Keller-box Method Research Articles

Magnetohydrodynamic Effects in Mixed Convection of Ferrofluid Flow at Lower Stagnation Point on Horizontal Circular Cylinder

In this paper, mixed convection of ferrofluid containing magnetite, Fe3O4 with ferroparticles suspended in water at the lower stagnation point on a horizontal circular cylinder is investigated. The partial differential equation which derived from the transformation of the dimensional governing equation and non-similarity transformation with the consideration of the effect of magnetohydrodynamic (MHD) are solved numerically by using Keller-box method. The influences of an external magnetic field on ferrofluid flow and heat transfer are then discussed. The results showed that the viscosity depends on the ferroparticle volume fraction and ferrofluid temperature. The uniform magnetic field that produced Lorentz force acts as a determiner of the trend of fluid movement and has the tendency to control the cooling rate of the surface.

Boundary Layer Flow of Dusty Williamson Fluid with Variable Viscosity Effect Over a Stretching Sheet

Numerical investigation of the boundary layer flow of Williamson fluid with the presence of dust particles over a stretching sheet is carried out by taking into account the variable viscosity effect and Newtonian heating boundary condition. The genuinely two-phase flow model which has been proved to be compatible to present the mutual relationship between non-Newtonian fluid and solid particles is considered in this present study. To be precise, the governing equations are initially transformed into ordinary differential equations through formulation process before proceeding further with the numerical computation by using Keller-box method. The resulting equations are then programmed in Matlab software. The obtained numerical results are validated with existing study found in open literature and a good agreement is achieved. The influence of pertinent parameters on velocity and temperature profiles, skin friction coefficient together with Nusselt number is presented in graphical and tabular forms. Results revealed that the increasing Williamson parameter decreases the fluid velocity of both fluid and dust phases. It is expected that the present numerical results could conceivably help in predicting the boundary layer problem arising in two-phase flow in the future.

Modelling and numerical computation for flow of micropolar fluid towards an exponential curved surface: a Keller box method

The numerical analysis of MHD boundary layer non-Newtonian micropolar fluid due to an exponentially curved stretching sheet is developed in this study. In the energy equation effects of viscous dissipation are included. For the mathematical description of the governing equations curvilinear coordinates are used. By utilizing exponential similarity variables, the modelled partial differential equations (PDEs) are reduced into ordinary ones. The resultant non-linear ODEs are numerically solved with two methods shooting and Keller box method. The study reveals that the governing parameters, namely, radius of curvature, material parameter, magnetic parameter, Prandtl number and Eckert number have major effects on the fluid velocity, micro-rotation velocity, surface friction, couple stress and heat transfer rate. The results indicate that the magnetic field diminishes the fluid velocity inside the hydrodynamics boundary layer whereas it enhances the temperature inside the thermal boundary layer. Microrotation profile decreases near the surface, as the magnetic parameter and radius of curvature increases but far away behavior is opposite. The material parameter enhances the velocity and microrotation profile whereas, opposite behaviors is noticed for the temperature distribution. Obtained outcomes are also compared with the existing literature and the comparison shows a good agreement with existing studies.

Thermal analysis on Darcy‐Forchheimer swirling Casson hybrid nanofluid flow inside parallel plates in parabolic trough solar collector: An application to solar aircraft

