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This study implements the comparative study of Casson and Williamson nanofluids by considering the impacts of linear thermal radiation and inclined magnetohydrodynamics. Here, we employ graphs to compare the variables affecting the behavior of non‐Newtonian and Newtonian fluids for a range of physical and dimensionless parameter values. The flow's coupled equations, which contain multiple independent variables, these equations can then be changed into a single independent variable by adding similarity variables and can be solved by applying the shooting method. The effects of thermomigration and Brownian motion on nonlinear flow equations are graphically examined. For an array of radiation parameter values, we have observed that the Newtonian fluid's concentration is lesser than that of both the non‐Newtonian fluids and also noticed that Newtonian fluids converge a little sooner than Casson and Williamson fluids. The primary innovation is shown in Table 1, where the mass transfer and heat transfer values are contrasted with the limiting circumstances of previous research findings that are documented in the literature. [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]

The present work is focused on the stagnation point flow over a Jeffery fluid flow through a stretching/shrinking sheet. The nanoparticles namely Cu are considered by exploiting water-based fluid. The governing nonlinear system of partial differential equations is converted into a system of highly nonlinear ordinary differential equations via a similarity transformation. Therefore, it is quite significant to assimilate the analytical extension of heat transfer and mass transfer fluid in the presence of porous under the influence of slip velocity. The analytical solution available for nondimensional momentum, heat transfer, and concentration profiles across the boundary layer is examination is done by checking the various values of physical parameters viz., Prandtl number, Radiation parameter, stretching/shrinking parameter, and mass transpiration for the flow and heat transfer which are obtainable through graphs. The present work on nanofluids flowing through stretching/shrinking surfaces has numerous applications in biomedicine, solar energy, cooling, nuclear system cooling, etc. [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]

The two‐dimensional mixed convective MHD flow with heat and mass transfer is investigated for its behavior with Dufour and Soret mechanisms over the porous sheet. The copper–alumina (Cu–Al2O3) hybrid nanoparticles are used in the base fluid water. The governing system of partial differential equations is converted into a system of ordinary differential equations via similarity transformations, obtaining the solution for velocity, temperature, and concentration fields in exponential form. The problem is demonstrated in the Darcy–Brinkman model, the impact of included parameters such as Richardson number, magnetic field, and Dufour numbers are studied for the obtained solution with the help of graphs. Increasing the magnetic field decreases both transverse and axial velocity profiles. Increasing the magnetic field and Richardson's number decreases the solution (Al2O3–H2O). Increasing the values magnetic field and Richardson's number decreases both transverse and axial velocity profiles. Increasing the values of the Dufour effect increases the axial and transverse velocity boundary layer. The magnetohydrodynamic hybrid nanofluid flow over porous media works efficiently in liquid cooling and, therefore, has significant applications in industrial heating and cooling systems, solar energy, magnetohydrodynamic flow meters and pumps, manufacturing, regenerative heat exchange, thermal energy storage, solar power collectors, geothermal recovery, and chemical catalytic reactors. [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 induced magnetic field is used to control the fluid motion and heat transfer in a variety of applications, such as in MHD devices, microfluidics, electrically conducting fluids in channels and in circular pipes, and clinical applications such as drug delivery and cooling of nuclear reactors. Henceforth this investigation aims to elucidate the behavior of viscoelastic (second-grade fluid) ternary nanofluid flow through a permeable stretching sheet with an induced magnetic field. The stretching surface is subjected to the Cattaneo-Christov heat and mass flux model to investigate heat and mass transfer properties. Solutions of reduced governing equations are obtained numerically via the shooting method and computed using the bvp-4c algorithm. The impacts of diverse active parameters such as porous medium, magnetic parameter, reciprocal magnetic Prandtl parameter, stretching parameter, HSS parameter, and relaxation time parameter for heat and mass flux are studied graphically. In addition, the values of significant engineering factors are calculated and comparative analysis is presented through bar graphs. It is seen that regular heat sink/source promotes thermal distribution and relaxation time for mass flux enhances the mass transfer rate between fluid flow and solid surface. [ABSTRACT FROM AUTHOR]

