Boundary Value Problem Solver Bvp4c Research Articles

Dissipative flow features of Carreau nanofluid with thermal radiation inside plane wall channel: Jeffery-Hamel analysis

The current article communicates a numerical investigation on laminar flow of dissipative generalized Newtonian Carreau nanofluid flowing through vertical conduit with converging and diverging plane walls. Thermal and concentration characteristics due to enthalpy change, activation energy, and non-linear thermal radiation have been examined in the presence of buoyancy forces. The channel walls for both temperature and volumetric fraction are assumed to be isothermal. The instability mechanism of nanofluids is reported using a two-phase nanofluid model, which works reasonably well for nanoparticle concentrations below a certain threshold. A Jeffery-Hamel (J-H) flow model is developed by assuming an incompressible purely radial flow of Carreau nanofluids with heat and mass transportation. Using the suitable non-dimensional variables, the resulting nonlinear partial differential equations are turned into a system of ordinary differential equations. The modified governing equations are then numerically solved using the built-in boundary value problem solver bvp4c, on the template form of commercial software MATLAB. The impacts of material, geometrical and thermophysical parameters governing the J-H problem are discussed and illustrated. Results indicate that higher buoyance forces incline the velocity profiles in converging enclosure, while a slight reduction is perceived in opposing forces. A significant decrease of wall heat transmission is reflected for larger values of activation energy and radiation parameter. For endorsing this communication, a comparison analysis is established with existing research and noticed a remarkable agreement. Practically, the flow inside converging and diverging channels are deployed in nuclear reactors that use plate-type nuclear energies, high heat-flux condensed heat exchangers, high-performance micro-electronic cooling systems, jets, rockets nozzles, and jet propulsion inlet.

Significance of Melting Heat Transfer and Brownian Motion on Flow of Powell–Eyring Fluid Conveying Nano-Sized Particles with Improved Energy Systems

The present study explores the characteristics of 2D MHD melting with reference to mass and heat transportation upon stagnation point Powell–Eyring nanofluid flow on an extensible surface. Melting is an important phenomenon that is involved in many procedures such as permafrost melting, solidification of slag, defrosting frozen ground etc., all of which are examples of soil freezing and melting that involve heat trafficking through a coil in a grounded pump. A mathematical model is developed for the boundary layer flow. The differential equations are solved through a numerical algorithm which makes use of the boundary value problem solver bvp4c, applying MATLAB software. The numerical variations of embedded parameters on velocity lineation, temperature figuration, and concentration delineation are represented graphically, as are the width of the boundary layer value and the delineation rate for the increasing velocity parameter. The velocity function shows a decremental response for M while the opposite behavior is seen against the concentration field.

MHD Stagnation Point Flow over a Nonlinear Stretching/Shrinking Sheet in Nanofluids

In this study, an investigation of the steady 2-D magnetohydrodynamiic (MHD) flow of stagnation point past a nonlinear sheet of stretching/shrinking within of a non-uniform transverse magnetic intensity in nanofluids had been analysed. Considered material of nanoparticles such as copper (Cu) in water base fluid with Pr = 6.2 to analyze the influence of volume fraction parameter of nanoparticles and the stretching/shrinking sheet parameter. The governing nonlinear partial differential equations (PDEs) are converted in to the nonlinear ordinary differential equations (ODEs) and use the boundary value problem solver bvp4c in Matlab program to solve numerically through the use of a similarity transformation. The impact of the parameter of the magnetic field on the coefficient of skin friction, the local number of Nusselt and the profiles of velocity and temperature are portrayed and explained physically. The analysis reveals that the magnetic field and volume fraction of nanoparticles affect the velocity and temperature. The dual solutions are achieved where for the shrinking sheet case and the solutions are non-unique, different from a stretching sheet.

Stability Analysis on Stagnation-Point Flow and Heat Transfer towards a Permeable Stretching/Shrinking Sheet with Heat Source in a Casson Fluid

This paper deals with a stagnation-point boundary layer flow and heat transfer of a Casson fluid towards a stretching/shrinking sheet. The main objective of the present study is to analyse the effects of the injection parameter and heat source on the velocity and temperature profiles as well as the skin friction coefficient and the Nusselt number. It is vital to study the heat transfer and fluid flow problems in the presence of injection and heat source effects due to a wide variety of applications in engineering and industry. The governing nonlinear partial differential equations are transformed into a system of nonlinear ordinary differential equations by using similarity transformation, before being solved numerically using the boundary value problem solver bvp4c routine in MATLAB. Dual solutions are found to exist for the shrinking sheet case, whereas the solution is unique for the stretching case. The stability analysis has been performed to determine the stable solution. It is shown that the first solutions are stable and physically reliable while the second solutions are not. Further, the present results have been compared with the previous published results for a particular case and the comparisons are found to be in good agreement. The local Nusselt number is decreases with an increase in heat source parameter. Rising values of the injection parameter has decreases both the skin friction coefficient and the local Nusselt number.

