Keller Box Finite Difference Method Research Articles

Convective Heat Transfer flow of MHD Hybrid Nanofluid over a Stretching sheet with velocity and thermal slip

In this study, we have investigated the convective heat transfer of a hybrid nanofluid with magnetohydrodynamic (MHD) properties over a permeable stretching sheet. The analysis takes into account the effects of velocity and thermal slip, and the hybrid nanofluid consists of two distinct nanoparticles—titanium dioxide (TiO2) and silver (Ag)—dispersed in water (H2O) as the base fluid. The use of Ag and TiO2 nano-composites is particularly significant in nano-biotechnology, especially in nano-medicine and cancer cell therapy, owing to their remarkable photocatalytic performances in handling pharmaceutical compounds and pollutants. The primary aim of this research is to explore how the behavior of the MHD hybrid nanofluid over a permeable stretching surface is influenced by velocity and thermal slip conditions. By employing appropriate similarity transformations, the governing partial differential equations (PDEs) are transformed into dimensionless nonlinear ordinary differential equations (ODEs), which are then analytically solved using the Keller-box finite difference method. The results indicate that the velocity profile decreases with an increase in velocity slip, while the temperature and concentration fields decrease with thermal slip. The study further analyzes the significant impact of various parameters, including velocity and thermal slip parameter, porosity parameter, magnetic parameter, Prandtl number, and Eckert number, on the velocity, temperature, and concentration profiles. These findings are presented in graphical and tabular form, showing excellent agreement with previous literature.

Radiation Effects on Inclined Magnetohydrodynamics Mixed Convection Boundary Layer Flow of Hybrid Nanofluids over a Moving and Static Wedge

Nowadays, hybrid nanofluids play an important role in heat transfer systems. They are a good alternative to increase the efficiency of heat transfer and save the energy. Thermal radiation and mixed convection flow of hybrid nanofluids past a permeable moving and stationary wedge were studied in this research. This research uses water as a base fluid to investigate the effects of silver (Ag) and magnesium oxide (MgO) nanoparticles. Similarity transformation techniques are used to convert the partial differential equations of hybrid nanofluids to ordinary differential equations, which is then solved numerically by applying the implicit finite difference Keller box method. The results of the research are illustrated graphically to show the behavior of velocity and temperature profiles, as well as skin friction and Nusselt number. Increasing the parameters of the aligned magnetic field, magnetic field interaction, mixed convection, and wedge angle parameter results in higher velocity profiles but lower temperature profiles. As the radiation parameter and the nanoparticle volume fraction increase, the temperature rises and the velocity decreases. With the exception of the radiation parameter, the skin friction and Nusselt number increase as the alignment angle of the magnetic field, the interaction of the magnetic field, the mixed convection, the wedge angle parameter, and the volume fraction of nanoparticle Ag and MgO rise. As a result of these findings, the velocity profiles and Nusselt numbers of moving wedges are higher, but the temperature profiles and skin friction are lower than those of stationary and moving against flow wedges. In addition, a comparison with previously published research is presented, with excellent agreement discovered. The results of this research will contribute to the field of knowledge in mathematics by bringing additional information for mathematician interested in future research on hybrid nanofluids.

Numerical analysis for the effects of boundary layer flow over an exponentially stretching cylinder embedded in a porous medium moving with nonlinear velocity

The objective of this study is to probe the effects of boundary layer flow over an exponentially stretching cylinder having nonlinear velocity placed in a porous medium. The modeled partial differential equations are transformed into nonlinear ordinary differential equations by applying suitable similarity transformations. A numerical solution is attained using the finite difference Keller–Box method. The physical parameters and skin friction coefficients are studied on the velocity profile. The fluid velocity and skin friction coefficient decrease with the increase in Re and κ, while a reverse behavior is reported for ɛ.

Multiple slip effects on nanofluid dissipative flow in a converging/diverging channel: A numerical study