The Casson viscous gas fluid flow with mass transpiration and radiation is demonstrated in this paper. A similarity transformation is utilized to interpret the representing construction of nonlinear PDEs by nonlinear ODEs. Then, using an exact analysis of the momentum equation, we constructed the dual solutions to the flow problem. This solution domain aids in the solution of the energy equation under the incomplete gamma function condition. The effect of radiation can also be seen in the heat equation. Physical parameters of interest, such as Brinkman number, porous media, thermal radiation number, induced slip, suction/injection and shear stress, are examined and graphically depicted. The gas slip velocity is used to simulate the slip flow model of the overall mass transfer on the moving sheet. Models of first- and second-order slip are introduced to induce the total mass transfer on the moving sheet. Additionally, the suction that causes the slip velocity rather than surface movement is investigated. The neighboring gases are forced to flow in the opposite direction of sheet movement by the mass suction-induced slip. As a result, the slip-induced suction and sheet movement cause the solution space to increase. The closed-form exact solutions are achieved for both the stretching and shrinking sheet cases. For stretched sheet instances, there is never a uniform solution. However, depending on the values of the Casson and wall mass transfer parameters, the solution in the case of the shrinking sheet may or may not exist, and if it does, it may or may not be unique or may have a dual nature. The research also demonstrates that stronger mass suction is required for a consistent flow of Casson fluid. The impacts of inverse Darcy number, induced slip parameter, Casson fluid and suction/injection on the flow and heat transfer properties of the fluid are examined under the influence of radiation, and the solution of each profile is shown in the form of figures along with the outcomes of the interface velocity and heat transfer rate at the surface. [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 research article, explores the influence of an inclined magnetic field on the fluid flow over a permeable stretching/shrinking surface with heat transfer. The study use water as a conventional base fluid, with graphene oxide (GO) and Aluminum oxide (Al2O3) nanoparticles submerged to create a nanofluid, the system of governing nonlinear partial differential equations converted into ordinary differential equations via suitable similarity conversions. This allow for the unique solution for stretching sheet/shrinking sheets to be obtained, along with the corresponding temperature solution in terms of the hypergeometric function, several parameters are included in the investigation and their contribution is graphically explained to examine physical characteristics such as radiation, inclined magnetic field, solution domain, volume fraction parameter, and temperature jump. Increasing the volume fraction and thermal radiation increases the thermal boundary layer, increasing the magnetic field parameter and inverse Darcy number increases the temperature and decays the velocity profile. The present work has many useful applications in engineering, biological and physical sciences, as well as in cleaning engine lubricants and thrust-bearing technologies. [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]

In the present analysis, we investigate the magneto hydrodynamics (MHD) Casson fluid flow on the stretching/shrinking surface with mass suction/injection. The modeled boundary problem leads to highly partial differential equations, which are then transformed to ordinary differential equations by utilizing similarity variables. Finally, the resulting ordinary differential equations are explained analytically by using different controlling parameters and corresponding boundary conditions. The results of various parameters, for example, porous medium parameter, magnetic parameter, suction/injection parameter can be analyzed using graphical form. The outcomes of the investigation reveals that, for the stretching scenario, transverse velocity and tangential velocity both drop as the values of the Casson fluid parameter rise, whereas for the shrinking case, transverse velocity rises. The present problem has demands in industrial and engineering applications for instance glass filament, paper and food manufacture, crystal growth and liquid films, and printing technology. [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 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]

The proposed planer layer dynamo physical model has real-world applications, especially in the Earth's liquid core. Thus, in this paper, an attempt is made to understand the finite amplitude convection when there exists a coupling between the Lorentz force and the Coriolis force. In particular, the effect of a horizontally applied magnetic field is studied on the Rayleigh–Bénard convection (RBC) that contains the electrically conducting fluid and rotates about its vertical axis. Free–free boundary conditions are assumed on the geometry. Attention is focused on the nonlinear convective flow behavior during the occurrence of cross rolls which are perpendicular to the applied magnetic field and parallel to the rotation axis. The visualization of cross rolls is achieved using the Fourier analysis of perturbations up to the O( ε 8 ). The relationship of the Nusselt number ( N u ) with respect to the Rayleigh number (R), the Ekman number (E), and the Elsasser number (Λ) is investigated. It is observed that E generates a strong damping effect on the flow velocity and on the heat transfer at high rotation rates. Using the heatline concept, it is observed that the temperature within the central regime is enhanced as the Λ increases. The results show that either E decreases or Λ increases, then the heat transfer rate increases. [ABSTRACT FROM AUTHOR]

