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Understanding the film lifting and draining of fluid on a vertical belt with surface tension is crucial for improving predictive models in coating and lubrication processes. This paper presents a theoretical study on the film lifting and drainage of a third-grade fluid with surface tension. The driving mechanisms on a vertical belt are the belt's upward movement, the gradient of surface tension, and gravity. The formulated nonlinear ordinary differential equation (ODE) is solved for a series-form solution using the Adomian decomposition method. Numerical computations are used to determine the stationary point placements and the thickness of the uniform film. The study elucidated that lift velocity shows a decreasing trend, while drainage velocity exhibits an increasing trend with increasing values of inverse capillary number C and Stokes number S t . The lift velocity shows an increase, whereas the drainage velocity demonstrates a decrease with an increase in the Deborah number De. With increasing values of S t and C, the stationary points shift away from the fluid–air interface, while an increase in De causes them to move towards the interface. Surface tension plays a role in supporting drainage and leads to a shift in the stationary points towards the belt. Newtonian and third-grade fluids are also compared in terms of velocity, stationary points, uniform film, and surface tension, providing insight into their behavior. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the analysis of Oldroyd 8-constant fluid flow with nanoparticle suspension via a porous media during the coating of wire is carried out. A constant magnetic field and electrically conducting fluid are considered. The governing equations thus obtained for the present model are converted to nonlinear differential equations using variables in dimensionless form. These equations are analytically solved. The influence of some parameters, like magnetic field parameter, porosity parameter, dilatant constant, pseudo-plastic constant and Brinkman number on velocity and temperature distributions are discussed graphically. For fluctuating viscosity, two models, Reynold's and Vogel's are considered. It is observed that the magnetic parameter and the Brinkman number increase, both temperature and velocity profiles show a retarding effect in both Reynold's and Vogel's models. [ABSTRACT FROM AUTHOR]

The underlying intension of this study is to oversee the flow and thermal management of hybrid nanofluid (Magnesium oxide+ Zirconium oxide+ polyethylene glycol) within a conical gap among a cone and a disk. Forced convection is examined for two situations (a) rotating disk and the stationary cone; (b) stationary disk and the rotating cone. The dimensionless differential system for the proposed model is numerically simulated with a shooting procedure (RK 4th order). Outcomes are explained through graph and tables. The main outcomes of this study are found to be – Rotating disk and stationary cone show higher temperature field for increasing estimations of the radiation parameter ( $ Rd $ R d) and the magnetic parameter ( $ M $ M) near the boundary. Hence, for heat enhancement, one can use the disk – cone model. Furthermore, increases in $ M,Rd,{\phi _1},{\phi _2} $ M , R d , ϕ 1 , ϕ 2 and $ n $ n shows an increment in Nusselt number at the cone surface $ ({N{u_c}}) $ (N u c ) compared to that at the disk surface $ ({N{u_d}}) $ (N u d ). The desired cooling in the disk and cone device could be attained by keeping the cone fixed and the disk rotating and by maintaining the surface temperature constant. It is perceived that improvement in the volume fraction of $ MgO $ M g O and $ \,Zr{O_2} $ Z r O 2 nanoparticles significantly enhanced velocity of base fluid and heat transfer rate. [ABSTRACT FROM AUTHOR]

This attempt demonstrates the nature of mass and heat transportation in non-Newtonian Reiner–Philippoff liquid flow over moving surface by considering zero and nonzero normal mass flux conditions and transverse magnetic field. Thermophoresis and Brownian movement is accounted due to the Buongiorno model of nanofluid. The expressions of flow are reframed through appropriate variables into pairs of self-similar expressions. Further, these self-similar expressions are executed numerically through the Shooting procedure (Runge–Kutta 4th order). Numerical benchmarks and graphical illustrations are prepared to visualize the role of influential constraints on interesting quantities. It is observed that Lorentz force is beneficial in obtaining effective velocity and heat transfer. Improvement in thermophoresis factor enriches temperature field in case of active control while reverse nature is observed in passive control situation. [ABSTRACT FROM AUTHOR]

The wire coating method is an engineering development to cover a wire for wadding, motorized forte and ecological protection. In wire coating analysis, moreover, the polymer extruded on the wire is hauled into interior of a die occupied with melted polymer. By considering this significance, the magneto-hydrodynamic flow and heat transmission of Oldroyd-8 constant fluid with suspension of nanoparticles in the wire coating development had been investigated. The fluid with fixed viscosity is considered in porous medium. The flow is conducted with uniform magnetic field. The arising physical governing system is modelled mathematically. The mathematical model is executed by incorporation of thermal radiation and nanoparticles (Embedded in water + hematite nanoparticles). The wire coating is scrutinized mathematically with four cases ((Ω = 0. 5 , ∅ = 0) , Ω = 0. 5 , ∅ = 0. 0 5 , Ω = − 0. 5 , ∅ = 0 , Ω = − 0. 5 , ∅ = 0. 0 5) with constant viscosity and also included in the Reynolds model for constant viscosity. The subsequent flow and heat transmission system were elucidated via the Runge–Kutta technique and the possessions of appropriate governing factors are presented in graphically. The outcome of the current investigation was equated with the previous available outcomes as a specific situation. The results were executed with nanofluid and without nanofluid as well as with positive and negative pressure gradients (Ω = 0. 5 , ∅ = 0) , Ω = 0. 5 , ∅ = 0. 0 5 , Ω = − 0. 5 , ∅ = 0 , Ω = − 0. 5 , ∅ = 0. 0. It is seen that the temperature circulation is augmented due to the upsurge in magnetic parameter M. It is interesting to note that the positive pressure gradient with nanofluid has less momentum distribution compared to rest of the cases. It is also noted that the with negative pressure gradient, the distribution is more compared to positive pressure gradient case. [ABSTRACT FROM AUTHOR]

In this study, the effects of suction and buoyancy are used to analyze the Ternary hybrid nanofluid flow on a stretching sheet through a porous media. Ternary hybrid nanofluid is (Carbon nanotubes) CNT + Graphene + Al 2 O 3 with base fluid as Water. Case-1 Buoyancy Assisting flow and Case-2 Buoyancy Opposing flow. Hybrid nanofluids have been used to speed up the heat transfer process. Nonlinear partial differential equations (PDEs) have been converted to ordinary differential equations (ODEs) using Lie group transformations. The ODE45, an algorithmic approach, has been using the aid of this built-in solver, and the resulting Ordinary differential equations were resolved. The general relationship between temperature, velocity, heat transfer rate, and shear stress on a stretchy surface is shown for a range of values of the significant factors. The temperature profiles have been rising with the impact of Da , f w . Using streamlines to examine the flow pattern of a fluid and a method of machine learning, in terms of modern language, an artificial neural network (ANN) made up of artificial neurons or nodes is known as a neural network. A neural network is a network or circuit of genetic neurons. [ABSTRACT FROM AUTHOR]