Triple Solutions of Carreau Thin Film Flow with Thermocapillarity and Injection on an Unsteady Stretching Sheet

Thin films and coatings which have a high demand in a variety of industries—such as manufacturing, optics, and photonics—need regular improvement to sustain industrial productivity. Thus, the present work examined the problem of the Carreau thin film flow and heat transfer with the influence of thermocapillarity over an unsteady stretching sheet, numerically. The sheet is permeable, and there is an injection effect at the surface of the stretching sheet. The similarity transformation reduced the partial differential equations into a system of ordinary differential equations which is then solved numerically by the MATLAB boundary value problem solver bvp4c. The more substantial effect of injection was found to be the reduction of the film thickness at the free surface and development of a better rate of convective heat transfer. However, the increment in the thermocapillarity number thickens the film, reduces the drag force, and weakens the rate of heat transfer past the stretching sheet. The triple solutions are identified when the governing parameters vary, but two of the solutions gave negative film thickness. Detecting solutions with the most negative film thickness is essential because it implies the interruption in the laminar flow over the stretching sheet, which then affects the thin film growing process.

Dual solutions for boundary layer flow and heat transfer of biomagnetic fluid over a stretching/shrinking sheet in presence of a magnetic dipole and a prescribed heat flux

This paper analyzes the steady boundary layer flow and heat transfer of biomagnetic fluid over a stretching/shrinking sheet with prescribed surface heat flux under the influence of a magnetic dipole. The governing equations are transformed into a set of ordinary differential equations (ODEs) by using similarity transformations. Numerical results are obtained using the boundary value problem solver bvp4c of MATLAB. The effects of various physical parameters on the velocity and temperature profiles as well as skin friction coefficient are discussed. The paper shows that dual solutions exist for certain values of stretching/shrinking sheet and suction parameters. Stability analysis is performed to determine which solution is stable and physically valid. Results of the stability analysis depict that the first solution (upper branch) is stable and physically realizable, while the second solution (lower branch) is unstable.

Slip Flow of an Unsteady Nanofluid Past a Stretching Surface in a Transverse Magnetic Field Using SRM

A theoretical and numerical examination has been completed to talk about the unsteady, two dimensional slip flow of a nanofluid of heat and mass transfer with transverse magnetic field was investigated in this paper. A stretching surface is used to investigate the flow. Obtained nonlinear equations are solved by Spectral Relaxation Method (SRM) technique and the results are verified by comparing the results obtained by using the Matlab in-built boundary value problem solver bvp4c, and the outcomes which are published in previous papers. The outcomes are exhibited pictorially and talked about difference coming about parameters. Expanding heat transfer rate, mass transfer rate and velocity slip raises velocity yet to be diminishes temperature and skin friction (surface shear stress).

Free open reference implementation of a two-phase PEM fuel cell model

In almost 30 years of PEM fuel cell modeling, countless numerical models have been developed in science and industrial applications, almost none of which have been fully disclosed to the public. There is a large need for standardization and establishing a common ground not only in experimental characterization of fuel cells, but also in the development of simulation codes, to prevent each research group from having to start anew from scratch. Here, we publish the first open standalone implementation of a full-blown, steady-state, non-isothermal two-phase model for low-temperature PEM fuel cells. It is based on macro-homogeneous modeling approaches and implements the most essential through-plane transport processes in a five-layer MEA. The focus is on code simplicity and compactness with only a few hundred lines of clearly readable code, providing a starting point for more complex model development. The model is implemented as a standalone MATLAB function, based on MATLAB’s standard boundary value problem solver. The default simulation setup reflects wide-spread commercially available MEA materials. Operating conditions recommended for automotive applications by the European Commission are used to establish new fuel cell simulation base data, making our program a valuable candidate for model comparison, validation and benchmarking. Program summaryProgram Title: MMM1DProgram Files doi:http://dx.doi.org/10.17632/2msdd4j84c.1Licensing provisions: BSD 3-clauseProgramming language: MATLABNature of problem: Steady-state, non-isothermal, two-phase simulation of the coupled through-plane transport of charge, heat and mass within the five-layer membrane electrode assembly of low-temperature proton exchange membrane fuel cells.Solution method: MATLAB’s boundary value problem solver bvp4c, a finite difference solver that implements the 3-stage Lobatto IIIa collocation method with automated mesh selection based on the residual.Additional comments: The complete source code and the license agreement can also be obtained from https://www.isomorph.ch.