The present paper investigates analytically a continuous stream of viscoelastic fluid and magnetohydrodynamic flow of second‐grade fluids owing to protracted sheets in a permeable medium with the help of the Cattaneo–Christov pattern. This idea is a new generalization of the classical Fourier law. Also, in this analysis, heat as well as mass transfer in second‐grade fluid past wall suction/injection is assumed. A few similarity transformations are used to simplify the addressing of boundary layer expressions. An analytical solution is obtained by applying the Appell hypergeometric properties. Furthermore, our work also describes an effect of the relaxation time variable, elasticity number, and Prandtl number together with temperature fields. Also, we studied the newly introduced parameter, that is, the thermal radiation parameter by Cattaneo, over a Fourier heat flux pattern. In addition, some physical presentation of the measurements is illustrated in the graphs. [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]

In this paper, the three-dimensional Casson flow of a ternary hybrid nanofluid over a porous linearly stretching/shrinking surface in the presence of an external magnetic field is considered. The surface deformation process is described by introducing two parameters of stretching/shrinking in the lateral directions. Using similarity transformations, the basic set of nonlinear partial differential equations is converted into ordinary differential equations. An exact analytical solution to this boundary value problem is obtained. The influence of the Casson parameter, magnetic field, porosity medium, and stretching/shrinking parameter, taking into account mass transpiration, on the velocity profiles and the skin friction coefficients is considered in detail. It has been established that the results obtained in some limited cases are in excellent agreement with the available data. Tables show the new results for the skin friction coefficients in the lateral directions (x and y) for different variants of surface deformation. [ABSTRACT FROM AUTHOR]

The impacts of radiation, mass transpiration, and volume fraction of carbon nanotubes on the flow of a Newtonian fluid past a porous stretching/shrinking sheet are investigated. For this purpose, three types of base liquids are considered, namely, water, ethylene glycol and engine oil. Moreover, single and multi-wall carbon nanotubes are examined in the analysis. The overall physical problem is modeled using a system of highly nonlinear partial differential equations, which are then converted into highly nonlinear third order ordinary differential equations via a suitable similarity transformation. These equations are solved analytically along with the corresponding boundary conditions. It is found that the carbon nanotubes can significantly improve the heat transfer process. Their potential application in cutting-edge areas is also discussed to a certain extent. [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]

The unsteady stagnation-point flow of a hybrid nanofluid over a stretching/shrinking sheet embedded in a porous medium with mass transpiration and chemical reactions is considered. The momentum and mass transfer problems are combined to form a system of partial differential equations, which is converted into a set of ordinary differential equations via similarity transformation. These ordinary differential equations are solved analytically to obtain the solution for velocity and concentration profiles in exponential and hypergeometric forms, respectively. The concentration profile is obtained for four different cases namely constant wall concentration, uniform mass flux, general power law wall concentration and general power law mass flux. The effect of different physical parameters such as Darcy number ðDa-1Þ, mass transpiration parameter VC ð Þ, stretching/shrinking parameter ðdÞ, chemical reaction parameter ð-Þ and Schmidt number ðScÞ on velocity and concentration profile is examined. Results show that, the axial velocity will decreases as the shrinking sheet parameter increases, regardless of whether the suction or injection case is examined. The concentration decreases with an increase in the shrinking sheet parameter and the chemical reaction rate parameter. [ABSTRACT FROM AUTHOR]

The present work discusses the laminar boundary layer flow of an electrically conducting Casson fluid due to a horizontal perforated sheet undergoing linear shrinking/stretching with mass transpiration. Navier's slip and second-order slip conditions are also imposed on the flow. The system is subjected to a transverse magnetic field. The non-Newtonian flow under consideration obeys the rheological equation of state due to the Casson model. The PDEs governing the bounder layer flow is reduced to a nonlinear boundary value problem in ODEs by utilizing appropriate similarity transformations and are expressed analytically. The similarity solution is found to be a function of the Casson parameter, magnetic parameter, mass suction/injection parameter, and the first/second-order slip parameters. Such a solution is either unique, or dual solutions exist in a region defined by the mass transfer induced slip parameter. The results of the present work are found to be an increase of the magnetic effects resulting in expansion of the unique solution region and contraction of the dual solution region for the flow due to the induced Lorentz force. In the unique solution region, an increase in magnitudes of mass suction induced slip and the first/second-order slip parameters result in a reduction of the wall shear stress in the shrinking sheet, while the wall shear stress with mass suction increases with the Casson and the magnetic effects. Similar results exist for the stretching sheet case with mass suction. However, only unique similarity solutions exist only for the case of stretching sheets with mass injection. The current work is a generalization of the classical works of Crane (1970) and Pavlov (1974) for a stretching sheet. Mass suction/injection induced slip enhances and achieves a dominant flow driven by reversing the flow direction of the moving sheet, which allows an adjacent flow against the sheet. The findings have possible industrial applications in fluid-based systems including stretchable/shrinkable things, automated cooling systems, power generation, microelectronics, and present new results to the problem. [ABSTRACT FROM AUTHOR]