The researchers explored the free convective flow of a hybrid nanofluid in a vertical microchannel with a rectangular cross‐section. Notably, both channel walls were heated alternately, and a transverse magnetic field was applied across the channel. The channel walls had unique properties, one of which was nonslip and the other was exceedingly hydrophobic. The major purpose was to investigate the effects of magnetism and superhydrophobicity on important flow parameters. The differential equations in the investigation were solved, producing accurate results. The study yielded some significant discoveries. First, when heated, the magnetic parameter reduced skin friction on both sides. Second, in both heating conditions, the magnetic field reduced flow rate and velocity. The flow rates in the two reported situations were similar at a crucial temperature jump coefficient. Furthermore, for low‐temperature jump coefficients, heating the superhydrophobic side reduced the Nusselt number whereas heating the nonslip side had no magnetic effect. The percentage change in the value of Nusselt number and velocity decreases continuously with increase in nonlinear density variation with temperature (NDT) parameter and magnetic parameter. The percentage increases in the value of skin friction with increase in temperature jump and slip length but decrease in the percentage of skin friction for the effect of magnetic term and NDT parameter. As the NDT parameter increases, the velocity percentage rises to 50.59% when the superhydrophobic surface is heated and to 84.30% when the nonslip surface is heated. The temperature jump is statistically significant for the value of the Nusselt number and skin friction for the no‐slip surface condition. These discoveries have practical consequences for the design and management of both tiny and large‐scale systems, with possible applications in microfluidics, microelectronics, nanoscience, and nanotechnology. [ABSTRACT FROM AUTHOR]

The utilization of a ternary hybrid nanofluid, a recent development in the realm of nanofluids, can result in improved heat transfer. In the ongoing study, a ternary hybrid nanofluid flow is utilized, and it is carried out atop a porous spinning disc, which is exposed to a magnetic field, heat generation, thermal radiation, and Casson fluid. In this study, Blood/Water are taken as base fluids and (Ag–Au–Al2O3) are considered as a ternary hybrid nanoparticle. Nanoparticles made of gold and silver are put to use in a vast number of industries and fields, including nanotechnology and medicine. As a result of surface effects and quantum effects, these precious metals exhibit unique features in nanoform that play a vital role in with optical, magnetic, chemical, and mechanical behavior. Al2O3 unique optical, physical and biochemical qualities make it worthwhile for numerous uses, including nanophotonic, catalysis and the fabrication of high-energy composites. A set of relevant similarity transformations is used to generate non-dimensional forms of controlling paired nonlinear Partial Differential Equations (PDEs). MATLAB is used to perform a numerical solution with ODE45. In addition, the velocity outline decreases, and the temperature profile increases slightly before decreasing over a revolving disk when the values of magnetic parameters are increased. The distribution and radiant heat components heat up as the level gets higher. Aim and objectives of the study: The aim of this study is to analyze the Comprehensive Scrutinization of Ternary hybrid and Casson flow in a conducting porous rotating disk with internal heating. The primary objective of this analysis is to increase awareness of the impending energy crisis among those working in the industrial and technological sectors. Ternary nanoparticles (NPs) have a wide range of applications, which lends credence to the developed model. For example, Al2O3 can be used in a variety of ways that benefit society, it is used in water purification to remove water from the gas streams and extend people's lives. [ABSTRACT FROM AUTHOR]

The economic and environmental benefits of dam flow need to be calculated theoretically using open-source field operation and manipulation (OpenFOAM). The potential consequences of a hypothetical dam failure depend mainly on the kind of collapse and the reservoir's capacity. Riverbed singularities develop during floods and change over time, affecting the dam-break pulse and the shape of hydrographs. Three-dimensional simulators are commonly used to detect incremental variations in dam-break flow. However, physical obstructions or quick changes in the bed channel's characteristics can significantly alter flood propagation and scenarios. To reduce damage to the spillway, flow speed is analyzed, and obstructions in the spillway bed are taken into account. Large eddy simulation is used to explain turbulent flow behavior on the dam spillway. The kinetic energy in a turbulent flow is determined at the input, obstructions, and exit. Using C++ programming and OpenFOAM simulation, velocity and turbulent kinetic energy calculations are shown graphically. The results of the study demonstrate a remarkable alignment with the previously documented findings, which adds to the credibility and validity of our research. [ABSTRACT FROM AUTHOR]

Passive systems that need exceptional apparent geometries, thermal packaging, or fluid additives. In passive procedures, a number of foremost efforts have been conducted. The passive methods are divided into two groups: geometry modification (e.g. nano/microchannels) and solid additives (e.g. micro/nano solid particles). In a recent development in microfabrication – the development of fabrication of minuscule buildings of micrometer measures – a variation of microstrategies including heat transfer have been industrialized. Viewing this, the heat and mass transfer characteristics of a free convective Casson fluid over an elastic stretching surface are numerically studied. Thermal radiation and non-Fourier flux are incorporated in the energy to balance the transformation. Similarity transformations are used to translate partial differential equations relating to momentum, diffusion, and energy into nonlinear ordinary differential equations. The shooting approach is used to achieve numerical solutions to these equations. The effect of various physical pertinent parameters including thermal radiation parameter, Brownian motion, thermophoresis, suction, curie parameter, thermal and diffusion buoyancy forces, thermal relaxation, stretching, or shrinking parameters on the concentration and temperature profiles is investigated. It is observed that the nonlinear case has highly mixed performance when compared to the linear case of temperature and concentration fields. [ABSTRACT FROM AUTHOR]

In this paper, the importance of mass and heat transfer characteristics of mixed convective conditions on nonlinear free convection flow embedded with non-Fourier flux and thermal radiation is numerically investigated. A partial Casson slip and mixed impact of thermophoresis and Brownian motion is considered at the elastic surface. To change the highly nonlinear partial differential equations into ordinary differential system, we used the similarity transformations and then the shooting method is employed to solve resulting equations. The significance of various physical parameters on the velocity, concentration and temperature profiles are investigated for both negative value of ferromagnetic interaction ( β 1 = − 0. 2) and positive value of ferromagnetic interaction ( β 1 = 0. 2). The governing physical aspects of coefficient of skin friction, the Sherwood and Nusselt numbers are obtained for various embedded parameters. Our findings concluded that the ferromagnetic interaction is positive and then more mass transfer rate is identified compared to the negative value of ferromagnetic interaction, but whereas in friction and heat transfer rate, it was completely mixed sense (i.e., different effects have different results). Brownian motion has less Nusselt number values in positive ferromagnetism compared to negative ferromagnetism and quite opposite behavior is observed in the presence of thermophoresis, this happened due to nonlinear flux and buoyancy. [ABSTRACT FROM AUTHOR]

