Permeable Wedge Research Articles

Multiple convection‐driven Falkner‐Skan flow of Carreau nanofluid along a permeable wedge: An analytical approach

AbstractThe impression of multiple convective conditions on Falkner‐Skan flow along a permeable static/moving wedge is demonstrated here. The Carreau nanofluid model depending on the power law index is incorporated with the effects of thermophoresis and Brownian motion. A suitable similarity conversion is consumed to bring out the nonlinear ordinary differential equations (ODEs) from the partial differential system. The transformed system is tackled via analytical and numerical procedure by homotopy perturbation method and RK‐5 with shooting technique, respectively. Numerical computations assert that the growth of concentration and temperature boundary layer widths are in an inversely proportional relation to the permeability of the medium, whereas reverse effect is observed for conduction convection and conduction diffusion parameters. Also, with the enhancement of Brownian motion parameter, the rate of heat transport gets reduced for static wedge by 14.25% whereas for moving wedge the reduction is 10.61%. Another significant observation shows that the growth in mass transport with conduction diffusion is 17.65% greater for moving wedge in contrast to static one in accordance with other governing parameters.

Effect of thermal radiation on engine oil nanofluid flow over a permeable wedge under convective heating

PurposeThe purpose of this paper is to examine the magneto hydrodynamic flow and heat transfer of nanofluids over a permeable wedge based on engine oil which is under the effects of thermal radiation and convective heating.Design/methodology/approachThe equations governing the flow are transformed into differential equations by applying similarity transformations. Keller box method is used to bring out the numerical solution.FindingsThe discovery interprets that temperature as well as the velocity of Ag-engine oil nanofluids are more noticeable than Cu-engine oil nanofluids. Thermal boundary layer increases for radiation parameter as well as Biot number. Fluctuations of co-efficient of drag skin friction as well heat transfer rate at the wall are also tested.Originality/valueTill now, no numerical studies are reported on the heat transfer enhancement of the permeable wedge under thermal radiation on engine oil nanofluid flow by considering convective heating.

On the Stability of MHD Boundary Layer Flow over a Stretching/Shrinking Wedge

The steady two dimensional magnetohydrodynamic (MHD) boundary layer flow and heat transfer over a stretching/shrinking permeable wedge is numerically investigated. The partial differential equations governing the flow and heat transfer are transformed into a system of ordinary differential equations using a similarity transformation. These equations are then solved numerically using the boundary value problem solver, bvp4c in Matlab software. It is found that dual solutions exist for a certain range of the shrinking strength. A stability analysis is performed to identify which solution is stable and physically reliable.

Dual solutions of a boundary layer problem for MHD nanofluids through a permeable wedge with variable viscosity

Considering the effect of variable viscosity and the phenomenon of flow separation, the MHD Cu/Ag-Water nanofluids through a permeable wedge are investigated. The governing equations of flow and energy are reduced by similarity transformations and then solved numerically by the shooting method. It is found that dual solutions exist for negative pressure gradient. Compared with the Ag-Water nanofluid, the flow separation occurs later for injection, while it occurs earlier for suction in the Cu-Water nanofluid. The outcomes also specify that suction and small variable viscosity parameters delay the separation for the two nanofluids.

MHD Falkner–Skan Flow of Casson Fluid Flow and Heat Transfer with Variable Property Past a Moving Wedge

In this paper, we have considered the classical problem effects of radiation, temperature dependent plastic dynamic viscosity and thermal conductivity with steady incompressible laminar free convective MHD Falkner–Skan flow of Casson fluid past a moving permeable wedge. By using suitable transformation, the governing PDEs corresponding to the momentum and energy equations are converted into non-linear coupled ODEs and solved by bvp4c method. The influence of various physical parameters like Casson fluid parameters radiation parameter, variable plastic dynamic viscosity parameter, variable thermal conductivity parameter, magnetic field parameter, local porosity parameter, thermal buoyancy parameter, Prandtl number, local Eckert number, moving wedge parameter on velocity and temperature profiles are discussed and display graphically. Local Nusselt number and skin friction coefficient are tabulated.

