Keller Box Method

The purpose of this paper is to introduce an effective strategy for solving boundary layer flow of an Eyring-Powell fluid over a stretching sheet in unbounded domain. This paper introduces a combination of Modified Generalized Laguerre, the quasi-linearization method, and Collocation method for solving boundary layer flow of an Eyring-Powell fluid over a stretching sheet in unbounded domain. Applying this technique leads to obtaining suitable results and less error. Moreover, the effect of different values of the Eyring-Powell fluid material parameters is investigated and the results show that by increasing the parameter (m), velocity increases, and by increasing the parameter (n), velocity decreases.

The purpose of this paper is to introduce an effective strategy for solving boundary layer flow of an Eyring-Powell fluid over a stretching sheet in unbounded domain. This paper introduces a combination of Modified Generalized Laguerre, the quasi-linearization method, and Collocation method for solving boundary layer flow of an Eyring-Powell fluid over a stretching sheet in unbounded domain. Applying this technique leads to obtaining suitable results and less error. Moreover, the effect of different values of the Eyring-Powell fluid material parameters is investigated and the results show that by increasing the parameter (m), velocity increases, and by increasing the parameter (n), velocity decreases.

This paper investigates the behavior of MHD stagnation point flow of Carreau fluid in the presence of infinite shear rate viscosity. Additionally heat transfer analysis in the existence of non-linear radiation with convective boundary condition is performed. Moreover effects of Joule heating is observed and mathematical analysis is presented in the presence of viscous dissipation. The suitable transformations are employed to alter the leading partial differential equations to a set of ordinary differential equations. The subsequent non-straight common ordinary differential equations are solved numerically by an effective numerical approach specifically Runge-Kutta Fehlberg method alongside shooting technique. It is found that the higher values of Hartmann number M ð Þ correspond to thickening of the thermal and thinning of momentum boundary layer thickness. The analysis further reveals that the fluid velocity is diminished by increasing the viscosity ratio parameter ðb Ã Þ and opposite trend is observed for temperature profile for both hydrodynamic and hydromagnetic flows. In addition the momentum boundary layer thickness is increased with velocity ratio parameter a ð Þ and opposite is true for thermal boundary layer thickness.

This paper investigates the behavior of MHD stagnation point flow of Carreau fluid in the presence of infinite shear rate viscosity. Additionally heat transfer analysis in the existence of non-linear radiation with convective boundary condition is performed. Moreover effects of Joule heating is observed and mathematical analysis is presented in the presence of viscous dissipation. The suitable transformations are employed to alter the leading partial differential equations to a set of ordinary differential equations. The subsequent non-straight common ordinary differential equations are solved numerically by an effective numerical approach specifically Runge-Kutta Fehlberg method alongside shooting technique. It is found that the higher values of Hartmann number M ð Þ correspond to thickening of the thermal and thinning of momentum boundary layer thickness. The analysis further reveals that the fluid velocity is diminished by increasing the viscosity ratio parameter ðb Ã Þ and opposite trend is observed for temperature profile for both hydrodynamic and hydromagnetic flows. In addition the momentum boundary layer thickness is increased with velocity ratio parameter a ð Þ and opposite is true for thermal boundary layer thickness.

Fins are commonly utilized to enhance (dissipate) heat in various engineering systems that include heat exchangers. In the present investigation, the impact of multi-boiling and thermo-geometric factors on a convective-radiative rectangular porous fin subjected to the temperature-dependent thermal conductivity of linear and non-linear variations is discussed extensively. The governing equations describing the problem were formulated with the aid of Darcy law. Similarity variables were employed to reduce the models to non-dimensional form. The solution of the governing dimensionless equation is approximated using the RK4 and spectral local linearization methods. Before parametric analysis, the agreement between the two numerical methods was established. Findings reveal that the non-linear variation of thermal conductivity shows better thermal efficiency than the linear variation. An improvement in the multi-boiling heat transfer parameter retards the temperature distribution of the fin. Furthermore, increasing the thermo-geometric parameter will result in a progressive decrease in the temperature of the fin. The results obtained in this work will aid in the design of heat exchangers and other heat transfer equipments.

The purpose of this research is to inspect the mixed convection flow of Eyring-Powell nanofluid over a linearly stretching sheet through a porous medium with Cattaneo-Christov heat and mass flux model in the presence of Hall and ion slip, permeability, and Joule heating effects. Proper similarity transforms yield coupled nonlinear differential systems, which are solved using the spectral relaxation method (SRM). The story audits show that the present research problem has not been studied until this point. Efficiency of numerous parameters on velocity, temperature, and concentration curves is exposed graphically. Likewise, the numerical values of skin friction coefficients, local Nusselt, and Sherwood numbers are computed and tabulated for some physical parameters. It is manifested that fluid velocities, skin friction coefficients, local Nusselt, and Sherwood numbers promote with the larger values of Eyring-Powell fluid parameter ε. It is also noticed that primary velocity promotes with larger values of mixed convection parameter λ, Hall parameter β e , and ion slip parameter β i , while the opposite condition is observed for secondary velocity, temperature, and concentration. Furthermore, comparative surveys between the previously distributed writing and the current information are made for explicit cases, which are examined to be in a marvelous understanding.