AbstractA mathematical model is developed for viscous slip flow and heat transfer in water/Ethylene glycol‐based nanofluids containing metallic oxide nanoparticles, through a converging/diverging channel. We adopt the single‐phase Tiwari–Das model. The governing equations are transformed to a set of similarity differential equations with the help of similarity transformation, before being solved numerically using Maple 20. Validation of the velocity gradient and temperature solutions is achieved with the second‐order implicit finite difference Keller Box method. Further validation is included for the special case of no‐slip nanofluid flow in the absence of viscous heating. The effects of the parameters, namely velocity slip, thermal jump, channel apex angle, Eckert number, Prandtl number, Reynolds number, and nano‐particle volume fraction on velocity, temperature, skin friction, and heat transfer rate are investigated in detail. It is found that with increasing velocity slip, for water‐TiO2 and ethylene glycol‐TiO2 nanofluids, the channel bulk flow is decelerated whilst with greater solid (nanoparticle) volume and in the presence of momentum slip, the flow is also retarded. With the increasing semivertex angle, the channel flow is generally accelerated. An increase in divergent semiangle leads to decelerate the flow from the centerline for the core flow region, whereas near and at the channel wall, it results in a weak acceleration. Higher temperatures are achieved with greater thermal slip values, for both water‐TiO2 and ethylene glycol‐TiO2 nanofluids, whereas for greater nanoparticle volume fraction, temperatures are weakly decreased for water‐TiO2 whereas a more significant decrease is observed for ethylene glycol‐TiO2 nanofluid. With a greater diverging channel angle, a substantial decrease in temperatures is caused by greater Reynolds numbers, and the reverse effect is computed for the converging channel. The novelty of the current work is that it extends previous studies to include multiple slip effects and viscous heating (Eckert number effects), which are shown to exert a significant influence on heat and momentum transfer characteristics. The study is relevant to certain pharmaco‐dynamics devices (drug delivery), next‐generation 3D nanotechnological printers, and also nano‐cooling systems in energy engineering where laminar flows in diverging/converging channels arise

Numerical Study of a Williamson Fluid Past a Semi-Infinite Vertical Plate with Convective Heating and Radiation Effects

In the present paper, we investigated mathematical model of the magnetohydrodynamic flow and heat transfer in an electro-conductive polymer on the externalsurface of a semi-infinite vertical plate under radial magnetic field. Thermal radiation andconvective heating effects are considered at the semi-infinite plate surface with modifiedboundary conditions. The Williamson viscoelastic model is employed which isrepresentative of certain industrial polymers. The non-dimensional, transformedboundary layer equations for momentum and energy are solved with the second orderaccurate implicit Keller box finite difference method under appropriate boundaryconditions. Validation of the numerical solutions is achieved via benchmarking withearlier published results. The influence of Williamson viscoelastic fluid parameter,magnetic body force parameter, convective heating, radiation parameter, stream wisevariable and Prandtl number on thermos-fluid characteristics are studied graphically. Themodel is relevant to the simulation of magnetic polymer materials processing.

Effect of thermal radiation on conjugate natural convection flow of a micropolar fluid along a vertical surface

This article investigates the behavior of conjugate natural convection over a finite vertical surface immersed in a micropolar fluid in the presence of intense thermal radiation. The governing boundary layer equations are made dimensionless and then transformed into suitable form by introducing the non-similarity transformations. The reduced system of parabolic partial differential equations is integrated numerically along the vertical plate by using an implicit finite difference Keller-box method. The features of fluid flow and heat transfer characteristics for various values of micropolar or material parameter, K, conjugate parameter, B, and thermal radiation parameter, Rd, are analyzed and presented graphically. Results are presented for the local skin friction coefficient, heat transfer rate and couple stress coefficient for high Prandtl number. It is found that skin friction coefficient and couple stress coefficient reduces whereas heat transfer rate enhances when the micro-inertia parameter increases. All the physical quantities get augmented with thermal radiation.

COMPUTATION OF EYRING-POWELL MICROPOLAR CONVECTIVE BOUNDARY LAYER FLOW FROM AN INVERTED NON-ISOTHERMAL CONE: THERMAL POLYMER COATING SIMULATION

Thermal coating of components with non-Newtonian materials is a rich area of chemical and process mechanical engineering. Many different rheological characteristics can be simulated for such coatings with a variety of different mathematical models. In this work we study the steady-state coating flow and heat transfer of a non-Newtonian liquid (polymer) on an inverted isothermal cone with variable wall temperature. The Eringen micropolar and three-parameter Eyring-Powell models are combined to simulate microstructural and shear characteristics of the polymer. The governing partial differential conservation equations and wall and free stream boundary conditions are rendered into dimensionless form and solved computationally with the KellerBox finite difference method (FDM). Validation with earlier Newtonian solutions from the literature is also included. Graphical and tabulated results are presented to study the variations of fluid velocity, micro-rotation (angular velocity), temperature, skin friction, wall couple stress (micro-rotation gradient) and wall heat transfer rate. With increasing values of the first Eyring-Powell parameter temperatures are elevated, micro-rotation is suppressed and velocities are enhanced near the cone surface but reduced further into the boundary layer. Increasing values of the second Eyring-Powell parameter induce strong flow deceleration, decrease temperatures but enhance micro-rotation values. An increase in non-isothermal power law index suppresses velocities, temperatures and micro-rotations i.e. all transport characteristics are maximum for the isothermal case (n =0). Increasing Eringen vortex viscosity parameter significantly enhances temperatures and also micro-rotations. The present numerical simulations find applications in thermal polymer coating operations and industrial deposition techniques and provide a useful benchmark for more general computational fluid dynamics (CFD) simulations.