As a try, this work has been focused in the way towards the effective contribution in the field of solar aviation using renowned nanotechnology. After realizing the causes and effects of traditionally used energy forms, the search of cost-efficient, eco-friendly, and most prominent renewable source leads us back to the solar utilities. Research era of solar radiation-powered aircraft has been in trend. Focusing on that, an efficient numerical model representing the flow and thermal aspects of a parabolic trough surface collector (PTSC) embedded on solar aircraft wings has been adopted for this study. As the first time with the note, an eminent and leading form of thermal efficient fluid of kind, the Casson hybrid nanofluid has been engaged with the expectations of enhanced performance in the solar aircraft wings. To test it, a trending reputable numerical scheme of the Keller-box method has been utilized and the parametrical studies were carried out. The upshots of those studies provide the affable proofs in favor of our expectations towards the improved solar wings with better thermal efficiency. The glimpse of those successes in the parametrical level has been showcased in the forms of tables and graphs. The lateral “x” direction significant about the inertial forces, suspended particle ratio, and skin resistance phenomena, while for the transverse fluidity in the “y” direction were has to be concern about the magnetic interactions, rotational coordinates, viscous nature of the fluid along with the porous states. The power of hybrid nanofluid combos was exposed in higher notes in a unique state of solar aircraft wings. Furthermore, the thermal efficiency of hybrid nanofluids over nanofluids got down to a minimal level of 6.1% and peaked up to 21.8%.

Entropy generation applications in flow of viscoelastic nanofluid past a lubricated disk in presence of nonlinear thermal radiation and Joule heating

Entropy generation is the loss of energy in thermodynamical systems due to resistive forces, diffusion processes, radiation effects and chemical reactions. The main aim of this research is to address entropy generation due to magnetic field, nonlinear thermal radiation, viscous dissipation, thermal diffusion and nonlinear chemical reaction in the transport of viscoelastic fluid in the vicinity of a stagnation point over a lubricated disk. The conservation laws of mass and momentum along with the first law of thermodynamics and Fick’s law are used to discuss the flow, heat and mass transfer, while the second law of thermodynamics is used to analyze the entropy and irreversibility. The numbers of independent variables in the modeled set of nonlinear partial differential equations are reduced using similarity variables and the resulting system is numerically approximated using the Keller box method. The effects of thermophoresis, Brownian motion and the magnetic parameter on temperature are presented for lubricated and rough disks. The local Nusselt and Sherwood numbers are documented for both linear and nonlinear thermal radiation and lubricated and rough disks. Graphical representations of the entropy generation number and Bejan number for various parameters are also shown for lubricated and rough disks. The concentration of nanoparticles at the lubricated surface reduces with the magnetic parameter and Brownian motion. The entropy generation declines for thermophoresis diffusion and Brownian motion when lubrication effects are dominant. It is concluded that both entropy generation and the magnitude of the Bejan number increase in the presence of slip. The current results present many applications in the lubrication phenomenon, heating processes, cooling of devices, thermal engineering, energy production, extrusion processes etc.

Numerical investigation of MHD radiative heat and mass transfer of nanofluid flow towards a vertical wavy surface with viscous dissipation and Joule heating effects using Keller-box method

In this paper, the effect of viscous dissipation and Joule heating on magneto-hydrodynamic (MHD) nanofluid flow towards a vertical sinusoidal wavy surface has been investigated numerically. The wavy surface is considered to be heated with uniform flux. Effects of radiation and nanoparticle volume concentration are also considered. The fluid flow over the surface geometry is designed through the highly non-linear partial differential equations, and governing equations must be transformed into a dimensionless non-similar system of equations to get a better solution by applying the highly efficient well-known Keller-box scheme. In this regard, the numerical solutions are obtained for a concentration of mass fluid, velocity profile, temperature profile, local skin friction number coefficient, local Nusselt number, and Sherwood number shown graphically. Connected parameters involve in the modelled problem such as heat generation parameter, radiative conduction flow parameter, the amplitude of the waviness of the surface, Prandtl number, Brownian motion parameter, Buoyancy ratio, magnetic parameter, Thermophoresis parameter and the Lewis number for momentum, temperature profile, and concentration profile of the fluid through different graphs are explained. In this process, certain iterations were performed for different step sizes in the direction of η=0.01 and x=0.005, respectively. So, keeping this in view to get the more accurate results with all parameters considered, 0≤M≤1, 0≤Rd≤1,0≤Q≤1,0.01≤α≤0.04, 0≤Le≤40, 0.1≤Nb≤0.7,0.1≤Nt≤0.7,0.1≤Nr≤0.7,0.1≤Ec≤0.4,0.1≤J≤0.4,. The obtained results illustrate that Nusselt number, Sherwood number, and velocity profile decreases respectively with the increase of waviness amplitude, but skin friction changes its behaviour to wavier. Also, temperature and concentration profiles increase respectively with enhancing the value of the amplitude of waviness. Finally, it is noted that Nusselt number and Sherwood number decrease for higher values of the Brownian parameter. But after a specific value of x, the trend of Sherwood number increases. However, the behaviour of skin friction decreases, but in a precise manner, it remains almost unchanged.