Little is known in the literature about the concept of nonuniform heat source/sink and higher-order chemical reaction for the dynamics of Oldroyd-B nanoparticles. erefore, the present article addresses the nonuniform heat source/sink and higher-order chemical reaction features in nonlinear mixed convection bidirectional MHD dynamics of Oldroyd-B nanoparticles with thermal radiation aspects through porous space. Strati cation e ects for both the temperature and concentration setups are also used in the mathematical model with the signi cance of random movement and thermodi usion of nanoparticles. Shape-preserving transformations have been employed to convert the transport equations into solvable forms. An innovative analytical tactic, namely, homotopy analysis method, has been adopted to nd the solution of the modeled problem. Behaviors of pertinent parameters on thermal and concentration pro les have been discussed through various graphs. Inspection of heat/mass transport against appropriate varieties of pertinent parameters has been made and explained physically. ermal pro le is augmented with the higher estimations of space and temperature-dependent heat source/sink links. Concentration pro le is diminished with the augmentation of higher-order chemical reaction parameter. Sherwood number is improved with the estimation of 0 ≤ β t ≤ 100 and is reduced with the growth of 0 ≤ β c ≤ 100. Nusselt number is declined with the upgraded amounts of 0 ≤ N b ≤ 3 and 0 ≤ N t ≤ 5.

In this exploration, we decided to investigate the significance of prescribed thermal conditions on unsteady 3D dynamics of waterbased radiative hybrid nanofluid with the impact of cylindrical-shaped nanosized particles (alumina (Al 2 O 3) and titania (TiO 2)). For physical relevancy, the impact of the Lorentz force is also included. The combination of suitable variables has been used to transform the transport equations into the system of ordinary differential equations and then numerically solved via the Keller-Box approach. Graphical illustrations have been used to predict the impact of the involved parameters on the thermal setup. Convergence analysis is presented via the grid independence approach. Skin frictions and local Nusselt numbers against various choices of involved parameters are plotted and arranged in tabular forms. It is observed through the present investigation that temperature distribution is increased with the higher choices of radiation parameter 0:0 ≤ R d ≤ 2:0 and decreased with the improvement in the choices of temperature maintaining indices (i.e., −2:0 ≤ r, s ≤ 2:0). Moreover, the thermophysical properties except specific heat for hybrid nanofluid are improved with the involvement of cylindrical-shaped nanoparticles. The temperature of the hybrid nanofluid is observed to be higher for variable thermal conditions as compared to uniform thermal conditions. Outfalls for a limited version of the report have been compared with a previous published paper.

In this communication, bidirectional flow of Casson nanomaterial driven by an unsteady moveable surface in the region of boundary layer is analyzed. Moreover, the significances of porous space, magnetic field, prescribed surface temperature (PST), and prescribed surface heat flux are also incorporated. Furthermore, the aspects of Brownian motion and thermophoresis are also comprised through Buongiorno nanofluid model. Governing equations are firstly transformed into system of ordinary differential equations by using a suitable combination of variables, and then computational assessment is made through Keller-Box method. Graphical illustrations for temperature distribution, concentration distribution, local Nusselt number and local Sherwood number against escalating amounts of pertinent parameters are presented. It is observed that escalating amounts of unsteady parameter and temperature controlled indices reduce the temperature distribution, as well as the concentration distribution. It is also observed that increasing amounts of Casson parameter enhances the rate of heat transfer, and reduces the rate of mass transfer for PST case. Finally, a comparison benchmark for limited case has been presented to validate the present methodology.

Bi-directional unsteady flow of nanomaterial has been explored in the existence of Cattaneo-Christov double diffusion. Brownian motion, thermophoresis and heat sources namely PST (prescribed surface temperature) and PHF (prescribed heat flux) are included. The obtained nonlinear systems are computationally solved by employing Keller box method. Graphs have been created for temperature and concentration fields for distinct amounts of involved parameters. Tables have been generated and discussed for skin friction, heat and mass transfer rates. It is assessed that temperature and concentration are reducing fields of thermal and relaxation factors for prescribed heat flux. Obtained effects are linked with the formerly published results and found favorable agreement.

Bi-directional unsteady flow of nanomaterial has been explored in the existence of Cattaneo-Christov double diffusion. Brownian motion, thermophoresis and heat sources namely PST (prescribed surface temperature) and PHF (prescribed heat flux) are included. The obtained nonlinear systems are computationally solved by employing Keller box method. Graphs have been created for temperature and concentration fields for distinct amounts of involved parameters. Tables have been generated and discussed for skin friction, heat and mass transfer rates. It is assessed that temperature and concentration are reducing fields of thermal and relaxation factors for prescribed heat flux. Obtained effects are linked with the formerly published results and found favorable agreement.

The present explore effort addresses the impact of radiation on MHD stream of an incompressible nano fluid due to an elongating elongating piece by warm and mass fluxes border situation. Similarity transformations are applied to attain the self-similar equations which are then solved numerically by means of shooting procedure alongside by means of 4th order Runge-Kutta method. Features of a variety of sundry constraints on the non-dimensional stream, thermal, nanoparticle volume fraction, local Nusselt & local Sherwood figures are visualized. Moreover the numerical values of friction factor, local Nusselt and Sherwood figures are also computed and analyzed.

This article investigates the magnetohydrodynamic flow of third grade nanofluid with thermophoresis and Brownian motion effects. Energy equation is considered in the presence of thermal radiation and viscous dissipation. Rosseland's approximation is employed for thermal radiation. The heat and concentration flux conditions are taken into account. The governing nonlinear mathematical expressions of velocity, temperature and concentration are converted into dimensionless expressions via transformations. Series solutions of the dimensionless velocity, temperature and concentration are developed. Convergence of the constructed solutions is checked out both graphically and numerically. Effects of interesting physical parameters on the temperature and concentration are plotted and discussed in detail. Numerical values of skin-friction coefficient are computed for the hydrodynamic and hydromagnetic flow cases.

In this paper stagnation boundary layer flow of nanofluid with mixed convection heat and mass transfer with Electrical Magneto Hydrodynamic (EMHD) effects over a second-order momentum slip boundary condition have been mathematically analysed. The governing equations are transformed by similarity variable and the problem becomes coupled third-order nonlinear coupled differential equations. We use fourth-order Runge Kuta method and shooting technique to find the solution. The effect of second-order momentum slip condition with linear thermal slip condition has determined. Variation of all nano energy conversion parameters depends on different factors has shown graphically. Some of the parameters possesses dual solution at different values of second-order velocity slip parameter (β1&β2).

