid transfer, filtration, and purification. The current study aims to describe heat and mass transfer in ternary nanofluid flow on a radially stretched sheet with activation energy. The velocity equation includes Darcy–Fochheimer porous media effects. The novelty of this study is enhanced by incorporating gyrotactic microorganisms which are versatile and in nanofluid can greatly improve the thermal conductivity and heat transfer properties of the base fluid, resulting in more efficient heat transfer systems. Furthermore, the governing PDEs are reduced to ODEs via appropriate similarity transformations. The influence of numerous parameters is expanded and physically depicted through the graphical illustration. As the Forchheimer number escalates, so do the medium's porosity and drag coefficient, resulting in more resistive forces and, as a result, lowering fluid velocity. It has been discovered that increasing the exponential heat source/sink causes convective flows that are deficient to transport heat away efficiently, resulting in a slower heat transfer rate. The concentration profile accumulates when the activation energy is large, resulting in a drop in the mass transfer rate. It is observed that the density of motile microorganisms increases with a rise in the Peclet number. Further, the results of the major engineering coefficients Skin-friction, Nusselt number, Sherwood number, and Microorganism density number are numerically examined and tabulated. Also, the numerical outcomes were found to be identical to the previous study. [ABSTRACT FROM AUTHOR]

This document is a correction notice for an article titled "Impact of MHD and Mass Transpiration on Rivlin–Ericksen Liquid Flow over a Stretching Sheet in a Porous Media with Thermal Communication" published in the journal Transport in Porous Media. The correction addresses errors in equations and parameters in the original article. The correction provides the correct versions of the equations and parameters, ensuring that the units on both sides of the equations match. The correction also mentions the inclusion of parameters Vw and L in the updated version of the article. The publisher, Springer Nature, remains neutral and unbiased in relation to jurisdictional claims and institutional affiliations. [Extracted from the article]

The exclusive behaviour of nanofluid has been actively emphasized due to the determination of improved thermal efficiency. Hence, the aim of this study is to highlight the laminar boundary layer axisymmetric stagnation point flow of Casson nanofluid past a stretching plate/cylinder under the influence of thermal radiation and suction/injection. Nanofluid comprises water and Fe3O4 as nanoparticles. In this article, a novel casson nanofluid model has been developed and studied on stretchable flat plate or circular cylinder. Adequate rational assumptions (velocity components) are employed for the transformation of the governing partial-differential equations into a group of non-dimensional ordinary-differential formulas, which are then solved analytically. The momentum and energy equations are solved through the complementary error function method and scaling quantities. Using various figures, the effects of essential factors on the nanofluid flow, heat transportation, and Nusselt number, are determined and explored. From obtained results, it is observed that the velocity field diminishes owing to magnification in stretching parameter B and Casson fluid parameter Λ . The temperature field increases by amplifying radiation N r , and solid volume fraction parameter ϕ . The research is applicable to developing procedures for electric-conductive nanomaterials, which have potential applications in aircraft, smart coating transport phenomena, industry, engineering, and other sectors. [ABSTRACT FROM AUTHOR]

The current studies analytically summarize the impact of magnetohydrodynamic and thermal radiation on the non-Newtonian continuous uniform motion of viscid non-compressible nanofluid across a penetrable stretching/shrinking sheet, even though accomplish Navier's first and second order slips along mass transpiration. Blood-bearing silver and copper nanomaterials have distinct flow and heat transfer properties when exposed to heat. Silver (Ag) as well as copper (Cu) nanoparticles are assumed to be present in blood as the non-Newtonian liquid; this fluid serves as the base. We anticipate that the current study will be useful in fields including food, petrochemical products, and medicines, as well as blood circulation, and highly beneficial for patients who are dealing with blood clotting in the uterus, which may result in infertility or cancer, to evaluate the blood flow in the tube. Employing the similarity conversion technique, the ruling partial differential equations are modified into a couple of non-linear ordinary differential equations. Then the transformed ordinary differential equations are analytically solved with the Laplace transformation and expressed in terms of an incomplete gamma function. The current analytical results are compared to previous studies. It is addressed how several physical features such as magnetic field M, Navier's first and second order slip, permeability, Prandtl number Pr, and radiation parameter affect non-dimensional velocity as well as temperature patterns through graphs. The results obtained reveal that there is an enhancement in the rate of heat transfer with the rise in nanoparticle volume fraction and radiation. The temperature distribution is also influenced by the presence of Prandtl numbers, radiation, solid volume fraction, permeability, and slip conditions. This shows that the solid volume fraction of nanoparticles can be used to control the behaviour of heat transfer and nanofluid flows. [ABSTRACT FROM AUTHOR]