Curvilinear flow of micropolar fluid with Cattaneo–Christov heat flux model due to oscillation of curved stretchable sheet

Abstract This paper aims to investigate the transfer of heat phenomenon in a hydromagnetic time dependent flow of micropolar fluid across an oscillating stretchable curved surface by using the Cattaneo–Christov heat flux model, which considers thermal relaxation time. An elastic curved surface that stretches back and forth causes the flow situation. The flow equations are derived as nonlinear partial differential equations by incorporating a curvilinear coordinates system, which is then solved analytically via the homotopy analysis method (HAM). The accuracy of the derived analytical results is also examined by using a finite-difference technique known as the Keller box method, and it is found to be in strong agreement. The influences of various physical characteristics such as material parameter, magnetic parameter, thermal relaxation parameter, a dimensionless radius of curvature, Prandtl number and ratio of surface’s oscillating frequency to its stretching rate parameter on angular velocity, fluid velocity, pressure, temperature, heat transmission rate, and skin friction and couple stress coefficient are depicted in detail with the help of graphs and tables. Furthermore, for the verification and validation of the current results, a tabular comparison of the published data in the literature for the case of flat oscillating surface versus curved oscillating surface is carried out and found to be in good agreement.

Study on heat transfer aspects of solar aircraft wings for the case of Reiner-Philippoff hybrid nanofluid past a parabolic trough: Keller box method

Solar energy is the major source of heat energy from the sun having immense utilization in technologies like photovoltaic cells, solar energy plates, solar street lights, solar water pumping, and so on. How to maximize the heat energy which is later on used as electrical energy for many purposes in solar aircraft is the hot topic of research in this decade. The current paper is about the study of entropy generation analysis and hybrid nano-solid particles impact on parabolic trough surface collector (PTSC) located inside the solar aircraft wings. homo/heterogeneous reactions have been taken. Non-Newtonian Reiner-Philippoff model along with porous medium and Darcy-Forchheimer effects have been taken into account for the present study. The heat transfer performance of the solar aircraft wings has been enhanced with the inclusion of effects like hybrid nanoparticles, thermal radiations, variable thermal conductivity, and viscous dissipation. By utilizing the proper conversions the demonstrated partial-differential equations are renovated into ordinary-differential formulas and tackled numerically with the utilization of well established numerical scheme termed as Keller-Box method. The impact of dimensionless sundry parameters on velocity, shear stress, temperature, homo/heterogeneous as well as surface drag and heat transfer rates are sketched through tables and figures. A positive variation in thermal radiation, thermal conductivity, and viscous dissipation effects enhances the heat transfer rate inside the solar aircraft wings.

Numerical Solutions of Hybrid Nanofluids Flow Via Free Convection Over a Solid Sphere

The purpose of the existing study is to examine how heat transfer enables consolidated by variations in the basic advantages of fluids in the existence of free convection with the assistance of suspended hybrid nanofluids. Iron-graphene oxide suspended in water as a hybrid nanofluid flow on a solid sphere is also considered in this work. The partial differential equations are gotten, for this problem, by transforming the mathematical governing equations using similarity equations (stream function). These partial differential equations are solved numerically by Keller-Box method and programmed by MATLAB program. the acquired numerical results are in excellent agreement with the preceding literature results. Graphical results of the influence of the hybrid nanofluid parameters on some physical quantities regarded to examine the behavior of hybrid nanofluid flow were attained, and they proved that hybrid nanofluid flow represents a more essential role in the operation of heat transfer than a regular nanofluid flow.