The fractional-order differential equations that exist in the field of science and engineering have been studied in this paper. The 2D fluid flow problems are recommended for the classical- and fractional-order analysis. It has been found that the nonlinear fractional-order problems are more realistic than the classical models to describe the proposed flow problems since the behavior of the stress of the model problem is not linear. The similarity variable in this study has been used in fractional form to transform the modeled governing equations from partial PDEs. The acquired equations are the nonlinear ordinary differential equations (ODEs) in fractional form. The electromagnetic field has also been imposed into the fluid motion to calculate the embedded constraints in the case of classical and noninteger orders. The nonlinear problems are attempted using the FDE-12 technique. The important physical phenomena including Nusselt number and skin friction are also calculated in case ...

Fins are commonly utilized to enhance (dissipate) heat in various engineering systems that include heat exchangers. In the present investigation, the impact of multi-boiling and thermo-geometric factors on a convective-radiative rectangular porous fin subjected to the temperature-dependent thermal conductivity of linear and non-linear variations is discussed extensively. The governing equations describing the problem were formulated with the aid of Darcy law. Similarity variables were employed to reduce the models to non-dimensional form. The solution of the governing dimensionless equation is approximated using the RK4 and spectral local linearization methods. Before parametric analysis, the agreement between the two numerical methods was established. Findings reveal that the non-linear variation of thermal conductivity shows better thermal efficiency than the linear variation. An improvement in the multi-boiling heat transfer parameter retards the temperature distribution of the fin. Furthermore, increasing the thermo-geometric parameter will result in a progressive decrease in the temperature of the fin. The results obtained in this work will aid in the design of heat exchangers and other heat transfer equipments.

Fins are commonly utilized to enhance (dissipate) heat in various engineering systems that include heat exchangers. In the present investigation, the impact of multi-boiling and thermo-geometric factors on a convective-radiative rectangular porous fin subjected to the temperature-dependent thermal conductivity of linear and non-linear variations is discussed extensively. The governing equations describing the problem were formulated with the aid of Darcy law. Similarity variables were employed to reduce the models to non-dimensional form. The solution of the governing dimensionless equation is approximated using the RK4 and spectral local linearization methods. Before parametric analysis, the agreement between the two numerical methods was established. Findings reveal that the non-linear variation of thermal conductivity shows better thermal efficiency than the linear variation. An improvement in the multi-boiling heat transfer parameter retards the temperature distribution of the fin. Furthermore, increasing the thermo-geometric parameter will result in a progressive decrease in the temperature of the fin. The results obtained in this work will aid in the design of heat exchangers and other heat transfer equipments.

The present article is intended to focus on the combined influence of Hall currents and thermal radiation on the magnetohydrodynamic flow of a nanofluid through a vertical rotating channel. The nanofluid is prepared by choosing the copper nanoparticles and water as a base fluid. The impact of mixed convection is also considered. The solution to the formulated problem is computed through a mathematical software Matlab. The physical interpretation of the problem is performed by preparing the graphical results. Consequences manifest that a decline in both primary and secondary velocity is produced via a rise in the strength of Lorentz forces. However, the disparate impact of the Hall parameter on primary and secondary velocity is noticed. Increasing the angular velocity of the vertical channel raises both velocity profiles. The tangent phase of the rate of heat transfer at the left vertical wall significantly decays for increment in the radiation parameter; however, it is directly related to the Prandtl number and the frequency parameter. Further, the temperature is intensified for a rise in the Prandtl number and the frequency parameter, and it decays for an increase in radiation parameters.

Fins are commonly utilized to enhance (dissipate) heat in various engineering systems that include heat exchangers. In the present investigation, the impact of multi-boiling and thermo-geometric factors on a convective-radiative rectangular porous fin subjected to the temperature-dependent thermal conductivity of linear and non-linear variations is discussed extensively. The governing equations describing the problem were formulated with the aid of Darcy law. Similarity variables were employed to reduce the models to non-dimensional form. The solution of the governing dimensionless equation is approximated using the RK4 and spectral local linearization methods. Before parametric analysis, the agreement between the two numerical methods was established. Findings reveal that the non-linear variation of thermal conductivity shows better thermal efficiency than the linear variation. An improvement in the multi-boiling heat transfer parameter retards the temperature distribution of the fin. Furthermore, increasing the thermo-geometric parameter will result in a progressive decrease in the temperature of the fin. The results obtained in this work will aid in the design of heat exchangers and other heat transfer equipments.

A typical base fluid such as water, oil or glycol is poor conductor of heat due to deficient thermo-physical properties. This deficiency is normally addressed by saturation of thermally strong conductive metallic nanoparticles such as Fe, Ti, Hg, Cu, Au into the base fluid resulting a stronger thermal conductivity, electric conductivity, heat and mass flux of the so formulated nanofluid. Nanoparticles having a diameter size less than 100 nano-meter are preferred in this formulation because these nano-sized particles stay suspended into the base fluid for a longer time-period. This communication aims to investigate the salient features of a nanofluid flow along a radiative Riga plate using Powell-Eyring model and convective boundary conditions. The flow model involves the effect of first order chemical reaction as well as the Brownian motion diffusion and Thermophoresis effects. Governing PDEs are transformed into ODEs using suitable transformations. HAM is applied for convergent series solutions to the boundary value problem. Impact of various parameters including Brownian motion, Thermophoresis, modified Hartman number, Lewis and Prandtl number on flow profiles is analyzed graphically. Parameters of physical interest like Skin-friction, Nusselt and Sherwood numbers are illustrated through numerical data. Effect of modified Hartman number is significant

This article investigates the magnetohydrodynamic flow of third grade nanofluid with thermophoresis and Brownian motion effects. Energy equation is considered in the presence of thermal radiation and viscous dissipation. Rosseland's approximation is employed for thermal radiation. The heat and concentration flux conditions are taken into account. The governing nonlinear mathematical expressions of velocity, temperature and concentration are converted into dimensionless expressions via transformations. Series solutions of the dimensionless velocity, temperature and concentration are developed. Convergence of the constructed solutions is checked out both graphically and numerically. Effects of interesting physical parameters on the temperature and concentration are plotted and discussed in detail. Numerical values of skin-friction coefficient are computed for the hydrodynamic and hydromagnetic flow cases.