Dual Solutions of MHD Three-Dimensional Flow Over a Permeable Stretching/Shrinking Surface with Velocity Slip and Thermal Radiation in a Nanofluid

The magnetohydrodynamic (MHD) three-dimensional flow of alumina (Al2O3)-water nanofluid over a permeable stretching/shrinking surface is numerically studied. The present model is devoted to the effects of velocity slip condition and thermal radiation parameter on the fluid flow and heat transfer. The governing partial differential equations are reduced into a system of ordinary differential equations using a similarity transformation, which are then solved numerically using the Matlab routine boundary value problem solver bvp4c. Dual solutions exist for a certain range of the suction parameter. A stability analysis is performed to determine which solution is stable and physically realizable. The numerical results for the skin friction coefficient and local Nusselt number are obtained and discussed. It is found that heat transfer rate increases as the velocity slip increases but the local Nusselt number decreases when the thermal radiation increases.

Stagnation-point flow towards a stretching/shrinking sheet in a nanofluid using Buongiorno's model

The stagnation point flow of a nanofluid towards a permeable stretching/shrinking sheet using the Buongiorno's model is studied. Numerical results are obtained using boundary value problem solver bvp4c in MATLAB for several values of the governing parameters. The numerical results show that dual (upper and lower branch) solutions exist for the shrinking case, while for the stretching case, the solution is unique. A stability analysis is performed to determine the physical realizable in practice of the dual solutions. It is found that the skin friction decreases when the sheet is stretched, but increases when the suction effect is increased. It is also found that increasing the thermophoresis parameter reduces the heat transfer rate at the surface, while increasing the Brownian motion parameter increases the mass transfer rate at the surface.

Axisymmetric stagnation-point flow over a stretching/shrinking plate with second-order velocity slip

The axisymmetric stagnation point flow over a stretching/shrinking surface with second-order slip and temperature jump is studied numerically. The governing partial differential equations are transformed into ordinary (similarity) differential equations. These equations along with the corresponding boundary conditions are solved numerically using a boundary value problem solver bvp4c in Matlab software. It is observed that dual (first and second) solutions exist for the similarity equations. The effects of different parameters on the velocity and the temperature distributions as well as the skin friction coefficient and the Nusselt number are analyzed and discussed.

The Magnetohydrodynamic Stagnation Point Flow of a Nanofluid over a Stretching/Shrinking Sheet with Suction

The magnetohydrodynamic (MHD) stagnation point flow of a nanofluid over a permeable stretching/shrinking sheet is studied. Numerical results are obtained using boundary value problem solver bvp4c in MATLAB for several values of parameters. The numerical results show that dual solutions exist for the shrinking case, while for the stretching case, the solution is unique. A stability analysis is performed to determine the stability of the dual solutions. For the stable solution, the skin friction is higher in the presence of magnetic field and increases when the suction effect is increased. It is also found that increasing the Brownian motion parameter and the thermophoresis parameter reduces the heat transfer rate at the surface.

Viscous and Joule Heating in the Stagnation Point Nanofluid Flow Through a Stretching Sheet With Homogenous–Heterogeneous Reactions and Nonlinear Convection

The combined effects of viscous and Joule heating on the stagnation point flow of a nanofluid through a stretching/shrinking sheet in the presence of homogeneous–heterogeneous reactions are investigated. The nanoparticle volume fraction model is used to describe the nanofluid. In this study, the density temperature relation is nonlinear which causes a nonlinear convective heat transfer. The surface of the sheet is assumed to be convectively heated with a hot fluid. The governing nonlinear differential equations are solved using the successive linearization method (SLM), and the results are validated by comparison with numerical approximations obtained using the Matlab in-built boundary value problem solver bvp4c and with existing results in literature. The nanofluid problem finds applications in heat transfer devices where the density and temperature relations are complex and the viscosity of the fluid has significant effect on the heat transfer rate.

A numerical scheme for solving a periodically forced Reynolds equation

A modified Reynolds equation is used to model the air-film in a high-speed squeeze-film bearing. The axial position of the bearing stator is prescribed as a finite amplitude periodic oscillation. A numerical approach is considered for solving the uncoupled and coupled periodic problems associated with this model. The uncoupled problem requires the computation of the squeeze-film dynamics when the rotor is held at a fixed axial position and the coupled problem incorporates the additional air–rotor interaction since the rotor position is unknown and modelled as a spring-mass-damper system. The details of a Fourier spectral collocation scheme are provided for the reduction of the modified Reynolds equation to a system of non-linear, first-order ordinary differential equations in space. Using the Matlab boundary value problem solver bvp4c this system of equations is solved to give the periodic pressure distributions and rotor heights. The high degree of accuracy in the spectral collocation scheme is demonstrated through comparison with an appropriate analytical solution. Further analysis indicates that the direct periodic solver is at least 10 times faster than the equivalent Crank–Nicholson finite-difference scheme. For changing values of a selected physical parameter the method of arc-length continuation is employed to track branches of solutions computed using the spectral collocation scheme. A selection of results is presented to demonstrate the range of accessible solutions and the robust nature of the numerical scheme. Copyright © 2010 John Wiley & Sons, Ltd.