Analyzed in this paper is an investigation of the impacts of the varying source of heat or thermal sink and viscous dissipation on Casson fluid flow from a moving wedge by considering thermal and momentum slips in a mathematical model. Regarding the regulation of concentration fields, we incorporated the Soret and Dufour phenomena. The outcomes are analyzed numerically by using the shooting technique. The governing parametric effects of the flow characteristics are revealed graphically. We also presented the local Sherwood and Nusselt number values for non-Newtonian fluid (Casson fluid) and Newtonian. It is discovered that the thermal and concentration boundary layer is more extensive in Casson fluid while equated with the Newtonian fluid. The thermal and mass transmission in Newtonian fluid is significantly larger than in non-Newtonian fluid. The heat source/sink, Eckert number and Dufour number boost the rate of thermal transport and depreciate the rate of mass transport for both the Newtonian and non-Newtonian fluid cases. [ABSTRACT FROM AUTHOR]

The use of heat transfers in heat exchangers, nuclear cooling, solar collectors, and electrical devices is crucial nowadays. Hybrid nanofluid can be used in these applications to get the best results because it encourages efficient heat transmission compared to conventional fluid. The effect of radiation and an unsteady Carreau hybrid fluid in addition to non-Fourier heat flux over a shrinking sheet is discussed in this paper. Using similarity transformations, the newly developed system of partial differential equations (PDEs) is converted into a set of ordinary differential equations (ODEs), that are next numerically addressed by utilizing shooting approach and the Runge–Kutta (RK) method. Tables and graphs are used to present utilizing the computational findings for nondimensional temperature, velocity, heat transfer rate, and friction between fluid and hybrid nanoparticles. Additionally, for certain physical factors, the physical quantities in numerical values were also presented (such as the friction factor and local Nusselt number). We made a restricted case comparison between the current findings and the body of prior research. Our finding confirms that the temperature profile is strengthened by the heat generation parameter and the effect of radiation. The porosity parameter decelerates the momentum boundary layer thickness near the plate. [ABSTRACT FROM AUTHOR]

Numerous researchers have been interested in nanofluids because of their improved thermal characteristics and heat transmission capabilities. Recently, it has been possible to create a novel nanofluid with exceptional thermal properties by combining ternary nanoparticles of various shapes. In this respect, it is believed that the thickness of the fin will change with the length of the fin and that the impacts of thermal radiation, convection on a heat transfer mechanism, and internal heat production in a fin wetted with ternary hybrid nanofluid will depend on the length of the fin. As a result, several fin profiles, including triangular, convex, and rectangular, have been taken into consideration. This study also investigates the comparison of heat and thermal energy fluctuations in both wet and dry conditions. In order to examine the porous nature, Darcy's model is required. With the aid of the Maple computer program, the resultant nonlinear partial differential equation and boundary conditions are non-dimensionalized and numerically resolved using the implicit finite difference approach, the graphic explanation of fin efficiency, and transient thermal response for different values of the essential parameters. The investigation yielded the novel discovery that the effectiveness of the fins is enhanced by the presence of a ternary hybrid nanofluid. Three fins with varied shapes have been compared in both wet and dry circumstances. The study has discovered that triangular fins have a quicker rate of temperature decline, whereas rectangular fins have a greater efficiency. The investigation's results have a significant impact on improving heat transmission in industrial operations. [ABSTRACT FROM AUTHOR]

In most of practical situation multiple slips (velocity, temperature and concentration) has importance in manufacturing, medicinal, imaging processes and design of materials. In view of this, the present work considered the multiple slips on Magneto hydrodynamic (MHD) dissipative non-Newtonian fluid (Casson fluid) above a curved type of a stretching surface are studied. Multiple slips with Casson fluid have significance in controlling the blood flow distribution in human and animal bodies. The R–K fourth-order via shooting technique is used to convert the nonlinear governing equations. The impact of governing parameters is shown and explored graphically. The present investigation explores the multiple slip effects in governing equations under the influence of the magnetic field; the fluid's temperature and velocity have reverse characteristics. In a few unique situations, the current findings have been in reasonable agreement with the current results. As opposed to when viscous dissipation is absent, the temperature distribution is higher when viscous dissipation is present. It helps us to decide depending on the industrial and manufacturing processes, whether dissipation has to be applied or not. The stretching parameter enhances the Skin friction coefficient and rate of heat transfer, while decreases the rate of mass transfer. Also, the fluid's temperature decreases because an increase in Casson fluid parameter enhances the plastic dynamic viscosity, which creates resistance in the fluid motion. [ABSTRACT FROM AUTHOR]

In the fields of hydrology, environmental engineering and thermal energy storage systems, the large temperature and concentration gradients lead to simultaneous diffusion and stratification phenomena. The non-orthogonal stagnation flows are significant due to their applications in many hydrodynamical processes for emergency shutdown cooling and energy harvesting. Therefore, the present study introduces the novel concept of thermal and solutal stratifications in a nanofluid's non-orthogonal stagnation point flow. In the concentration and energy balance equations, the diffusive heats resulting from simultaneous mass and heat transport are also taken into account. The surface of stagnation is assumed to be oscillating and stretching linearly. The mathematical model developed under these assumptions is made dimension-free using appropriate variables. The numerical results of the problem are computed using the Lobatto-IIIa formula-based finite difference bvp4c scheme. The numerical results are verified by comparing them to the findings of an earlier study, and the results are discovered to be in good agreement. The graphical illustrations depict significant alterations in the temperature and concentration plots with stratification parameters. The findings showed that the mass transfer is greatly increased by the solutal stratification's ( ϵ 2 ) supremacy over the thermal stratification ( ϵ 1 ). In conjunction with the obliqueness parameter (γ ), the skin friction coefficient is lowered. [ABSTRACT FROM AUTHOR]