In this study, an incompressible two-dimensional Oldroyd-B nanofluid steady flow past a stretching sheet considering the outcomes of magneto-hydrodynamics (MHD) and porous medium with magnetic, electrical, and thermal radiation effects is investigated. Using a similarity transformation, the governing equations in the form of partial differential equations (PDEs) are converted into a nonlinear ordinary differential equations (ODEs) system. The acquired system is numerically solved by the finite element method (FEM). The effects of physical parameters like Deborah numbers “ β 1 ” and “ β 2 ”, Brownian motion “ N b ”, thermophoresis parameter “ N t ”, Prandtl parameter “ Pr ”, Lewis number “ L e ”, thermal conductivity “ k ”, dynamic viscosity “μ”, magnetic and electric effects as “ M ” and “ E 1 ”, and thermal radiation effect “ Rd ” on the flow are studied in detail. For higher N b values, regional Nusselt numbers are increasing in magnitude. The local Sherwood number’s size rises fo...

The nonlinear convective flow of Eyring-Powell nanofluid using Catteneo-Christov model with heat generation or absorption term and chemical reaction rate over nonlinear stretching surface is analyzed. The simultaneous nonlinear partial differential equations governing the boundary layer flow are transformed to the corresponding nonlinear ordinary differential equations using similarity solution and then solved using Galerkin finite element method (GFEM). The impacts of pertinent governing parameters like Brownian diffusion, thermophoresis, mixed convection, heat generation or absorption, chemical reaction rate, Deborah numbers, Prandtl number, magnetic field parameter, Lewis number, nonlinear stretching sheet, and Eyring-Powell fluid parameters on velocity field, temperature, and nanoparticle concentration are given in both figures and tabular form. The result shows that the rise in chemical reaction rate will improve mass transfer rate and reduce heat transfer rate and local buoyan...

In this exploration, we decided to investigate the significance of prescribed thermal conditions on unsteady 3D dynamics of water-based radiative hybrid nanofluid with the impact of cylindrical-shaped nanosized particles (alumina ( Al 2 O 3 ) and titania ( Ti O 2 )). For physical relevancy, the impact of the Lorentz force is also included. The combination of suitable variables has been used to transform the transport equations into the system of ordinary differential equations and then numerically solved via the Keller-Box approach. Graphical illustrations have been used to predict the impact of the involved parameters on the thermal setup. Convergence analysis is presented via the grid independence approach. Skin frictions and local Nusselt numbers against various choices of involved parameters are plotted and arranged in tabular forms. It is observed through the present investigation that temperature distribution is increased with the higher choices of radiation parameter 0.0 ≤ R ...

An analysis is carried out to investigate the two dimensional flow of an electrically conducting Newtonian fluid over a permeable stretching cylinder. The main objective of this study is to analyze the thermo-physical properties on the axisymmetric flow with Prandtl variable in the presence of internal heat generation/absorption. Unlike typical studies, the temperature dependent physical properties are discussed in highly coupled velocity and temperature fields. The solution of the problem is computed numerically with Chebyshev Spectral Newton’s Iterative Scheme (CSNIS) and results are shown through graphs for various emerging parameters. Moreover the exchange of energy from the surface to the fluid is examined through variations in Nusselt number. To verify our results, comparisons are made with the published studies which provides authentication of our present results and numerical scheme.

Little is known in the literature about the concept of nonuniform heat source/sink and higher-order chemical reaction for the dynamics of Oldroyd-B nanoparticles. Therefore, the present article addresses the nonuniform heat source/sink and higher-order chemical reaction features in nonlinear mixed convection bidirectional MHD dynamics of Oldroyd-B nanoparticles with thermal radiation aspects through porous space. Stratification effects for both the temperature and concentration setups are also used in the mathematical model with the significance of random movement and thermodiffusion of nanoparticles. Shape-preserving transformations have been employed to convert the transport equations into solvable forms. An innovative analytical tactic, namely, homotopy analysis method, has been adopted to find the solution of the modeled problem. Behaviors of pertinent parameters on thermal and concentration profiles have been discussed through various graphs. Inspection of heat/mass transport a...

In this study, we discussed the enhancement of thermal conductivity of elasticoviscous fluid filled with nanoparticles, due to the implementation of radiation and convective boundary condition. The flow is considered impinging obliquely in the region of oblique stagnation point on the stretching surface. The obtained governing partial differential equations are transformed into a system of ordinary differential equations by employing a suitable transformation. The solution of the resulting equations is computed numerically using Chebyshev spectral newton iterative scheme. An excellent agreement with the results available in literature is obtained and shown through tables. The effects of involving parameters on the fluid flow and heat transfer are observed and shown through graphs. It is importantly noted that the larger values of Biot number imply the enhancement in heat transfer, thermal boundary layer thickness, and concentration boundary layer thickness.

In this paper stagnation boundary layer flow of nanofluid with mixed convection heat and mass transfer with Electrical Magneto Hydrodynamic (EMHD) effects over a second-order momentum slip boundary condition have been mathematically analysed. The governing equations are transformed by similarity variable and the problem becomes coupled thirdorder nonlinear coupled differential equations. We use fourth-order Runge Kuta method and shooting technique to find the solution. The effect of second-order momentum slip condition with linear thermal slip condition has determined. Variation of all nano energy conversion parameters depends on different factors has shown graphically. Some of the parameters possesses dual solution at different values of second-order velocity slip parameter (1 & 2).