A brief review of numerical methods for heat and mass transfer of Casson fluids

This work presents a brief review of various numerical methods applicable to Casson fluid by studying various types of research papers within the last 10 years. The past research conducted in different cases by various researchers is used as a primary source to know about numerical methods used to solve the governing equations. The review is helpful in searching the literature which explores the effect of the important controlling parameters on Casson flow profiles. Comparisons with the traditional methods are also given and the results are carefully verified. It can be observed that some old methods among all different type of numerical methods are stable and popular among researchers like Shooting Technique, Runge–Kutta method and Keller Box Method etc. Results are truly verified for each of such cases.

Heat Transmission Reinforcers Induced by MHD Hybrid Nanoparticles for Water/Water-EG Flowing over a Cylinder

The assumptions that form our focus in this study are water or water-ethylene glycol flowing around a horizontal cylinder, containing hybrid nanoparticles, affected by a magnetic force, and under a constant wall temperature, in addition to considering free convection. The Tiwari–Das model is employed to highlight the influence of the nanoparticles volume fraction on the flow characteristics. A numerical approximate technique called the Keller box method is implemented to obtain a solution to the physical model. The effects of some critical parameters related to heat transmission are also graphically examined and analyzed. The increase in the nanoparticle volume fraction increases the heat transfer rate and liquid velocity; the strength of the magnetic field has an adverse effect on liquid velocity, heat transfer, and skin friction. We find that cobalt nanoparticles provide more efficient support for the heat transfer rate of aluminum oxide than aluminum nanoparticles.

Numerical solution of micropolar fluid flow via stretchable surface with chemical reaction and melting heat transfer using Keller-Box method

The main theme of this research is to find the numerical results of stagnation point flow of micropolar fluid over a porous stretchable surface due to the physical effects of internal heat generation/absorption, melting heat transfer and chemical reaction via Keller-Box method (KBM). The graphs and tables are depicted and explained for various embedded parameters. The range of melting heat transfer parameter is 0≤M≤3, the range of chemical reaction parameter is 0≤Kr≤1 whereas the values of space-temperature dependent heat source/sink parameters lies in −0.4≤Q≤0.4 and −2≤Q∗≤2. The upshots of the current problem illustrate that at fluid-solid interface, rate of HMT (heat and mass transfer) declined on escalating the values of stretching parameter. Moreover, as the values of internal heat source/sink parameter increases, heat transfer rate declines at fluid-solid interface.

Diffusion-thermo Effects in Stagnation Point Flow of Second Grade Fluid past a Stretching Plate

Transmission of heat and mass in boundary layer flows over stretching surfaces play a significant role in metallurgy and polymer industry. In Current article the assisting and opposing flow of a second grade fluid towards a stretching sheet is analyzed to examine the heat and mass transfer in stagnation point boundary layer flow. Different flow parameters such as concentration, surface temperature and stretching velocity are supposed to variate linearly. The basic transport equations are transformed into non-linear ordinary differential equations by means of boundary layer approximation and similarity transmutations, which are then solved by employing nonlinear shooting (NLS) and Keller-box methods (KBM). These techniques are very useful for solving boundary-layer problems and are applicable to other general situations than that presented current study. The outcomes of velocity, temperature, concentration profile, skin-friction coefficient, heat and mass transfer coefficients are analyzed briefly in graphical and tabular formats. The mass transmission rate was found to be in direct relation with Schmidt number. Moreover, we predict that a rise in Prandtl number leads to a decline in temperature and thermal layer of boundary thickness for both supporting and contrasting flows. The outcomes of this article are important for the analysts in the field of second grade fluids. We believe that the article is very well prepared and the results are original and useful from both theoretical and application point of views.