In the recent decades, the increasing energy demands and its applications have seen the focus shifting to the hybrid nanofluid flows but so much is still left to be investigated. This analysis is executed to explore the hydro-magnetic flow to investigate the incompressible flow and heat transfer towards a stretching surface with velocity and thermal slips. The scaling similarity transformations are created using Lie group analysis and employing these to convert nonlinear partial differential equations to the nonlinear ordinary differential equations. Here, after converting equations from dimensional to non-dimensional, we will use the BVP4C solver (MATLAB) for plotting the graphs to analyze how distinct non-dimensional parameters affect the skin friction and Nusselt number transfer rate, case 1 graphene + CNT + aluminum oxide with base fluid as water and case 2 magnesium oxide + zirconium oxide + copper oxide with water as base fluid, here taking nanoparticles without different shapes. The hybrid nanofluid temperature profile has mixed behavior, and the velocity profile increases when M rises. The hybrid nanofluid temperature profile curvature has composite behavior when Pr rises. The link between several independent or predictor variables and one dependent or criterion variable has been examined using multilinear regression analysis (MLR). When coefficient values for many variables are subject to change, it can forecast a wide range of outcomes. [ABSTRACT FROM AUTHOR]

The spatial heterogeneity in the porosity and permeability of the porous media is noticed in the catalytic reactors, composite materials, turbomachinery, groundwater remediation, filters, oil recovery, physiological processes, biological tissues and arteries. In the complex porous arrangement, the fluid follows a preferred path constituting a channelling effect. This channelling phenomenon occurring in such porous beds is considered here for the investigation with the novel inclusion of quadratically stretchable but convectively heated walls of the channel. In this study, the flow and temperature modulations in the hybrid ferrofluid flow due to the combined influence of Kelvin and Lorentz forces are examined. The heterogeneous porous channel is stretched quadratically in the fluid flow direction. The Carman–Kozeny correlation is used to estimate the permeability of the medium by taking the exponential variations in the porosity across the width of the channel. The mathematical model for the problem is developed and is further reduced to a self-similar form with the help of proper similarity transformations. The Chebyshev pseudospectral quasi-linearization scheme is utilized to obtain the optimal numerical information. A numerical survey is performed in the form of a comparative analysis between the heterogeneous porous medium (HePM) and homogeneous porous medium (HoPM). The interesting convection transfer mode is found to be dominant for HePM, whereas the substantial conduction transfer mode is noted for HoPM. The Nusselt number is pronounced with the uplifted values of the ferromagnetic number and nonlinear stretching constant. However, it is hampered with the Hartmann number and bead diameter number. [ABSTRACT FROM AUTHOR]

Over the last 10 years, heat transfer performance in immediate cooling and heating applications has grown into the foremost concern for heat transfer practitioners in Engineering and manufacturing practices. Henceforward, the study in new heat transfer fluids is extremely intense and challenging. This study examines flow and thermal management in axisymmetric hydrodynamic pure water-based hybrid solid nanoparticles in a flow induced by a swirling cylinder with Fourier Heat source. Flow and heat transfer are analyzed and compared for CNT+Graphene+Al 2 O 3 and Paraffin wax+Sand+AA7072 hybrid nanofluid flow. Shooting technique (R-K 4th order) is applied to work out the flow equations numerically. Simulated results are exhibited through graphs and tables. The computational results are statistically validated with the published research work and a modest concurrence is found. The main outcome of this study is found to be in Multi-regression analysis, where the Λ w.r.t Pr has higher domination compared to Φ w.r.t Pr. Also, it is interesting to know that Φ w.r.t Re has more rate of heat transfer compared to Φ w.r.t M. As the volume fraction rises, the size of the particle is less and Reynolds number dominated the flow, due to this, a decrement is seen in the friction values. Overall, it is observed that heat transfer rate is higher in CNT+Graphene+Al2O3 compared with Paraffin wax+Sand+AA7072. [ABSTRACT FROM AUTHOR]

Nowadays, the heat thrust liquid electric heater is generally used in profitable applications since it protects upto 2–3 times the energy of ordinary liquid electric heater. The heat pump makes use of a refrigerant for its process. The small-temperature refrigerant engrosses permitted heat from full of atmosphere midair in the evaporator which is crushed by an extremely well-organized electrical compressor to an extraordinary-temperature and high-pressure vapor refrigerant. For entire heat transfer connoisseurs, heat transfer performance in cooling and heating applications has become a top priority. Hence, research towards new heat transfer fluids is extremely intense and challenging. This investigation examines flow and heat transfer analysis in axisymmetric magnetohydrodynamic flow polyethylene glycol (PEG)-based nanofluid, hybrid nanofluid and ternary hybrid nanofluid flow induced by a swirling cylinder. Flow and heat transfer are analyzed and compared for three cases PEG-based copper oxide, magnesium oxide and zirconium oxide ternary nanofluid (PEG + ZrO2 + CuO + MgO), PEG-based copper oxide (PEG + CuO) nanoparticles and PEG-based zirconium and magnesium oxide hybrid nanofluid (PEG + ZrO2 + MgO). Shooting technique (R–K fourth-order) is employed to work out the flow equations numerically. Simulated results are displayed through graphs. The computational results are validated with the published research work and a modest concurrence is found. The main outcome of this study is found to be as follows: It is interesting to note that C f x Re (x R) is lesser in nanofluid case compared with ternary and hybrid nanofluid cases. It is found that C f ϕ Re (G H R) is more in ternary hybrid nanofluid compared with hybrid and nanofluid cases. Overall, it is observed that heat transfer rate is higher in nanofluid compared with ternary and hybrid nanofluid cases whereas lesser rate of heat transfers in ternary nanofluid case. [ABSTRACT FROM AUTHOR]

This study examines the flow of an incompressible flow over a linear stretching surface with the inclusion of momentum and thermal slip conditions. A scaling set of alterations is applied to the governing system for both with and without magnetic field situations. The physical system being leftover invariant caused by some associations surrounded by the transformations. Later we find the absolute invariants 3rd -order ODEs for the linear momentum equation and two 2nd order ODEs consistent with the energy and concentration are obtained. The equations that coincide with the boundary circumstances are elucidated mathematically. The physical pertinent parameters as shown in graphs and the friction factor, Nusselt number and Salts 1 and 2 Sherwood numbers are shown in surface plots. We observed that the momentum slip parameter decelerates the skin friction coefficient in the presence of a magnetic field and enhances in the absence of the magnetic field parameter. The thermal slip parameter enhances the Nusselt number in both the presence and absence of magnetic field parameter. Finally, the thermal and concentration buoyancy ratio parameters are shown to upsurge the friction factor, Nusselt and Salts 1 and 2 Sherwood numbers in both cases of M = 0 and M =1. [ABSTRACT FROM AUTHOR]