Our main emphasis in this manuscript was to scrutinize the aspects of activation energy for magnetized Cross nanofluid subjected to cylindrical surface. Formulation for energy expression is developed through heat sink-source phenomenon. More specifically, Velocity of Cross liquid is deliberated by considering infinite shear rate viscosity and Lorentz's force. The considered Cross nanoliquid expression (Buongiorno relation) comprises thermophoretic and Brownian movement mechanisms. Moreover, Chemical processes are deliberated subjected to appliance of activation energy. Bvp4c algorithm is implemented to tackle the nonlinear structure. Outcomes for Sherwood number, Nusselt number, skin fraction, temperature, concentration and velocity are presented in this manuscript. Our results revealed that temperature of Cross nanoliquid intensifies for larger thermophoretic parameter. Moreover, nanoliquid concentration dwindles for greater estimation of activation energy parameter.

An unsteady double diffusive mixed convection boundary layer flow over a vertically stretching sheet in the presence of suction/injection is investigated in this paper. The governing partial differential equations are reduced by applying suitable transformations to a set of nonlinear ordinary differential equations, which is solved by the Keller box method. The influence of various flow parameters on the velocity, temperature, and species concentration profiles of the fluid is studied. The effect of some problem parameters on the skin friction coefficient in the presence of suction/injection is considered.

This article addresses 3D flow of Oldroyd-B liquid in the presence of nanomaterials induced by a linearly extendable surface. Analysis is carried out in rotating frame. The novel characteristics in regards to Brownian dispersion and thermophoresis are retained. Heat transfer via convective process is considered. Developed constraint with zero nanoparticles flux at boundary is employed. Problem formulation is made for boundary-layer and low magnetic Reynolds parameter approximations. Suitable nondimensional variables lead to strong nonlinear ordinary differential system. The strongly nonlinear differential expressions are solved through optimal homotopy analysis method. The optimal arrangement articulations of velocities, temperature, concentration and transfer of heat rate (local Nusselt number) are explained by means of plots by utilizing the different estimations of included variables. It is noticed that heat transfer rate significantly enhances for the higher values of Biot and Prandtl numbers.

This research paper intends to investigate the 3D flow of Prandtl liquid in the existence of improved heat conduction and mass diffusion models. Flow is created by considering linearly bidirectional stretchable sheet. Thermal and concentration diffusions are considered by employing Cattaneo-Christov double diffusion models. Boundary layer approach has been used to simplify the governing PDEs. Suitable nondimensional similarity variables correspond to strong nonlinear ODEs. Optimal homotopy analysis method (OHAM) is employed for solutions development. The role of various pertinent variables on temperature and concentration are analyzed through graphs. The physical quantities such as surface drag coefficients and heat and mass transfer rates at the wall are also plotted and discussed. Our results indicate that the temperature and concentration are decreasing functions of thermal and concentration relaxation parameters respectively.

In the present work comparative study of thermal analysis of a rectangular block (1x1x1unit³) is done by taking different material (i.e. Copper, Steel, Brass) for the conduction and convective boundary condition. Best material is selected under the boundary condition, by using FEM based software (ANSYS 11). Temperature and thermal heat flux contours are plotted for different materials. The result is based on temperature, thermal heat flux variation in the material, contour of temperature distributions and heat flux.

Present work addresses the heat/mass transfer mechanism for higher order mixed convective flow of Williamson nanofluid impinging over a nonlinear bidirectional elongating surface with power-law heat/mass fluxes. For scientific relevancy, we also investigated the combined influence of chemical reaction, nonlinear emission of radiation and hydro-magnetic environment on the flow pattern. Novel combination of scaling transformations has been instigated to metamorphose the principal equations into a set of highly nonlinear coupled ordinary-differential equations and then Keller-Box method (KBM) has been utilized to attain the solution of modeled problem. Convergence of the solution has been discussed via grid independence methodology. Controls of sundry dominant parameters on temperature and concentration variations have been discussed via various plots, whereas variations in heat/mass transfer rates are explained through tabular arrangements. Our analysis shows that the Nusslet number and the Sherwood number are developed up-to 38% with the positive estimations of power-law index 0:1 n 0:5, whereas thermal gradient h 0 ð0Þ is reduced up-to 5% with the appraisal of thermomigration of nanoparticles 0:4 N t 0:8. Rate of heat transference is improved up-to 24% and the rate of mass transference is improved up-to 5% with the inclusion of nonlinear thermal radiation aspects 1:1 h w 1:5. The validation of the solution has been made by comparing the results for restricted cases with the available literature.

The increasing global demand for energy necessitates devoted attention to the formulation and exploration of mechanisms of thermal heat exchangers to explore and save heat energy. Thus, innovative thermal transport fluids require to boost thermal conductivity and heat flow features to upsurge convection heat rate, and nanofluids have been effectively employed as standard heat transfer fluids. With such intention, herein, we formulated and developed the constitutive flow laws by utilizing the Rossland diffusion approximation and Stephen’s law along with the MHD effect. The mathematical formulation is based on boundary layer theory pioneered by Prandtl. Governing nonlinear partial differential flow equations are changed to ODEs via the implementation of the similarity variables. A well-known computational algorithm BVPh2 has been utilized for the solution of the nonlinear system of ODEs. The consequence of innumerable physical parameters on flow field, thermal distribution, and soluta...

In this paper, a steady two-dimensional Magnetohydrodynamic (MHD) mixed convection stagnation point flow of an incompressible, viscous and electrically conducting micropolar fluid toward a stretching/shrinking vertical surface with prescribed surface heat flux is investigated. The effects of induced magnetic field and the radiative heat flux are taken into account. The transformed differential equations are solved numerically by a finite-difference scheme, known as Keller-box method. The results for skin friction, heat transfer and induced magnetic field coefficients are obtained. The velocity, microrotation and temperature distribution for various parameters are shown graphically. The present results are compared with existing results in literature and establish to be in good conformity.