Study of the MHD Flow of Casson Nanofluid in the Presence of Oxides Nanoparticles Based C2H6O2/H2O Under Constant Heat Flux Boundary Condition

In this study, a numerical investigation of free convection in magneto-hydrodynamic (MHD) Casson nanofluid flow with heat transfer over a stretching sheet and with constant heat flux boundary condition is done. A magnetic field is considered normal to the stretching sheet. Two types of nanoparticles with various conductivities, such as iron oxide and graphene oxide are suspended in ethylene glycol/water-based Casson nanofluid. An implicit finite difference scheme known as Keller-box method has been employed in order to solve the set of nonlinear partial differential equations resulted from the governing equations, namely continuity, momentum, and energy equations. The effects of the MHD Casson nanofluids parameters on the physical properties such that local skin friction coefficient, local Nusselt number, velocity, and temperature have been displayed and discussed. The study has revealed that ethylene glycol has higher velocity and less temperature than water with different values of nanoparticle volume fraction, magnetic, and Casson parameters with the existence of Fe3O4/GO nanoparticles. Comparisons with previous studies show that the proposed method is in good agreement with other results.

Keller box study for inclined magnetically driven Casson nanofluid over a stretching sheet: single phase model

The current analysis signifies the Casson nanofluid stream with inclined magnetohydrodynamics, viscous dissipation, joule heating, thermal radiation, particle shape factor effects over a moving flat horizontal surface with entropy analysis. This study presents a novel idea regarding the implementation of Tiwari and Das model on Casson fluid model by considering ethylene glycol as a base fluid. Tiwari and Das model considers nanoparticles volume fraction for heat transfer enhancement instead of the Buongiorno model which heavily relies on thermophoresis and Brownian diffusion effects for heat transfer analysis. The velocity slip and convective slip boundary conditions have been employed at the surface of the sheet. By utilizing the suitable transformations, the modeled PDEs (partial-differential equations) are renewed in ODEs (ordinary-differential equations) and treated these equations numerically with the help of classical Keller box method. Two different classes of nanofluids, Copper-Ethylene Glycol () and Molybdenum Disulfide-Ethylene Glycol () have been taken into considering for our analysis. The behaviour of surface drag coefficient and Nusselt number for the varied values of various sundry parameters is designed via tables. Our findings show that an increase in the Reynolds and Brinkman numbers increased the overall entropy of the system.

Computational single‐phase comparative study of a Williamson nanofluid in a parabolic trough solar collector via the Keller box method

A solar thermal collector collects heat by absorbing sunlight and converting this incoming radiation into useful thermal energy. The parabolic trough solar sollector (PTSC) is widely installed in concentrated solar power technology with a temperature range of 325-700 K. The thermodynamic efficiency of the PTSC can be intensified by using nanofluids owing to their enhanced thermophysical properties. In this paper, the two-dimensional flow over a stretched sheet in PTSC under Williamson nanofluid effects is explored. With the help of useful transformations, the partial differential equations arising from boundary layer equations of the Williamson nanofluid model are changed into nonlinear ordinary differential equations. The Keller box numerical scheme is then used to find the solution of the resulting nonlinear ordinary differential equations. Two nanofluids, copper-engine oil (Cu-EO) and alumina-engine oil (Al2O3-EO) are considered to address the performance analysis of PTSC in this study. It is observed that the parameters of the porous media reduced the heat transfer rate but increased the velocity gradient. In order to examine the effect on various performance parameters of the device, the nanoparticle concentration is also varied. With the increase in the Reynolds and Brinkman numbers, the entropy is found to increase. The thermodynamic performance of the Cu-EO nanofluid is remarkably better than that of the Al2O3-EO nanofluid under the same conditions. The thermal efficiency of Cu-EO over Al2O3-EO is seen with a minimum of 1.5% and a maximum of 9.1%. The theoretical simulations documented here may be effective in improving solar thermal energy systems.