For entire heat transfer practitioners from the last ten years, heat transmission performance in cooling and heating applications has become foremost concern. Hence, research towards innovative heat transference fluids is enormously powerful and stimulating. This study examines flow and thermal management in axisymmetric magneto hydrodynamic Polyethylene glycol (PEG) based hybrid nanofluid flow induced by a swirling cylinder. Flow and heat transfer is analyzed and compared for PEG+ Cu2 O+ MgO and PEG+Graphene+ Cu + Ag hybrid nanofluid flow. Shooting technique (R-K 4th order) is applied to work out the flow equations numerically. Simulated results are demonstrated via graphs. The computational results are validated with the published research work and found a modest concurrence. The foremost outcome of this investigation is found to be the axial, swirl and radial velocities in hybrid nanofluid are observed to decay with improvement in Reynolds number, nanofluid volume fraction and magnetic parameter. Platelet shaped nanoparticle colloidal suspension exhibit more decaying axial, swirl and radial velocity compared to spherical shaped nanoparticle colloidal suspension. It is detected that heat transmission rate is higher in PEG + Cu2 + O + MgO Hybrid nanofluid compared with PEG + Graphene + Cu + Ag Hybrid nanofluid. For cooling purpose one can adopt PEG+ Cu2 O + MgO hybrid nanofluid. [ABSTRACT FROM AUTHOR]

The dynamical behaviour and thermal transportation feature of mixed convective Casson bi-phasic flows of water-based ternary Hybrid nanofluids with different shapes are examined numerically in a Darcy- Brinkman medium bounded by a vertical elongating slender concave-shaped surface. The mathematical framework of the present flow model is developed properly by adopting the single-phase approach, whose solid phase is selected to be metallic or metallic oxide nanoparticles. Besides, the influence of thermal radiation is taken into consideration in the presence of an internal variable heat generation. A set of feasible similarity transformations are applied for the conversion of the governing PDEs into a nonlinear differential structure of coupled ODEs. An advanced differential quadrature algorithm is employed herein to acquire accurate numerical solutions for momentum and energy equations. Results of the conducted parametric study are explained and revealed in graphs using bvp5c in MATLAB to solve the governing system. The solution with three mixture compositions is provided (Type-I and Type-II). Al2O3 (Platelet), GNT (Cylindrical), and CNTs (Spherical), Type-II mixture of copper (Cylindrical), silver (Platelet), and copper oxide (Spherical). In comparison to Type- I ternary combination Type-II ternary mixtures is lesser in terms of the temperature distribution. The skin friction coefficient is more in Type-1 compared to Type-2. [ABSTRACT FROM AUTHOR]

The heat and mass transfer of two immiscible fluids in an inclined channel with thermal diffusion, vicious, and Darcy dissipation is studied. The first region consists of a clear fluid, and the second one is filled with a nanofluid saturated with a porous medium. The behaviors of Cu-H2O, In-H2O, and Au-H2O nanofluids are analyzed. The transport properties are assumed to be constant. The coupled non-linear equations of the flow model are transformed into the dimensionless form, and the solutions for the velocity, temperature, and concentration are obtained by the regular perturbation technique. Investigations are carried out on the flow characteristics for various values of the material parameters. The results show that the velocity and temperature of the fluids enhance with the thermal Grashof number, solutal Grashof number, and Brinkman number while decrease with the porosity parameter and solid volume fraction. [ABSTRACT FROM AUTHOR]

This work analyzes the two-dimensional flow of an incompressible magneto-hydrodynamic fluid over linear stretching sheet in the presence of suction or injection and convective boundary conditions. A scaling group transformation method is applied to the flow governing equations. The system remains invariant due to the relation between the transformation parameters. Upon finding three absolute invariants, third-order ordinary differential equations (ODEs) corresponding to momentum equation and second-order ODEs corresponding to energy and diffusion equations are derived. Shooting technique (R-K fourth-order) is applied to work out the flow equations numerically. MATLAB is used for the simulation and the results are exhibited through graphs. The computational results are validated with the published research work and a modest concurrence was found. The main outcome of this study is found to be that raising values of f si and Γ 1 decline the friction, whereas Γ 1 and Γ 2 show the opposite (increasing). The rising values of f si and M in addition to M and Γ 2 show a decline in friction factor. The Nusselt number values are improved as raising values of Pr versus f si and Pr versus M. It is very clear the monotonically increasing M versus f si and strictly increasing f si versus M cases. It is very clear the mass-transfer rate is smoothly improved L e 1 versus L e 2 and strictly increased L e 1 versus  Γ 1 . [ABSTRACT FROM AUTHOR]

Compared to (pure) methanol, methanol-based nanofluids are more effective in several aspects like thermophysical properties. For example, specific heat and thermal conductivity are increased at higher temperature. Researchers experimentally discovered that methanol-based hybrid nanofluid ( CH 3 OH + SiO 2 + Al 2 O 3 ) is useful to enhance the absorption rate of CO 2 in a tray column absorber. Despite numerous reports on hybrid nanofluid via curved stretching sheet, so far, no work is done on the impact of increasing Brinkman number, activation energy on the dynamics of methanol conveying silica and alumina when thermal radiation and cross-diffusion effects are profoundly significant. In the present paper, the examination of methanol-based hybrid fluid flow across a bended (curved) stretching sheet with cross-diffusion effects (Soret–Dufour numbers) and activation energy is made. An analysis of entropy generation on the fluid flow is also visualized. Suitable similarity transformations are applied to modify the leading equations as nonlinear ODEqns. Shooting procedure (R–K 4th order based) is performed to unriddle the consequent equations. Outcomes are explicated via graphs in two cases, i.e., hybrid fluid (methanol + silica + alumina) and nanofluid (methanol + silica). The important outcomes are: curvature parameter ameliorates the velocity but minifies the friction factor and escalation in entropy generation is recognized for the acclivity in Brinkman number and magnetic field parameter. Moreover, we witnessed that activation energy parameter steps up the concentration profile, nanoparticle volume fraction parameters minimizes the friction factor, reaction rate parameter ameliorates concentration, thermal radiation elevates the temperature and Dufour number and Brinkman number are useful to minimize the heat transfer rate. It is interesting to mention that influence of magnetic field on velocity profile depends on the value of nanoparticle volume fraction. Further, our results are validated with previous results for surface drag force under limited conditions and we detected a respectable accord. [ABSTRACT FROM AUTHOR]