Study of magneto-hydrodynamics (MHD) impacts on an axisymmetric Casson nanofluid flow and heat transfer over unsteady radially stretching sheet

The current study is presented to probe the impacts of thermal radiation and mixed convection of axisymmetric Casson fluid flow in the presence of magnetic field along with nano-particles. The flow is provoked because of unsteady radially stretching sheet. The solution of transformed ordinary differential equations is attained by utilizing Keller-box finite difference method (IFDM). The influence of different parameters on flow features is sketched through various graphs and results are tabulated and argued briefly. The convection conditions on the wall temperature and concentration are also exerted in the study. Finally, a comparison is established with existing literature to support our results and a good agreement is found which corroborates our work. A decline in both velocity and temperature is witnessed with the rise in Casson parameter. It is also observed that with the rise Brownian motion and Thermophoresis parameters, the temperature of the fluid rises but concentration profile decreases in start and then start increasing. The fluid concentration in its boundary layer decreases with the increase of heavier species, the parameter of the reaction rate and the exponent of power law for fluid having Prandtl number = 10.0, 15.0, and 20.0. Moreover, the fluid velocity depreciates with the rise in unsteadiness parameter while a significant decrease in temperature is noted. The outcomes of study are quite useful in fabrication of magnetic nano-materials and high temperature treatment of magnetic nano-polymers.

Mathematical Modeling of Natural Convection in a Third-Grade Viscoelastic Micropolar Fluid from an Isothermal Inverted Cone

A mathematical model is developed to investigate the nonlinear, isothermal steady-state free convection boundary layer flow in an incompressible third-grade viscoelastic micropolar fluid from a vertical isothermal cone. The non-Newtonian fluid exhibits both viscoelastic and micro-structural characteristics and is representative of certain polymers. The transformed conservation equations for mass, linear momentum, angular momentum and heat are solved numerically subject to realistic boundary conditions using the second-order accurate implicit finite-difference Keller-box method. The numerical code is validated with previous studies and also with a Nakamura tridiagonal method. Detailed interpretation of the computations is included. The present simulations are of interest in chemical engineering systems and solvent and low density polymer materials processing.

Hydromagnetic Flow of Heat and Mass Transfer in a Nano Williamson Fluid Past a Vertical Plate With Thermal and Momentum Slip Effects: Numerical Study

Abstract In this article, the study of heat, momentum and mass (species) transfer in an electro-conductive polymer on the external surface of a vertical plate. The effects of Brownian motion and thermophoresis are incorporated in the model in the presence of both heat and nanoparticle mass transfer convective conditions. The Williamson viscoelastic model is employed which is representative of certain industrial polymers. The non-dimensional, transformed boundary layer equations for momentum and energy are solved with the second order accurate implicit Keller box finite difference method under appropriate boundary conditions. The influence of Weissenberg number, magnetic body force parameter, thermal slip parameter, hydrodynamic slip parameter, stream wise variable and Prandtl number on thermo fluid characteristics are presented graphically and discussed. A weak elevation in temperature accompanies increasing Weissenberg number whereas a significant acceleration in the flow is computed near the plate surface. Rate of heat transfer is reduced with increases the Weissenberg number. The study is relevant to enrobing processes for electric-conductive nano-materials, of potential use in aerospace, smart coating transport phenomena and other industries.

Radiative and magnetohydrodynamics flow of third-grade viscoelastic fluid past an isothermal inverted cone in the presence of heat generation/absorption

A mathematical analysis to investigate the non-linear, isothermal, steady state, free convection boundary layer flows of an incompressible third-grade viscoelastic fluid past an isothermal inverted cone in the presence of magnetohydrodynamic, thermal radiation, and heat generation/absorption. The transformed conservation equations for linear momentum, heat, and mass are solved numerically subject to the realistic boundary conditions using the second-order accurate implicit finite-difference Keller-box method. The numerical code is validated with the previous studies. Detailed interpretation of the computations is included. The present simulations are of interest in chemical engineering systems and solvent and low-density polymer material processing.