This article is concerned with the two-dimensional flow of Powell-Eyring fluid with variable thermal conductivity. The flow is caused due to a stretching cylinder. Temperature dependent thermal conductivity is considered. Both numerical and analytic solutions are obtained and compared. Analytic solution is found by homotopy analysis method. Numerical solution by shooting technique is presented. Discussion to different physical parameters for the velocity and temperature is assigned. It is observed that the velocity profile enhances for larger magnetic parameter. It is also further noted that for increasing the value of Prandtl number temperature profile decreases.

The nonlinear convective flow of Eyring-Powell nanofluid using Catteneo-Christov model with heat generation or absorption term and chemical reaction rate over nonlinear stretching surface is analyzed. The simultaneous nonlinear partial differential equations governing the boundary layer flow are transformed to the corresponding nonlinear ordinary differential equations using similarity solution and then solved using Galerkin finite element method (GFEM). The impacts of pertinent governing parameters like Brownian diffusion, thermophoresis, mixed convection, heat generation or absorption, chemical reaction rate, Deborah numbers, Prandtl number, magnetic field parameter, Lewis number, nonlinear stretching sheet, and Eyring-Powell fluid parameters on velocity field, temperature, and nanoparticle concentration are given in both figures and tabular form. The result shows that the rise in chemical reaction rate will improve mass transfer rate and reduce heat transfer rate and local buoyan...

An amalgamation of base fluid (usually oil, water, ethylene, glycol, glycerol, etc.) and metallic tiny particles (usually Cu, SiO2, Al2O3, etc.) having diameter less than 100 nm is designated as nanofluid. In last few decades, nanofluids have gained an incredible height owing to their industrial importance and frequent applications in thermal insulations, thermal-power plants, agriculture, purification of oil, sterilization processes, powder technology, automatic transmission of liquids, thermo-siphons, energy preservation, nano-emulsification, nanolubricants, nanofluid detergent, modulators, nuclear reactors and hydrothermal conduction, etc.  Bidirectional flows of various fluids impinging over expanding surfaces with the inclusion of nano-sized/tiny particles have a variety of significances in engineering and industrial sectors. The insertion of nano-sized/tiny particles into base liquids enhances the ability of thermal conductivity and usually these are used to overcome the poor thermal enactment of base liquids.
Boundary layer flows impinging over an expanding surface have a prominent role in metal and plastic industries, like: thinning and annealing of wires, manufacturing of rubber and plastic sheets, condensation process along a liquid film, drawing of stretchable sheets and aerodynamic extrusion of plastic films. Moreover, coating of sheets with tiny particles has an indispensable role in the enhancement of heat conveying phenomenon. Thus, we decided to investigate the boundary layer flows with the involvement of tiny particles. It is further inspected that steady bidirectional flow problems are much investigated as compared to the unsteady bidirectional flow problems. In the view of such facts, the core objective of present thesis is to explore bidirectional flow problems of nanofluids affecting over an unsteady elongating obstacle with the features of prescribed heat sources. The present thesis is structured as follows:
Chapter one consists on the literature review and fundamental concepts of liquid mechanics. Mathematical forms of the law of mass, momentum, energy and concentration conservations in Cartesian frame are presented. Boundary layer equations for various liquids (Newtonian, Carreau, Prandtl and Eyring-Powell) in the view of unsteady bidirectional flow are presented, mathematically. Details of nanofluid models and dimensionless numbers are also discussed in this chapter. Basic concept of Keller-Box method for numerical simulations, its advantages and coding procedure through an example from literature are also included in this chapter.
Chapter two explores the unsteady dynamics of a nanomaterial impinging over a bidirectional elongating obstacle with the features of porous medium, magnetic environment and internal heat production/consumption. Random movement and thermodiffusion of tiny/nano-sized particles are also in action. The characteristic of prescribed surface temperature (PST) is also used to maintain the surface temperature. The modeled transport equations are converted into dimensionless forms with the help of similarity transformations before finding the numerical solution via Keller-Box method. Comparison is also constructed with the previously published work and outcomes are in decent acceptance with antecedent published outcomes. The results for reduced Nusselt and Sherwood numbers against parametric values of involved constraints are described through tables. Thermal and concentration fields are discussed and anatomized through several graphical illustrations. The contents of this exploration are published in Special Topics and Reviews in Porous Media: An International Journal 10 (2019) 457-473.
Chapter three discloses bidirectional flow of a nanomaterial with unsteady Cattaneo-Christov double diffusion in the environment of variable thermal conditions. Cattaneo-Christov fluxes are used to investigate the thermal and concentration relaxations. Novel features of random movement and thermo-migration of tiny-sized particles are also retained. Similarity conversions are used to create the system of non-linear ordinary differential equations. The acquired/obtained nonlinear system is computationally tackled by recalling Keller-Box method. Graphical illustrations have been prepared for thermal and concentration setups for distinct choices of impacted constraints. Tables have been formed and deliberated for coefficients of skin friction, temperature gradient and concentration gradient. Obtained outcomes are connected with the formerly available outcomes and found promising agreement. The contents of this investigation are published in Results in Physics 15 (2019) 102581.
Chapter four presents the relationship between unsteady Cattaneo-Christov double diffusion, random movement and thermodiffusion of tiny-sized particles in the existence of convectively heating and zero-mass flux at the bidirectionally stretchable wall. Combination of similarity expressions is implemented to transform the prevailing partial differential equations into ordinary differential equations and then solved, numerically, by using Keller-Box simulation technique. Thermal and concentration phases are discussed with the influences of involved constraints, graphically. The contents of this exploration are recently published in Ain Shams Engineering Journal (2021).
Chapter five describes the unsteady bidirectional flow of magneto-Carreau nanomaterial impacting over an expanding surface with the aspects of thermal radiation and variable thermal conditions. An appropriate set of transforming variables is adopted to metamorphose the transport equations into ordinary differential equations and then simulated via Keller-Box method. Features of the involved constraints for thermal, concentration and velocity phases are graphically addressed. Simulations of effective drag forces, thermal gradient and concentration gradient have been made through tabular arrangements and explained in detail. A marvelous agreement is found by comparing the outcomes with the available reported work for a limited situation.  The contents of this assessment are published in Heat Transfer 49 (2020) 3456-3476.
Chapter six frames an unsteady dynamics of a magnetized Eyring-Powell nanofluid impacting over the bidirectional expanding obstacle with the features of prescribed thermal conditions and radiation aspects. Again, transport equations are transmuted into ordinary differential equations with the proper arrangement of similarity variables and then the obtained set of equalities has been solved, numerically. Various tables and plots have been generated to illustrate the rate of heat as well as rate of mass transferences. 
Finally, the whole parametric exploration has been validated through a comparison benchmark. The contents of this inspection are published in Case Studies in Thermal Engineering 21 (2020) 100689.
Chapter seven describes the computational study concerning with the unsteady flow of Eyring-Powell magneto nanoliquid over a bidirectionally deformable surface. Transference of activation/Arrhenius energy is used in the improvement of binary-chemical-reaction. Nonlinear significance of thermal radiation is also incorporated in the transport equation of energy. Investigation has been carried out through convective Nield's boundary restrictions. Firstly, useful combination of variables has been implemented to metamorphose the transport equations into ordinary-differential-equations. Later on, Keller-Box approach has been adopted to simulate the physical problem. Physical interpretations of obtained results are also described for temperature and mass concentration distributions through various graphs. Rate of heat transportation has been explained through tabular data for acceptable ranges of involved engineering parameters. Convergence analysis and error estimations of the numerical solution are also presented through various mesh refinement levels of the computational domain. Finally, comparison benchmarks with the restricted cases have been presented for the validation of the results obtained through the present parametric investigation. The contents of this analysis are published in International Journal of Modern Physics C 31 (2020) 2050156.
Chapter eight determines the bidirectional dynamics of Prandtl nanofluid with the aspects of dissipation function, Ohmic hating and magnetic coating. Thermodiffusion and random movement of nano-sized/tiny particles are also explored with the help of Buongiorno model. Prescribed heat/mass flux conditions have been imposed at the stretchable obstacle. An appropriate set of scaling variables has been used to metamorphose the transport equalities into dimensionless forms and then simulated via Keller-Box approach. The features of miscellaneous arising constraints on thermal and temperature phase have been addressed through various plots, while tabular arrangements have been used to inspect the rate of heat/mass transferences. The efficiency of the applied method has been predicated by incorporating error analysis. The contents of this study are published in Heat Transfer 49 (2020) 4801-1819.