Compared with methanol (pure) and blood, methanol‐ and blood‐based nanofluids are significant with numerous characteristics, including thermophysical effects and immersion rate of CO2 in tray column absorber. Blood‐based liquids show importance in biological treatments like digestion as well as breathing systems. Mixed convective conditions and suction or injection are helpful in controlling the heating or the cooling in industrial applications. Keeping this in view, we examined the physical perception of entropy generation (EG) in a mixed convective hybrid nanofluid (HBN) flow with chemical reaction, cross‐diffusion, and transpiration in a curved surface with activation energy and thermal radiation. Transformed principal equations are determined with the assistance of a fourth‐order Runge–Kutta method with a shooting technique. We obtained solutions for two different cases, that is, for hybrid (mixture of cases) (magnetite + hematite + blood) and (magnetite + hematite + methanol). Obtained results are discussed via graphs and tables. Also, we discussed the effects of the Bejan number and the EG. We found that the EG appears more conspicuously in the case of magnetite + hematite + methanol compare to magnetite + hematite + blood case. We also found that in the magnetite + hematite + methanol case the flow and heat transfer characteristics are more pronounced compared with that in the case of magnetite + hematite + blood. This helps us to choose magnetite + hematite + blood or magnetite + hematite + methanol compositions in the manufacturing industry. Furthermore, the rate of heat transfer can be enhanced up to 16.42% in the case of methanol‐based HBN while the rate of heat transfer can be enhanced up to 19.16% in the case of blood‐based HBN. [ABSTRACT FROM AUTHOR]

Hybrid nanofluids combine two or more nano properties with a base fluid such as water ethylene. Usually, this helps enhance the heat transfer rate; in this article, using new similarity transformations created by Lie group analysis, the governing nonlinear partial differential equations are transformed into a system of connected nonlinear ordinary differential equations. The resulting design is numerically solved using a BVP4C solver with the shooting method (MATLAB). The magneto hydrodynamic flow of an incompressible fluid and the rate of heat and mass transfer were investigated for two cases, with various nanoparticle shapes including cylindrical, spherical, and platelet. Case 1 was CNT (1%), graphene (1%), and aluminum oxide (1%), and Case 2 was copper (1%), silver (1%), and cobalt ferrite (1%). When the Hartmann number rises, velocity and temperature exhibit inverse behavior: the velocity profile increases, and the temperature profile decreases. When the suction rises, the velocity and temperature profiles both increase. Optimization techniques were used from response surface methodology (RSM) to set factorial variables so that the response met the desired maximum or minimum value. Factorial methods like ANOVA were used to model the response, but they were expanded to simulate the effects in terms of extrapolation. [ABSTRACT FROM AUTHOR]

The study of convective heat transfer in differently shaped fins with radiation, internal heat generation and variable thermal conductivity was considered. The energy equation of the model was converted into the dimensionless form by adopting the proper variables, which was later solved using the differential transformation method. The impact of the parameters on the thermal performance, efficiency and heat transfer of the fins was analyzed graphically and also by performing thermal analysis on the fins. It was noticed that there was a significant effect on the thermal performance of the fins with different shapes, and also the heat transfer rate of the fin increased for improved values of the internal heat generation and radiation parameters. The exponential profile showed better results than other profiles, and the results obtained were supported by thermal analysis using ANSYS software. [ABSTRACT FROM AUTHOR]

In current days, hybrid models have become more essential in a wide range of systems, including medical treatment, aerosol particle handling, laboratory instrument design, industry and naval academia, and more. The influence of linear, nonlinear, and quadratic Rosseland approximations on 3D flow behavior was explored in the presence of Fourier fluxes and Boussinesq quadratic thermal oscillations. Ternary hybrid nanoparticles of different shapes and densities were also included. Using the necessary transformation, the resulting partial differential system is transformed into a governing ordinary differential system, and the solution is then furnished with two mixed compositions (Case-I and Case-II). Combination one looked at aluminum oxide (Platelet), graphene (Cylindrical), and carbon nanotubes (Spherical), whereas mixture two looked at copper (Cylindrical), copper oxide (Spherical), and silver oxide (Platelet). Many changes in two mixture compositions, as well as linear, quadratic, and nonlinear thermal radiation situations of the flow, are discovered. Case-1 ternary combinations have a wider temperature distribution than Case-2 ternary mixtures. Carbon nanotubes (Spherical), graphene (Cylindrical), and aluminum oxide (Platelet) exhibit stronger conductivity than copper oxide (Spherical), copper (Cylindrical), and silver oxide (Platelet) in Case 1. (Platelet). In copper oxide (Spherical), copper (Cylindrical), and silver (Platelet) compositions, the friction factor coefficient is much higher. The combination of liquids is of great importance in various systems such as medical treatment, manufacturing, experimental instrument design, aerosol particle handling and naval academies, etc. Roseland's quadratic and linear approximation of three-dimensional flow characteristics with the existence of Boussinesq quadratic buoyancy and thermal variation. In addition, we combine tertiary solid nanoparticles with different shapes and densities. In many practical applications such as the plastics manufacturing and polymer industry, the temperature difference is remarkably large, causing the density of the working fluid to vary non-linearly with temperature. Therefore, the nonlinear Boussinesq (NBA) approximation cannot be ignored, since it greatly affects the flow and heat transport characteristics of the working fluid. Here, the flow of non-Newtonian elastomers is controlled by the tension of an elastic sheet subjected to NBA and the quadratic form of the Rosseland thermal radiation is studied. [ABSTRACT FROM AUTHOR]

An electrically conducting three-dimensional non-Newtonian Carreau fluid induced by bidirectional movement of sheet is demonstrated in this work. The aspects of viscous heating, radiation and first-order chemical reactions are executed in energy transport and mass species expressions. The new variables are defined to restructure the mathematical problem into single independent variable equations (ordinary differential equations, ODE's). The re-framed ODE's are exploited numerically through the utilization of MATLAB bvp4c package. The derived results of velocities, temperature and species distributions are executed graphically and interpreted physically for both shear-thinning and shear-thickening cases. Also, the numerical results for coefficients of skin friction, rate of mass transfer in terms of Sherwood number and rate of heat transfer in terms of Nusselt number are presented in tables. The influence of local Weissenberg numbers on Sherwood and Nusselt numbers for shear-thinning fluids is reversed to shear-thickening fluids. The thermal radiation parameter enhances the fluid temperature, and chemical reaction parameter decelerates the fluid concentration. [ABSTRACT FROM AUTHOR]