This article focuses on MHD flow and heat transfer of Jeffrey fluid due to a stretching/shrinking surface with carbon nanotubes, considering the effects of thermal radiation, heat source/sink parameters, and Navier's slip. Generally, solids offer higher thermal conductivity than fluids. To offer higher thermal conductivity, a new type of nanofluid is formed by suspending two types of carbon nanotubes (CNTs), i.e. single-wall carbon nanotubes (SWCNTs) and multi-wall carbon nanotubes (MWCNTs), which act as nanoparticles, into the base fluid, water. It is intended to enhance the thermal conductivity and mechanical properties of the base fluid. The structure of the problem is an equation of momentum and temperature, which are then converted into a set of ODEs to imitate the MHD flow of carbon nanotubes. The magnetic parameter, radiation parameter, and Navier slip effect significantly affect the structure of the solution to the problem. Carbon nanotubes act as nanoparticles that enhance the heat performance and mechanical properties more than the base fluid, so they have many applications in electronics and transportation. The velocity and temperature profiles, skin friction coefficient, and Nusselt number are observed and discussed through graphs. The results reveal that for stretching case, velocity profile increases with increasing the magnetic field, while the opposite trend observed in shrinking case. We notice that the SWCNT Nanofluids are better nanofluids than the MWCNT Nanofluids. We study from these final results that the usage of CNTs in most cancerous therapies can be more useful than all sorts of nanoparticles. [ABSTRACT FROM AUTHOR]

The heat transfer of Casson viscous gas flows on a linear stretching/shrinking sheet is systematically investigated. In this study, the effects of inclined magnetic force, induced slip, mass transpiration, and thermal radiation are considered. The elementary equations are solved using suitable similarity variables, which map to nonlinear ordinary differential equations. These ordinary differential equations are solved using an analytical method. An exact solution is obtained for the velocity equation. The physical parameters of interest like induced slip parameter, suction/injection, Chandrasekhar number, radiation number, and shear stress are analyzed and presented graphically. In summary, our findings demonstrate that mass transfer-induced slip has a non-negligible impact on flows driven by a moving sheet. Mass transfer-induced slip may even be able to dominate the flow-driven effect of the moving sheet by changing the flow directions to flow against the sheet motion and also thermal radiation parameter increases as thermal boundary layer increases. [ABSTRACT FROM AUTHOR]

The current work shows an incline MHD and Casson fluid flow with a mixed convective boundary layer with hybrid nanofluid Cu–Al2O3/water flow over a stretching/shrinking sheet. The present study is analyzed using an Al2O3–Cu/H2O hybrid nanofluid with a fixed Prandtl number of 6.8. The governing equation of highly nonlinear partial differential equations is converted into ordinary differential equations using exact similarity transformations. Moreover, the radiation effects are also permitted with help of Rosseland's approximation. The subsequent system of equations is then investigated analytically with appropriate boundary conditions. The outcomes of this topic can be addressed using a graphical representation with many parameters like radiation, heat source/sink, stretching/shrinking mass transpiration so on. The research shows that the solution depicts a unique explanation for stretching/shrinking sheets and that the explanation demonstrates the dual flora focused on some stretching/shrinking sheet parameters. The nanoparticles are disseminated in water, which serves as the base fluid. Graphs are also used to study the effects of the magnetic parameter, mass transpiration, and heat source/sink parameter on the velocity profile. It has a wide range of uses in the polymer sector, power generators, flow meters, and pumps, among others. The results indicate that the solution illustrates an inimitable solution for the stretching sheet and that the explanation manifests the dual flora aimed at some parameters for the stretching/shrinking sheet. The hybrid nanofluid has significant features improving the heat transfer process and is extensively developed for manufacturing industrial uses. It was found that the basic similarity equations admit two phases for both stretching/shrinking surfaces. [ABSTRACT FROM AUTHOR]