The present communication addresses MHD radiative nanomaterial flow of Ree-Eying fluid between two coaxially rotating disks. Both disks are stretchable. Buongiorno model is used for nanofluids. Nanofluid aspects comprise random motion of particles (Brownian diffusion) and thermophoresis. MHD fluid is considered. Furthermore, dissipation, radiative heat flux and Ohmic heating effects are considered to model the energy equation. Total entropy rate is calculated through implementation of second thermodynamics law. Series solutions are developed through homotopy analysis method. Impacts of physical parameters on the velocity, temperature, entropy and concentration fields are discussed graphically. Skin friction coefficient and heat and mass transfer rates are numerically calculated through Tables 2-4. It is noticed that the velocity of liquid particles decreases versus higher estimations of magnetic parameter while it enhances via larger rotational parameter. Temperature field significa...

A recently invented algorithm known as the grasshopper optimization algorithm (GOA) was employed to optimize diesel engine performance and emission operated with ternary fuel (ethanol-biodiesel-diesel) blends. Using the regression modelling over these experimental results; the mathematical equations between the factors i.e., ethanol ratio (vol%), biodiesel ratio (vol%), engine load (Nm)) and the responses i.e., BSFC (g/kWh), BTE (%), HC (ppm), CO 2 (%), NOx (ppm), CO (%) were calculated. Grasshopper optimization algorithm was then run through these regression equations to calculate the optimum factor levels. The confirmation results suggested that the BTE was maximized and the other responses were minimized successfully. For the ANOVA results, under the 95% confidence level with α = 5% (=0.05), the p-value for all the regression models was less than 0.05, which indicated the significance of the regression models. In terms of the performance tests of the models, the regression models good fit the given observations with a low prediction error. The grasshopper optimization algorithm showed that ethanol-biodiesel-diesel blend in the ratio of 10%, 7.5%, 82.5% run at 7 Nm engine load gave the optimum results for diesel engine performance and emission characteristics. These findings have important implications for the potential of grasshopper optimization algorithm to improve engine performance and emission characteristics.

This article analyzes the Darcy Forchheimer 2D thin film fluid of nanoliquid. Flow of nanoliquid is made due to a flat unsteady stretchable sheet. In nanoliquids, nanomaterial is in form of CNTs (carbon nanotubes). Also, in present analysis, single walled carbon nanotubes (SWCNTs) are accounted as nanoparticles. The classical liquid 'water' is treated as based liquid. The flow in permeable region is characterized by Darcy-Forchheimer relation. Heat transport phenomena are studied from convective point of view. The transformation of partial differential set of equations into strong ordinary differential frame is formed through appropriate variables. Homotopy Analysis Method (HAM) scheme is executed for solving the simplified set of equations. In addition, a numerical analysis (ND-Solve) is utilized for the convergence of the applied technique. The influence of some flow model quantities like P r (Prandtl number), λ (unsteadiness factor), k (porous medium factor), F (Darcy-porous medium factor) on liquid velocity and thermal field are scrutinized and studied through sketches. Certain physical factors like f (0) (friction factor coefficient) and −θ (0) (rate of heat transport) are first derived and then presented through tables.