Numerical exploration of thermal radiation and Biot number effects on the flow of a non‐Newtonian MHD Williamson fluid over a vertical convective surface

AbstractA theoretical and computational study of the magnetohydrodynamic flow and free convection heat transfer in an electroconductive polymer on the external surface of a vertical plate under radial magnetic field is presented. The Biot number effects are considered at the vertical plate surface via modified boundary conditions. The Williamson viscoelastic model is employed which is representative of certain industrial polymers. The nondimensional, transformed boundary layer equations for momentum and energy are solved with the second‐order accurate implicit Keller box finite difference method under appropriate boundary conditions. Validation of the numerical solutions is achieved via benchmarking with earlier published results. The influence of Weissenberg number (ratio of the relaxation time of the fluid and time scale of the flow), magnetic body force parameter, stream‐wise variable, and Prandtl number on thermo fluid characteristics are studied graphically and via tables. A weak elevation in temperature accompanies increasing Weissenberg number, whereas a significant acceleration in the flow is computed near the vertical plate surface with increasing Weissenberg number. Nusselt number is reduced with increasing Weissenberg number. Skin friction and Nusselt number are both reduced with increasing magnetic field effect. The model is relevant to the simulation of magnetic polymer materials processing.

Hydromagnetic Flow and Heat Transfer in a Williamson Non-Newtonian Fluid from a Horizontal Circular Cylinder with Newtonian Heating

A theoretical and computational study of the magneto hydrodynamic flow and free convection heat transfer in an electro-conductive polymer on the external surface of a horizontal circular cylinder under radial magnetic field is presented. The Williamson viscoelastic model is employed which is representative of certain industrial polymers. Newtonian heating is incorporated via appropriate boundary conditions as this represents better actual thermal materials processing operations. The non-dimensional, transformed boundary layer equations for momentum and energy are solved with the second order accurate implicit Keller box finite difference method under appropriate boundary conditions. Validation of the numerical solutions is achieved via benchmarking with earlier published results. The influence of Weissenberg number (ratio of the relaxation time of the fluid and time scale of the flow), magnetic body force parameter, stream wise variable and Prandtl number on thermofluid characteristics are studied graphically and via tables. A weak elevation in temperature accompanies increasing Weissenberg number whereas a significant acceleration in the flow is computed near the cylinder surface with increasing Weissenberg number. Nusselt number is reduced with increasing Weissenberg number. Skin friction is increased whereas Nusselt number is reduced with greater stream wise coordinate. The study is relevant to smart coating transport phenomena.

STEFAN BLOWING, NAVIER SLIP, AND RADIATION EFFECTS ON THERMO-SOLUTAL CONVECTION FROM A SPINNING CONE IN AN ANISOTROPIC POROUS MEDIUM

Thermal radiation features in many high temperature materials processing operations. To evaluate the influence of radiative flux on spin coating systems, we consider herein the thermo-solutal (coupled heat and mass transfer) in steady laminar boundary layer natural convection flow from a rotating permeable vertical cone to an anisotropic Darcian porous medium. Surface slip effects are also included in the model presented. The conservation equations are rendered into self-similar form and solved as an ordinary differential two-point boundary value problem with surface and free stream boundary conditions using MAPLE 17 software. The transport phenomena are observed to be controlled by ten parameters, viz primary and secondary Darcy numbers (Dax and Da), rotational (spin) parameter (NR), velocity slip parameter (a), suction/injection parameter (S), thermal slip parameter (b), mass slip parameter (c) buoyancy ratio parameter (N), and conduction-radiation parameter (Rc). Tangential velocity and temperature are observed to be enhanced with greater momentum slip whereas swirl velocity and concentration are reduced. Increasing swirl Darcy number strongly accelerates both the tangential and swirl flow and also heats the regime whereas it decreases concentrations. Conversely a rise in tangential Darcy number accelerates only the tangential flow and decelerates swirl flow, simultaneously depressing temperatures and concentrations. Increasing thermal slip accelerates the swirl flow and boosts concentration but serves to retard the tangential flow and decrease temperatures. With higher radiation contribution (lower Rc values) temperatures are elevated and concentrations are reduced. Verification of the MAPLE 17 solutions is achieved using a Keller-box finite difference method (KBM). A number of interesting features in the thermo-fluid and species diffusion characteristics are addressed. Key words: Stefan blowing; Spinning cone; MAPLE 17; Anisotropic

MHD flow of tangent hyperbolic fluid over a stretching cylinder: Using Keller box method

A numerical solution of MHD flow of tangent hyperbolic fluid model over a stretching cylinder is obtained in this paper. The governing boundary layer equation of tangent hyperbolic fluid is converted into an ordinary differential equation using similarity transformations, which is then solved numerically by applying the implicit finite difference Keller box method. The effects of various parameters on velocity profiles are analyzed and discussed in detail. The values of skin friction coefficient are tabulated and plotted in order to understand the flow behavior near the surface of the cylinder. For validity of the model a comparison of the present work with the literature has been made.