Present study explains about unsteady Casson nanoliquid film flow over a surface moving with velocity U w = λ x / t . The governing momentum equation is reduced to ODE by using corresponding similarity transformation, which is then tackled by employing numerical technique. The problem is analysed for both two-dimensional film flow and axisymmetric film flow. The exact solution is derived which satisfies the governing equation. It is noted that solution exists only for a specified scale of the moving surface parameter λ . ie., λ ≥ - 1 / 2 for two-dimensional flow and λ ≤ - 1 / 4 for axisymmetric flow. The velocity increases first and reaches the maximum velocity and then decreases to the boundary condition. Streamlines are also analysed for both axisymmetric and two-dimensional flow patterns by considering the stretching ( λ > 0 ) and shrinking wall conditions ( λ < 0 ). Study has been made for large values of wall moving parameter λ . The aim of this investigation is to analyse the Casson nanoliquid film flow which finds applications in industries like coating of sheet or wire, laboratories, painting, many more. [ABSTRACT FROM AUTHOR]

The present investigation is carried out to predict the flow characteristics of a micropolar liquid that is infused with ternary nanoparticles across a stretching/shrinking surface under the impact of chemical reactions and radiation. Here, three dissimilarly shaped nanoparticles (copper oxide, graphene and copper nanotubes) are suspended in H2O to analyse the characteristics of flow, heat and mass transfer. The flow is analysed using the inverse Darcy model, while the thermal analysis is based on the thermal radiation. Furthermore, the mass transfer is examined in light of the impact of first order chemically reactive species. The considered flow problem is modelled resulting with the governing equations. These governing equations are highly non linear partial differential equations. Adopting suitable similarity transformations partial differential equations are reduced to ordinary differential equations. The thermal and mass transfer analysis comprises two cases: PST/PSC and PHF/PMF. The analytical solution for energy and mass characteristics is extracted in terms of an incomplete gamma function. The characteristics of a micropolar liquid are analysed for various parameters and presented through graphs. The impact of skin friction is also considered in this analysis. The stretching and rate of mass transfer have a large influence on the microstructure of a product manufactured in the industries. The analytical results produced in the current study seem to be helpful in the polymer industry for manufacturing stretched plastic sheets. [ABSTRACT FROM AUTHOR]

Hybrid nanofluids (HNFs) comprise combinations of different nanoparticles suspended in base fluid. Applications of such nanofluids are rising in the areas of energy and biomedical engineering including smart (functional) coatings. Motivated by these developments, the present article examines theoretically the magnetohydrodynamic coating boundary layer flow of HNFs from a stretching sheet under the transverse magnetic field in porous media with chemically reactive nanoparticles. Darcy's law is deployed. Momentum slips of both first and second order are included as is solutal slip. The transformed boundary value problem is solved analytically. Closed form solutions for velocity are derived in terms of exponential functions and for the concentration field in terms of incomplete Gamma functions by the application of the Laplace transformation technique. The influence of selected parameters e.g. suction/injection, magnetic field and slips on velocity and concentration distributions are visualized graphically. Concentration magnitudes are elevated with stronger magnetic field whereas they are suppressed with greater wall solutal slip. Magnetic field suppresses velocity and increases the thickness of the hydrodynamic boundary layer. The flow is accelerated with reduction in inverse Darcy number and stronger suction direct to reduce in skin friction. The concentration magnitudes are boosted with magnetic field whereas they are depleted with increasing solutal slip. The analysis provides a good foundation for further investigations using numerical methods. [ABSTRACT FROM AUTHOR]

The current work studies the motion of viscoelastic liquid saturated with carbon nanotubes over a stretching surface in a Darcy porous medium analytically below an influence of Cattaneo-Christov heat flux. The carbon nanotubes (CNTs) act as nanoparticles which are then appended into the base fluid. Water and kerosene are used as a base fluid with two types of CNTs, namely, Single-wall carbon nanotubes and Multiwall carbon nanotubes. Carbon nanotubes possess a wide range of industrial and biomedical applications including energy production, nuclear reactor cooling, and galaxy cooling applications because they can expand the thermal and mechanical properties of base things. As a result, the carbon nanotubes used in the mentioned fields are being investigated for their potential in heat transfer applications. Governing equations formulated using the Partial differential equations have converted to Ordinary differential equations exhausting the appropriate comparison transformation process. An influence of some relevant constraints on velocity and temperature is evaluated in details. The Cattaneo-Christov heat transfer model is utilized to investigate the heat transfer individualities with varying thermal conductivity consuming the attributes of the Appell hypergeometric function. The impacts of the emerging parameters on the profiles are depicted through graphical representations and analytically constructed tables. Considering its usefulness in modulating temperature distribution in different industrial application, including solar collector design, electronic cooling, building ventilation, etc. According to our findings, the temperature profile exhibits an enhancement with the thermal radiation parameter and the viscous-elastic fluids. In addition, when compared to the classical Fourier's law of heat conduction, the temperature profile and thermal boundary layer thickness for the Cattaneo-Christov heat flux model are lower. [ABSTRACT FROM AUTHOR]