With the fast improvement of the industry and the utilization of inventive strategies, scientists want to think over the steady blending convection of water-based nanofluids past static wedges. The Buongiorno model with convection is applied. Also, incorporated the Brownian motion and thermophoresis. The attention is on the nature of mixed wedge-formed convective heat and mass transfer of the nanofluid flow. Utilizing comparable change, the governing partial differential equations (PDEs) are reduced to ordinary differential equations (ODEs) solved by the R-K Gill method. The physical quantities of velocity, temperature and concentration fields, as well as diffusion and thermal transfer rates with friction factor coefficients, is discussed. The investigation demonstrated that the temperature convergence of the liquid was higher within the sight of the thermophoresis parameter and Biot numbers. It has been seen that divider pressure increments with expanding wedge and mixed convection parameters. [ABSTRACT FROM AUTHOR]

The present study aims to investigate time-dependent two-dimensional Sakiadis ow of quiescent fluid. This ow is induced by stationary at plate through uniform free-stream (Blasius ow). The variable conductivity and viscosity ratio parameters as well as non-linear chemical reaction were taken into account in the mathematical equations. Similarity variables were employed in the governing transport expressions to convert them into the ordinary differential system. The transformed system is numerically computed using Runge-Kutta scheme and based on shooting criteria. The results regarding concentration, velocity, and temperature distributions were also studied through plots. Moreover, mass and heat transfer rates and friction factor were discussed in detail. According to the findings, the constraint of chemical reaction slowed down the rates of friction factors and heat transportation in the case of the Sakiadis-Blasius ow and enhanced the mass transportation rate in both cases. In addition, the rate of mass transportation was smaller in Sakiadis ow than that of Blasius ow. The obtained results of the rate of heat transfer were in excellent agreement with those from the literature. [ABSTRACT FROM AUTHOR]

In this paper, we have analysed a binary chemical nanofluid dissipative flow (in two cases i.e., 50% EG + 50% water/silica and 50% EG + 50% water/graphene oxide) due to a curved stretching sheet with activation energy. Appropriate transformations yield the nonlinear ordinary differential system. Shooting procedure (R-K 4th order based) is executed to solve the resultant equations. Graphical illustrations thoroughly demonstrate the features of the involved pertinent parameters. We have deliberated the behaviour of the alike parameters on the rate of transfers (heat and mass) and surface drag force (skin friction coefficient) by means of tables. This investigation reveals that (a) reaction rate parameter and temperature difference parameter are helpful to ameliorate the mass transfer rate (b) concentration enhances for higher estimation of activation energy variable (c) increasing the volume fraction of nanoparticles reflects an escalation in temperature (d) heat transfer rate enhancement is recognized for the influence of heat transfer Biot number. At the end this study, we came to know that the EG-Water + Graphene Oxide mixture has more heat transfer rate compared to EG-Water + Silica mixture. This outcome helps to conclude that, whenever the more heat transport required in manufacturing and industries, we can take the EG-Water + Graphene Oxide mixture. [ABSTRACT FROM AUTHOR]

Adding variety of nanoparticles to the base fluid is current technique in order to boost the thermal performance of conventional fluids and mononanofluids. The forthright intention of the present investigation is to analyze numerically the up-to-date progress in flow and heat transport nature of magnetohydrodynamic, radiative Newtonian fluid, water-based Al2O3 nanofluid, water-based graphene nanofluid and water based Al2O3 + graphene hybrid nanofluid due to convectively heated stretching sheet. The flow equations are transformed by applying appropriate transformations into a pair of self-similarity equations. Further similarity equivalences are numerically solved through Runge–Kutta based shooting method. Graphs and tables are structured to analyze the behavior of sundry influential variables. From this study it is found that rate of heat transfer for Graphene + water is 2.921934, Al2O3 + H2O + Graphene is 2.250658 and Al2O3 + H2O is 3.260554. From this we conclude that water based Al2O3 + graphene hybrid nanofluid can be opted for cooling performance. Water based Al2O3 nanofluid significantly enhance convection heat transfer performance over a stretching sheet. Friction at the wall for Graphene + water is (− 1.719525), Al2O3 + H2O + Graphene is (− 2.256614) and Al2O3 + H2O is (− 1.959539). From this we conclude that water based Al2O3 + graphene hybrid nanofluid shows lower wall friction rate compared to other two mixture compositions. [ABSTRACT FROM AUTHOR]

Nonlinear convective flow and heat transfer characteristics are analyzed between stationary nonporous and porous rotating disks utilizing graphene nanoparticles in a water and ethylene glycol base fluid. Heat transfer characteristics are analyzed via incorporating thermal radiation and heat absorption/generation. The governing fluid equations are computed numerically using Runge–Kutta based shooting technique after employing appropriate transformations. Characteristics of sundry variables are elaborated graphically as well as through the construction of Table for water base and ethylene glycol based graphene nanoparticles. It is observed that improvements in nonlinear convection variable owing to temperature and heat generation variable improve wall friction in radial direction. Improvement in Hartman number decreased wall friction in radial and tangential directions along with Nusselt number in graphene/ethylene glycol and graphene/water nanofluid. Ethylene glycol based graphene nanofluid takes less time for execution as compared to water based nanofluid. [ABSTRACT FROM AUTHOR]

The current article discusses the heat transfer characteristics of ferromagnetic liquid over an elastic surface with the thermal radiation and non-Fourier heat flux. In most of the existing studies, the heat flux is considered as constant, but whereas we incorporated the non-Fourier flux to get the exact performance of the flow. Also, we excluded the PWT and PHF cases to control the boundary layer of the flow. The governing equations related to our contemplate are changed into non-linear ordinary differential equations (ODE's) by utilizing appropriate similarity changes, which are at the point enlightened by Runge-Kutta based shooting approach. The equations are broken down concerning boundary conditions and to be explained prescribed wall temperature (PWT) and prescribed heat flux (PHF) cases. The impacts of diverse non-dimensional physical parameters on velocity and temperature profiles are laid out graphically. Also, the assortment of skin friction and local Nusselt number for both PWT and PHF cases for various assessments of non-dimensional parameters have been sorted out. Towards the wrap-up of the examination, we suspect that the friction factor coefficient is higher in the PWT case compared to the PHF case. This result helps to conclude that the flux conditions are useful for cooling applications. [ABSTRACT FROM AUTHOR]

The performance of copper selenide and effectiveness of chemical catalytic reactors are dependent on an inclined magnetic field, the nature of the chemical reaction, introduction of space heat source, changes in both distributions of temperature and concentration of nanofluids. This report presents the significance of increasing radius of nanoparticles, energy flux due to the concentration gradient, and mass flux due to the temperature gradient in the dynamics of the fluid subject to inclined magnetic strength is presented. The non-dimensionalization and parameterization of the dimensional governing equation were obtained by introducing suitable similarity variables. Thereafter, the numerical solutions were obtained through shooting techniques together with 4th order Runge–Kutta Scheme and MATLAB in-built bvp4c package. It was concluded that at all the levels of energy flux due to concentration gradient, reduction in the viscosity of water-based nanofluid due to a higher radius of copper nanoparticles causes an enhancement of the velocity. The emergence of both energy flux and mass flux due to gradients in concentration and temperature affect the distribution of temperature and concentration at the free stream. [ABSTRACT FROM AUTHOR]