Non-Darcy natural convection flow for non-Newtonian nanofluid over cone saturated in porous medium with uniform heat and volume fraction fluxes

Purpose – The purpose of this paper is to investigate the effect of uniform lateral mass flux on non-Darcy natural convection of non-Newtonian fluid along a vertical cone embedded in a porous medium filled with a nanofluid. Design/methodology/approach – The resulting governing equations are non-dimensionalized and transformed into a non-similar form and then solved numerically by Keller box finite-difference method. Findings – A comparison with previously published works is performed and excellent agreement is obtained. Research limitations/implications – The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. It is assumed that the cone surface is preamble for possible nanofluid wall suction/injection, under the condition of uniform heat and nanoparticles volume fraction fluxes. Originality/value – The effects of nanofluid parameters, Ergun number, surface mass flux and viscosity index are investigated on the velocity, temperature, and volume fraction profiles as well as the local Nusselt and Sherwood numbers.

Spectral method for solving high order nonlinear boundary value problems via operational matrices

A spectral method based on operational matrices of Bernstein polynomials using collocation method is elaborated and employed for solving nonlinear ordinary and partial differential equations with multi-point boundary conditions. First, properties of Bernstein polynomial, operational matrices of integration, differentiation and product are introduced and then utilized to reduce the given differential equation to the solution of a system of algebraic equations. This new approach provides a significant computational advantage by converting the given original problem to an equivalent integro-differential equation which implies all boundary condition. Approximate solution is achieved by expanding the desired function in terms of a Bernstein basis and employing operational matrices. Unknown coefficients are determined by collocation. The method is compared with modified Adomian decomposition method, Birkhoff-type interpolation method, reproducing kernel Hilbert space method, fixed point method, finite-difference Keller-box method, multilevel augmentation method and shooting method. Illustrative examples are included to demonstrate the high precision, validity and good performance of the new scheme even for solving nonlinear singular differential equations.

Numerical Modeling of Non-Similar Mixed Convection Heat Transfer over a Stretching Surface with Slip Conditions

In this paper, the heat transfer effect on the steady boundary layer flow of a Casson fluid past a stretching surface in the presence of slip conditions was analyzed. The stretching surface is maintained at a constant temperature. The boundary layer conservation equations, which are parabolic in nature, are normalized into non-similar form and then solved numerically with the well-tested, efficient, implicit, stable Keller-box finite difference scheme. The resulting equations are solved numerically by using the Kellerbox finite-difference method, and the expressions for velocity and temperature are obtained. They satisfy all imposed initial and boundary conditions and reduce to some well-known solutions for non-Newtonian fluids. Numerical results for velocity, temperature, skin friction and Nusselt number are shown in various graphs and discussed for embedded flow parameters. It is found that both velocity and temperature decrease with an increase of the Casson fluid parameter.

A STUDY ON MHD BOUNDARY LAYER FLOW OVER A NONLINEAR STRETCHING SHEET USING IMPLICIT FINITE DIFFERENCE METHOD

The problem of the MHD boundary layer flow of an in compressible viscous fluid over a non-linear stretc hing sheet is discussed. Using similarity transformation governing equations are transformed in to nonlinear ordinary different ial equation. The Implicit finite difference Keller box method is applied to find the numerical solution of nonlinear differential equat ion. Graphical results of fluid velocity have been presented and discussed for the related parameters.

On the Accuracy of a Finite-Difference Method for Parabolic Partial Differential Equations with Discontinuous Boundary Conditions

Although the numerical solution of parabolic partial differential equations (PDEs) is widely documented, the effect of discontinuous boundary conditions on numerical accuracy is not. This article employs the Keller box finite-difference method to study the effect of such discontinuities when solving the linear one-demensional transient heat equation. We demonstrate that this formally second-order-accurate scheme can lose accuracy, but that an analytical understanding of the behavior of the solution helps in providing an accuracy-restoring formulation. Benchmark computations are presented that will provide guidance in the numerical solution of nonlinear parabolic PDEs for which there are no closed-form analytical solutions.