The purpose of this study is to examine the variables that influence both the Circular Economy (CE) and pro-environmental behavior in steel recycling manufacturing companies in Thailand. To develop and examine a model using structural equation modeling (SEM), we used a set of CE indicators proposed in previous studies. The variables identified in this study were determined through responses from stakeholders in the steel manufacturing and scrap management industries. A questionnaire-based survey was used to analyze the study target's (foundries, castings, and steel recycling manufacturers) behavior. Respondents contended that they value long-term relationships with suppliers. Quality control has a sizable impact on lean manufacturing techniques. Recycling costs and environmental concerns both have an effect on how behavioral control is perceived. It is critical for society to be educated about the circular economy and its practices. Environmental commitment is the most critical ...

Mixed convection peristaltic flow of Jeffrey nanofluid in a channel with compliant walls is addressed here. The present investigation includes the viscous dissipation, thermal radiation and Joule heating. Whole analysis is performed for velocity, thermal and concentration slip conditions. Related problems through long wavelength and low Reynolds number are examined for stream function, temperature and concentration. Impacts of thermal radiation, Hartman number, Brownian motion parameter, thermophoresis, Joule heating and slip parameters are explored in detail. Clearly temperature is a decreasing function of Hartman number and radiation parameter.

An attempt is made to examine the classical Von Karman flow problem for a second grade fluid by using a generalized non-similarity transformation. This approach is di_erent from that of Von Karman’s evolution of the flow in such a way that the physical quantities are allowed to develop non-axisymmetrically. The three-dimensional equations of motion for the second grade fluid are treated analytically yielding the derivation of the exact solutions for the velocity components. The physical interpretation of the velocity components, vorticity components, shear stresses and boundary layer thickness are also presented.

Offshore photovoltaic energy is possibly the most important future step in the harnessing of solar energy. Since no long-term offshore photovoltaic installation exists to date, various unknowns are still present, creating a research gap. For instance, floating structures will have some type of response to incoming waves. This response is highly dependent on the design of the floating structure. This response will have some effect on the insolation on offshore photovoltaic systems installed on floating structures. This research presents a simulation tool that would allow an offshore system designer to assess this effect in order to minimize it and thus, optimize the energy yield of the system. Furthermore, this simulation tool was verified with an experimental setup simulating sinusoidal wave responses and the results are presented in this research. Finally, a parametric analysis was performed taking days close to the 21 st of each month of the year for photovoltaic installations fac...

Unsteady electrohydrodynamic hybrid nanofluid Al 2 O 3 ‐ Cu / H 2 O past a convective heat stretched/shrinked sheet is examined. A stagnation point fluid flow with velocity slip constrains and heat source or sink is deliberated. The combined set of PDEs is translated into ODEs by including approved similarity transformations. HAM is applied for the solution to the obtained nonlinear system. The magnetic input factor, Prandtl number, electric field factor, Eckert number, heat source factor, and unstable factor are the governing parameters. The impact of these factors on the temperature and velocity profiles features of the problem is considered with explanation. Intensification in values of electric and magnetic fields parameters enhanced the heat transfer rate. The greater Prandtl number lessens the temperature. Amplification in temperature is perceived for Eckert parameter. The heat transferred rate of hybrid nanofluid in the entire domain increases as the heat source increases, wh...

A time-dependent convectional flow of a two-phase nanofluid over a rotating cone with the impact of heat and mass rates is elaborated in this article. The instability in the flow field is induced by the cone angular velocity that depends on the time. The Navier–Stokes self-similar solution and the energy equations are obtained numerically. Here, the achieved solution is not only for Navier–Stokes equations but also for the equations of the boundary layers. In this work, the concentration, Brownian motion, and thermal buoyancy effects have important significance. We have assumed viscous dissipation with heat-absorbing fluid. Similarity answers for spinning cones with divider temperature boundary conditions give an arrangement of nonlinear differential conditions that have been handled numerically. The MATLAB methodology BVP4C is used to resolve the reduced structure of nonlinear differential equations numerically. Observation for skin friction and Nusselt number is also taken into ac...

In this article, the Hall and ion-slip effects on a mixed convection flow of an electrically conducting nanofluid over a stretching sheet in a permeable medium have been discussed. Using the similarity transformations, the partial differential equations corresponding to the momentum, energy, and concentration equations are transformed to a system of nonlinear ordinary differential equations which are solved numerically using a spectral relaxation method (SRM). The effects of significant parameters on the velocities, temperature, and concentration profiles are analyzed graphically. Moreover, the results of the skin friction coefficients, local Nusselt number, and Sherwood number are determined numerically. The results of the analysis showed that the velocity profile in the flow direction increases with an increase in mixed convection parameter λ, Hall parameter βh, and ion-slip parameter βi, and it decreases with an increase in the magnetic parameter M. Furthermore, temperature and c...

Simultaneous impacts of non-linear radiation and magnetohydrodynamics in Marangoni convection nanoliquid are addressed Novel aspects of activation energy and space dependent heat source are addressed. Nanoliquid attributes Brownain movement and thermophoresis diffusion. NDSolve base shooting technique is employed for the numerical simulation. Aspects of various embedded variables are focused on velocity, heat and mass transport distributions via graphical interpretations. Moreover temperature gradient at the surface is estimated and analyzed. Our study identified that exponential based space heat source (ESHS) parameter significantly enhanced the thermal field. Activation energy and temperature difference parameters decrease the nanoparticles concentration. Moreover temperature gradient enhances for higher Marangoni ratio parameter, Hartmann number, dimensionless activation energy and thermophoresis parameter.