In this investigation, the flow of an unsteady mixed convection boundary layer viscous nanofluid on a stretchable sheet is considered. The flow examination is affected by a magnetic field. The reason for the examination exhibited is to create models for nanomaterials that incorporate the Brownian movement and thermophoresis phenomena. The created nonlinear standard differential condition is illuminated numerically utilizing the Runge"Kutta"Gill technique and the start program. The different factors of speed, temperature, and concentration are reported and discussed. The examination shows that the speed, temperature, and concentration are lower in contrast with the consistent stream on account of an assisting flow, whereas the opposite situation is noticed in the opposing flow case. The effects of Brownian movement and thermophoresis in the concentration case are totally different. [ABSTRACT FROM AUTHOR]

The forthright intention of the present investigation is to analyze the up-to-date progress in Jeffrey nanofluid flow past an electromagnetic sheet by utilizing the properties of nonlinear convection, radiation, convective boundary condition, zero mass flux condition and Arrhenius activation energy. The flow equations are transformed by applying appropriate transformations into a pair of self-similarity equations. Further similarity equivalences are numerically solved through Runge–Kutta-based shooting method. Graphs and tables are structured to analyze the behavior of sundry influential variables. The results acquired showed good agreement with the previous notable works. Through this study we observed that improvement in Lorentz force in the positive x-direction strengthens the momentum, which intensifies the transfer of heat energy from the boundary, resulting in reduced fluid temperature. [ABSTRACT FROM AUTHOR]

In this article an investigation is done on hydromagnetic effects on electrically conducting fluid past an exponentially accelerated infinite vertical plate with exponentially varying temperature and concentration. The influence of thermal diffusion and radiation absorption is considered in this analysis. The problem is governed by coupled non-linear partial differential equations which are solved by finite difference method. The plate temperature is increasing linearly with time and the concentration level near the plate is increased. Among the effects of various In this article an investigation is done on hydromagnetic effects on electrically conducting fluid past an exponentially accelerated infinite vertical plate with exponentially varying temperature and concentration. The influence of thermal diffusion and radiation absorption is considered in this analysis. The problem is governed by coupled non-linear partial differential equations which are solved by finite difference method. The plate temperature is increasing linearly with time and the concentration level near the plate is increased. Among the effects of various 284 South East Asian J. of Mathematics and Mathematical Sciences physical parameters on the velocity, temperature, concentration, skin friction, Nusselt number and Sherwood number are broadly discussed with the help of graphs and table. [ABSTRACT FROM AUTHOR]

The present study emphases on the three-dimensional (3D) Casson nanofluid flow across a slendering sheet in porous layers by considering the thermophoresis and Brownian motion effect. The proposed mathematical model has a tendency to characterise the effect of the non-uniform heat source/sink. In the present simulation, the graphene–water-based nanoparticles have been used at two different temperatures namely 10 and 50 °C. The nonlinear ordinary differential equations are solved using the Runge–Kutta Feldberg integration method. The characteristics of velocity, temperature and concentration boundary layers in the presence of graphene–water nanoparticles are presented for different physical parameters such as heat source/sink parameter, thermophoresis parameter, Brownian motion parameter, Casson fluid parameter, porosity parameter, volume fraction and velocity power index parameter. Moreover, the friction factor coefficients, Nusselt number and Sherwood number are also estimated and discussed for aforesaid physical parameters. It is found that there is a significant increase in the thermal and concentration boundary layer thickness when the strength of the thermophoresis parameter is increased. In contrast, thermal boundary layer increases with the rise in the Brownian motion parameter, while the reverse trend holds true for concentration field. In addition, the rate of heat and mass transfer rate are higher in case of graphene–water nanoparticle at 50 °C compared to 10 °C temperature. [ABSTRACT FROM AUTHOR]

A theoretical study on the effect of magnetohydrodynamic field on the classical Blasius and Sakiadis flows of heat transfer characteristics with variable conditions and variable properties are studied in this paper. Hydromagnetic flows and heat transfer in porous media have been considered extensively in recent years due to their occurrence in several engineering processes such as compact heat exchangers, metallurgy, casting, filtration of liquid metals, cooling of nuclear reactors and fusion control. The governing partial differential equations are converted into ordinary differential equations using the suitable similarity transformations. The transformed boundary layer equations like momentum and energy equations are solved numerically by using Runge–Kutta method. The effect of governing physical parameters over the velocity and temperature distributions is demonstrated graphically. Moreover, the local friction factor coefficient and rate of heat transfer in terms of Nusselt number are computed and discussed through tables. We validated the present solutions with already existing studies under limited cases. It is found that the thermal buoyancy parameter enhances the velocity depreciates the temperature field for both the Sakiadis and Blasius flow cases. The influence of Biot number increases skin friction coefficient and local Nusselt number for both Sakiadis and Blasius flow cases.The rate of heat transfer for temperature ratio parameter is high in Blasius flow case when compared with Sakiadis flow case. [ABSTRACT FROM AUTHOR]

In this investigation, we compared the effect of unsteady magneto hydrodynamic Carreau fluid in the interruption of graphene nanoparticles with heat generation over stretching and shrinking surface. The simulation is performed by combination of graphene nanoparticles into base water. Dispersion of graphene nanoparticles into base fluid finds several applications, such as biosensors, detection and purification of cancer cells, biocompatibility, and bio-imaging. The arising set of dimensional partial differential equations (PDEs) is converted into a set of non-dimensional ordinary differential equations (ODEs) by suitable similarity transformations, and then solved by the help of Runge-Kutta with shooting technique. The numerical results for non-dimensional velocity and temperature are displayed with graphs. We also analyzed the physical quantities (friction factor and local Nusselt number) for various values of non-dimensional physical parameters mathematically, and they are presented through tables. We also compared the present results with existing literature under some limited case. At the end of our analysis, we noted that radiation amplifies temperature profiles of both stretching and shrinking sheet cases. But, compared to shrinking sheet stretching sheet gains higher values. Heat generation parameter enriches temperature profiles in both stretching and shrinking sheets; compared to stretching sheet, shrinking sheet gains higher values. [ABSTRACT FROM AUTHOR]