The present work is focused on the slip flow of an electrically conductive viscoelastic fluid over a porous stretching/shrinking surface with mass transpiration and radiation. The closed-form analytical solutions for non-dimensional highly nonlinear governing equations consist of conservation of momentum, species, and energy are derived in the form of hypergeometric functions using similarity transformations. Heat and mass transfer analysis is also performed with numerous parameters that are transformed into the Kummer's function of the first kind. The presence of some physical phenomena such as magnetic field, viscoelasticity, and suction/injection affect the flow field characteristics, whereas thermal radiation, Prandtl number, and Schmidt number affect the temperature field and concentration. Furthermore, the analysis is performed for two heating processes, namely the power law surface temperature, concentration, and power law wall heat flux and wall mass flux. This problem arises in a large class of industrial manufacturing processes such as polymer extrusion, wire drawing, and the drawing of plastic sheets. [ABSTRACT FROM AUTHOR]

A steady-state, two-dimensional flow of Rivlin-Ericksen magnetohydrodynamics (MHD) fluid flow induced by stretching of the sheet of porous medium considering heat and mass transfer is investigated in the present analysis. The fluid flow is influenced by a uniform magnetic field. The inverse Darcy model, as well as thermohydrodynamic characteristics, is taken into account. Within thermal analysis effects of temperature-dependent heat source/sink, viscous dissipation, heat generation due to the elastic deformation, and thermal radiation are considered. Mass transfer is concentrated to chemically reactive diffusive species by means of first-order chemical conversion rate. The similarity transformations are employed to convert highly non-linear governing partial differential equations into a set of ordinary differential equations. Then the analytical results of the temperature and mass transfer equations are expressed in the form of Kummer's function for two different cases namely prescribed surface temperature and prescribed heat flux cases. The presented closed-form analytical solution of this research can be used as a benchmark solution for the results of numerical methods and can find possible industrial and technological applications in fluid-based systems involving shrinkable/stretchable materials. A steady-state 2D flow of Rivlin-Ericksen MHD fluid flow induced by stretching of the sheet of porous medium considering heat and mass transfer is investigated in the present analysis. The fluid flow is influenced by a uniform magnetic field. The inverse Darcy model, as well as thermo-hydrodynamic characteristics, are taken into account. Within thermal analysis effects of temperature-dependent heat source/sink, viscous dissipation, heat generation due to the elastic deformation, and thermal radiation are considered. Mass transfer is concentrated to chemically reactive diffusive species by means of first-order chemical conversion rate. The similarity transformations are employed to convert highly non-linear governing partial differential equations into a set of ordinary differential equations. Then the analytical results of the temperature and mass transfer equations are expressed in the form of Kummer's function for two different cases namely prescribed surface temperature and prescribed heat flux cases. The presented closed-form analytical solution of this research can be used as a benchmark solution for the results of numerical methods and can find possible industrial and technological applications in fluid-based systems involving shrinkable/stretchable materials. [ABSTRACT FROM AUTHOR]

An investigation of 2D laminar magnetohydrodynamics couple stress hybrid nanofluid is performed with inclined magnetic field over the stretching/-shrinking surface embedded in porous media. The quiescent water-based nanofluid is subjected to the external force by pulling the two ends of the sheet with equal magnitude and opposite forces. The governing nonlinear partial differential equations (PDEs) are converted into nonlinear ordinary differential equation (ODE) by adopting the suitable similarity transformation and obtains the analytical solution for velocity. The hydrodynamic characteristics are investigated in the presence of different physical parameters such as couple stress parameter, magnetic parameter, mass transpiration, stretching/shrinking parameter, porosity and magnetic parameter with the help of graphs. The thermal properties are enhanced by adding the copper and alumina nanoparticles to the base fluid. The result of this study has important applications in industrial and scientific are such as polymer extrusion process, liquid crystal solidification, cement artificial foams production, ceramics, metal spinning, and roofing shingles and also have possible technological applications in fluid-based systems involving stretchable/shrinkable materials. [ABSTRACT FROM AUTHOR]