Free Convection Boundary Layer Flow over a Horizontal Circular Cylinder in Al2O3-Ag/Water Hybrid Nanofluid with Viscous Dissipation

In this paper, the mathematical model of free convection boundary layer flow of horizontal circular cylinder immersed in Ag/Water nanofluid and Al2O3-Ag/Water hybrid nanofluid are considered. The governing non-linear partial differential equations are first transformed to a more convenient way before being solved numerically using the Keller-box method. The numerical values for the reduced Nusselt number and the reduced skin friction coefficient are obtained and illustrated graphically as well as temperature profiles and velocity profiles. Effects of the Prandtl number, Eckert number and nanoparticle volume fraction are analyzed and discussed. It is found that the Nusselt number for Al2O3-Ag/Water hybrid nanofluid is comparable with Ag/Water nanofluid with a reduction in skin friction coefficient. The preliminary results reports here are important as a reference in exploring the potential of hybrid nanofluid to reduce the production cost compared to the used of metal nanofluid.

Zero velocity regions near forward and rare points of circular cylinder: A heat and mass transfer study

This paper examines the unsteady two-dimensional boundary layer heat and mass transfer features of viscoelastic fluid flow near the neighborhood of forward and rare stagnation points. The heated circular cylinder is immersed in a viscoelastic fluid and the forward and rare stagnation points occurs near downward and upward directions of the cylinder respectively. The velocity near the surface of the cylinder is assumed to be zero and at very large distance from cylinder a uniform free stream velocity is spontaneously started in vertical direction. A non-similar transformation reduces the set of partial differential equations into non-similar partial differential equations. The transformed non-linear partial differential system is solved by using Keller box method. This technique shows well-behaved solutions for the steady-state (large time) and transient (small time) flows near the boundary layer region. Moreover, the body force present in this phenomena is in the form of convection which gives assisting (greater than zero) and opposing flows (less than zero). Parameters like the viscoelastic parameter, the convection parameter and the stagnation point parameter (forward and rare stagnation points) are found to control the flow field. Moreover, the Schmidt and the Prandtl numbers are used to control the concentration and temperature distributions. Features of several parameters on the non-dimensional concentration, temperature, local Nusselt and Sherwood numbers are viewed through graphs (2-d and 3-d) and tables. Contour images are plotted in order to visualize the three dimensional surface into two dimensional plane.

Chemical reaction impact on MHD natural convection flow through porous medium past an exponentially stretching sheet in presence of heat source/sink and viscous dissipation

This study investigates the viscous dissipation impact on free convection MHD flow through a porous medium over an exponentially stretching surface in presence of chemical reaction. The basic governing PDEs are converted into non-linear ODE's by using similarity transformations and then using the Keller-box method, numerical solutions are obtained. The flow features of boundary layers along with the bounding surface are identified and analysed using diagrams. It is noted that the increase in the Eckert number, Radiation and Magnetic parameter (M) increases the temperature profiles, while the increase in the chemical reaction parameter, porosity and Schmidt number decreases the concentration profile. To validate the results, a comparative study between the present study and previously published results for a particular case is conducted and good agreement is found between them.

Numerical Analysis of Mixed Convection Flow Past a Symmetric Cylinder with Viscous Dissipation in Viscoelastic Nanofluid

Research on the nanofluid becomes trending amongst researchers especially in the industrial and engineering field due to its important and extensive applications. Therefore, the present study aims to investigate numerically the impact of viscous dissipation conducted by sodium carboxymethyl cellulose (CMC-water) nanofluid containing copper nanoparticles at room temperature with convective boundary conditions (CBC). The Tiwari and Das model was selected in this study and the transformed boundary layer equations for momentum and energy subject to the appropriate boundary conditions were numerically solved by employing numerical scheme, namely the Keller-box method. The results were analysed in detail and presented graphically for the velocity, temperature, skin friction coefficient as well as the heat transfer coefficient. The obtained results indicated that there was no significant effect for velocity and temperature profiles when values of Eckert number increased. However, it is significant for skin friction and heat transfer coefficient profiles. In the meantime, the thermal conductivity of the fluid may increase by increasing the concentration of nanofluid.