The consequence of magnetohydrodynamics (MHD) flow on entropy generation analysis and thermal radiation for carbon nanotubes via a stretched surface through a magnetic field has been discovered. The governing partial differential equations are altered into ordinary differential equations with the aid of the similarity variable. Here, water is considered the base fluid with two types of carbon nanotubes, such as single-wall carbon nanotubes (SWCNTs) and multi-wall carbon nanotubes (MWCNTs). This domain is used in the energy equation, and then it is solved analytically and transferred in terms of hypergeometric function. The existence and nonexistence of solutions for stretching are investigated. Some of the primary findings discussed in this article show that the presence of carbon nanotubes, magnetic field, and Eckert number develop heat transfer in nanofluids and heat sources and that Eckert number reduces entropy formation. Different regulating parameters, such as Casson fluid, mass transpiration, thermal radiation, solid volume fractions, magnetic constraint, and heat source/sink constraint, can be used to analyze the results of velocity and temperature profiles. The novelty of the current study on the influence of magnetic field entropy analysis on CNTs flow with radiation, is that elastic deformation is the subject of this research, and this has not previously been examined. Higher values of heat sources and thermal radiation enhance the heat transfer rate. The study reveals that thermal radiation, Casson fluid; mass transpiration, Darcy number, and Prandtl number increase, and that decrease in the buoyancy ratio, magnetic parameter, and volume fraction decrease the values of the buoyancy ratio, and also control the transfer of heat. [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 invention of carbon nanotubes (CNT) has a wide range of industrial and medical applications. The notion of boundary layer flow is used in medicine, particularly in nanomedicine, and the use of magnetic fields is used to treat cancer tumour growth. The governing PDEs are altered into ODEs with the help of suitable transformations. The mass transfer of a chemically reactive species and the flow of MHD over a stretching plate subjected to an inclined magnetic field are investigated, and analytical solutions for velocity in terms of exponential function and temperature field in terms of incomplete Gamma function are obtained using the Laplace transformation. We investigate the variation of physically important parameters with varying suction, magnetic field, and slip using the analytical results. The differences in velocity and temperature profiles are explored in relation to a number of physical parameters. MWCNT nanofluids have higher effective velocities than the SWCNT deferred nanofluids, and this might assist in industrial applications and medical benefits. Earlier research tells us that carbon nanotubes are likely quicker than nanoparticles at achieving the same tumour instance. As a result, in the presence of CNTs or nanoparticles, the magnetic field can also act as a source. We found that SWCNTs nanofluids are better nanofluids than MWCNTs nanofluids. [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]

Thermosolutal Marangoni boundary layer flows are of great interest due to their applications in industrial applications such as drying of silicon wafers, thin layers of paint, glues, in heat exchangers, and crystal growth in space. The present analysis deals with the effect of chemical radiation and heat absorption/generation of the viscous fluid flow on a thermosolutal Marangoni porous boundary with mass transpiration and heat source/sink. The physical flow problem is mathematically modeled into Navier–Stokes equations. These nonlinear partial differential equations are then mapped into a set of nonlinear ordinary differential equations using similarity transformation. The analytical solutions for velocity, temperature, and concentration profiles are rigorously derived. The solutions so obtained are analyzed through various plots to demonstrate the effect of various physical parameters such as mass transpiration parameter Vc, inverse Darcy number Da−1, Marangoni number Ma, Schmidt number Sc, chemical reaction coefficient (K), Prandtl number (Pr), thermal radiation parameter (NR), and the heat source/sink parameter (I) on the momentum/thermal boundary, and their physical insights are also reported. [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]

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]