The influence of inclined magnetic field and heat and mass transfer of a hydromagnetic fluid on stretching/shrinking sheet with Stefan blowing effects and radiation has been investigated. The elementary viscous equations for momentum, heat and mass transfer, which are highly nonlinear partial differential equations, are mapped into highly nonlinear ordinary differential equations with the help of similarity transformation. The subsequent highly nonlinear differential equation is solved analytically. The exact solution of heat and mass transfer appearances is found in terms of the incomplete gamma function. The species and temperature boundary conditions are assumed to be a linear function of the distance from the origin. Further, the impact of various parameters, such as Chandrasekhar number, thermal radiation, inclined Lorentz force and mass transpiration on velocity and temperature summaries, are conferred in detail. [ABSTRACT FROM AUTHOR]

The steady magnetohydrodynamics (MHD) incompressible hybrid nanofluid flow and mass transfer due to porous stretching surface with quadratic velocity is investigated in the presence of mass transpiration and chemical reaction. The basic laminar boundary layer equations for momentum and mass transfer, which are non-linear partial differential equations, are converted into non-linear ordinary differential equations by means of similarity transformation. The mass equation in the presence of chemical reaction is a differential equation with variable coefficients, which is transformed to a confluent hypergeometric differential equation. The mass transfer is analyzed for two different boundary conditions of concentration field that are prescribed surface concentration (PSC) and prescribed mass flux (PMF). The asymptotic solution of concentration filed for large Schmidt number is analyzed using Wentzel-Kramer-Brillouin (WKB) method. The parameters influence the flow are suction/injection, superlinear stretching parameter, porosity, magnetic parameter, hybrid nanofluid terms, Brinkman ratio and the effect of these are analysed using graphs. [ABSTRACT FROM AUTHOR]

A steady two-dimensional nanofluid flow along a flat surface is investigated with two types of physical models. A theoretical and numerical study is performed for three different nanofluids, in which Al2O3, TiO2, and Fe3O4 particles are dispersed in a water-based fluid. We find numerical results using Computational Fluid Dynamics (CFD) simulations and the Blasius similarity transformations. The effect of the amount of solid particles of up to a concentration of 4% on the flow and heat transfer characteristics is also analysed. The skin friction and the Nusselt number increase with the volume fraction of the nanoparticles. These properties are different for the Al2O3 water-based nanofluid and the other two nanofluids. The effect of the volume fraction of the nanofluid on the change in the skin friction coefficient and Nusselt number is investigated by both methods. The differences of the results obtained with the use of the two models are analysed depending on the nanoparticle concentration and the method. [ABSTRACT FROM AUTHOR]

Boundary layer flows of an incompressible conductive fluid past a porous stretching sheet are widely investigated in connection with their applications in chemical and engineering processes involving stretchable materials, like cooling of a molten liquid, extrusion processes, manufacturing of polymer fibers, etc. The flow problem is mathematically modeled into nonlinear partial differential equations governing the momentum and heat transfer in the boundary layer, which are transformed into nonlinear ordinary differential equations via similarity transformations. The analytical solutions for the energy equations are found through transformation of the governing thermal boundary layer problems into confluent hypergeometric differential equations. The temperature profile is analyzed for two types of boundary heating processes: with a prescribed surface temperature (PST) and with a prescribed heat flux (PHF), which are quadratic in nature. The effects of the viscous dissipation term or Eckert number, Prandtl number, inverse Darcy number, and Chandrasekhar number on velocity profiles are analyzed graphically using various plots. [ABSTRACT FROM AUTHOR]

The paper analyzes steady laminar boundary layer flow of low-conductivity Casson fluid over a stretching/shrinking sheet subjected to a transverse magnetic field in the presence of suction/injection when the fluid far away from the surface is at rest. This flow problem is mathematically modelled and the non-Newtonian fluid under consideration obeys the rheological equation of state by the Casson model. A similarity transformation converts the governing nonlinear partial differential equations into nonlinear ordinary differential equations, which are solved analytically. Using the stream function and velocity components, these results are analyzed in dependence on the Casson fluid parameters, Chandrasekhar number, and mass transpiration parameters. [ABSTRACT FROM AUTHOR]

The present study reveals the analytical model to inquiry impact of an inclined magnetic field and radiation on Boussinesq–Stokes suspension flow over a porous flat surface in the presence of mass suction/injection and viscous dissipation. By adopting a system of nonlinear partial differential equations to model the entire physical situation, a proper similarity variable may turn the system of equations into nonlinear ordinary differential equations and is solved analytically. The impact of emerging flow parameter on velocity, temperature, and local skin friction coefficient is described comprehensively through graphs. The findings also suggest that momentum boundary layer thickness diminishes with decreases magnetic field strength in the presence of suction/injection case, and thermal boundary layer thickness accelerates with radiation and Eckert number parameter. Nonetheless, there are several applications for this research in a variety of engineering domains and technology, for instance geophysics, polymer processing, total energy consumption, electric engines, blood flow measures, pumps, and flow metres. [ABSTRACT FROM AUTHOR]

The present study examines the effect of the Hiemenz stagnation point and magnetic field on the ternary nanofluid flow on the permeable stretching/shrinking surface. Ternary nanofluid composites are formed by dissolving silicon dioxide (SiO2), aluminum oxide (Al2O3), and titanium dioxide (TiO2) in base fluid water (H2O), and applying these nanoparticles to the water (base fluid) will enhance heat transfer. Furthermore, it studied the heat transmission process using variable thermal conductivity of the thermal radiation. It is noted that there is a lack of study on ternary nanofluids in the circumstances of the Hiemenz stagnation point and radiation with porous media. The novelty of the present problem is to examine the influence of magnetic field and radiation over ternary nanofluid flow. Governing equations of velocity and temperature are converted to a set of nonlinear ordinary differential equations via suitable transformations, and the obtained equations are solved using the boundary conditions, energy equation with radiation effect solved analytically by using error function and hypergeometric function. Significant physical characteristics like mass transpiration, Schmidt number, magnetic parameter, and thermal radiations, volume fraction can be discussed using the graphical analysis. The outcomes of the investigation reveal that increasing the magnetic field enhances temperature and decreases the momentum of the fluid flow. Increasing the volume fraction and thermal radiation increases the thermal boundary layer, and velocity decreases by increasing in inverse Darcy parameter, ternary nanofluids significantly increase thermal conductivity and can be used as coolants for radiators due to their improved thermal performance. Finally, adding a magnetic field causes a moving conductive fluid to conduct current, which in turn creates forces on the fluid. This is particularly useful for green and sustainable development in a variety of engineering applications as well as biomedical disciplines like medication delivery and energy efficiency. [ABSTRACT FROM AUTHOR]

MHD flow of an electrically conducting Newtonian fluid over a super linear stretching sheet in the presence of suction/injection and Navier slip is studied using modified Adomain decomposition method (MADM) and Padé approximants. Governing nonlinear partial differential equations are transformed into nonlinear ordinary differential equations using an appropriate similarity transformation. The transformed equations are solved analytically by the modified ADM and Padé approximation. The modified ADM for solving nonlinear differential equations is purely and solely the traditional Taylor's series method. Padé approximants are applied to increase the convergence of the given series. The developed analytical technique is verified comprehensively. It is found that Navier's slip condition can lead to a non-essential growth of the boundary layer thickness and a decrease in the axial and transverse velocities. [ABSTRACT FROM AUTHOR]

On the basis of Darcy–Brinkman model, linear stability analysis is used to study bio-thermal convection in a suspension of gyrotactic microorganisms in a highly porous medium heated from below. A Galerkin method is performed to solve the governing equations generating a correlation between the traditional thermal Rayleigh number and the critical value of the bioconvection Rayleigh number. The effects of three variables including the bioconvection Péclet number, the gyrotaxis number and the modified Darcy number on both the wave number and the critical bioconvection Rayleigh number are analyzed and shown graphically. Results indicate that the critical bioconvection Rayleigh number becomes larger with increasing Darcy number. [ABSTRACT FROM AUTHOR]

The preliminary aim of this article is to investigate the effect of magnetohydrodynamic (MHD) flows of a viscous fluid due to a superlinear stretching sheet. These boundary layer flows arise in the industrial processes such as polymer extrusion processes, metal spinning, glass blowing and heat exchangers. The representing frameworks of highly nonlinear partial differential equations are mapped to nonlinear ordinary differential equations with a constant coefficient via similarity transformation and are solved analytically. The results are analyzed by means of various plots to provide the comparison and found to be in better agreement with the classical results of Crane and Pavlov. The viscous fluid due to a superlinear stretching sheet in the presence ofMHDhas enormous amount of nonlinearity in conducting the solution area with different arrangements. [ABSTRACT FROM AUTHOR]