Nonlinear radiation effects on flow of nanofluid over a porous wedge in the presence of magnetic field

PurposeThe purpose of this paper is to assess the flow of a nanofluid over a porous moving wedge. The passive control model along with the magnetohydrodynamic (MHD) effects is used to formulate the problem. Furthermore, in energy equation, the non-linear thermal radiation has also been incorporated. The equations governing the flow are transformed into a set of ordinary differential equations by using suitable similarity transforms. The reduced system of equations is then solved numerically using a well-known Runge–Kutta–Fehlberg method coupled with a shooting technique. The influence of parameters involved on velocity, temperature and concentration profiles is highlighted with the help of a graphical aid. Expressions for skin-friction coefficient, local Nusselt number and Sherwood number are obtained and presented graphically.Design/methodology/approachNumerical solution of the problem is obtained using the well-known Runge–Kutta–Fehlberg method.FindingsThe analysis provided gives a clear description that the increase in m and magnetic parameter M results in an increased velocity profile. Both these parameters normalize the velocity field. Radiation parameter, Rd, increases the temperature and concentration of the system so does the temperature ratio θω reduces the heat transfer rate at the wall for both stretching and shrinking wedge.Originality/valueIn the study presented, the flow of nanofluid over a moving permeable wedge is considered. The solution of the equations governing the flow is presented numerically. For the validity of results obtained, a comparison is also presented with already existing results. To the best of the authors’ knowledge, this investigation is the first of its kind on the said topic.

Solution of Falkner- Skan Unsteady MHD Boundary Layer Flow and Heat Transfer Past a Moving Porous Wedge in a Nanofluid

The present problem deals with the effect of various dimensionless parameters on the forced convection unsteady MHD boundary layer flow and heat transfer over a permeable wedge stretching surface in a nanofluid. The effects of Brownian motion and thermophoresis are considered. The solution for the velocity, temperature and nanoparticle concentration depends on various dimensionless parameters like Prandtl number Pr, velocity ratio parameter λ, permeability parameter K*, unsteady parameter A, pressure gradient parameter β, magnetic parameter M. The local similarity transformation is used to convert the governing partial differential equations into coupled third order non-linear ordinary differential equations. These equations are numerically solved using suitable method along with shooting technique. Numerical results are obtained for distributions of velocity, temperature and nanoparticle concentration, as well as, for the skin friction, local Nusselt number and local Sherwood number for several values of governing parameters. The numerical results are shown graphically and also in a tabular form. The velocity decreases for increasing values of velocity ratio parameter but increases for magnetic parameter, unsteady parameter, permeability parameter and pressure gradient parameter. The temperature profiles decreases for thermophoresis parameter Prandtl number and velocity ratio parameter whereas the temperature remains unchanged for Brownian motion parameter. Again, nanoparticle concentration profiles increases for Brownian motion parameter but decreases for thermophoresis parameter, Lewis number and velocity ratio parameter. Besides, the results of rate of velocity and rate of heat transfer are compared with previously published work for the validity and accuracy and found to be in good agreement.

Second Law Analysis of Flow, Heat and Mass Transfer Past a Nonlinearly Stretching Permeable Wedge with Temperature Jump and Chemical Reaction

Abstract Second law analysis (entropy generation) for the steady two-dimensional laminar forced convection flow, heat and mass transfer of an incompressible viscous fluid past a nonlinearly stretching porous (permeable) wedge is numerically studied. The effects of viscous dissipation, temperature jump, and first-order chemical reaction on the flow over the wedge are also considered. The governing boundary layer equations for mass, momentum, energy and concentration are transformed using suitable similarity transformations to three nonlinear ordinary differential equations (ODEs). Then, the ODEs are solved by using a Keller’s box algorithm. The effects of various controlling parameters such as wedge angle parameter, velocity ratio parameter, suction/injection parameter, Prandtl number, Eckert number, temperature jump parameter, Schmidt number, and reaction rate parameter on dimensionless velocity, temperature, concentration, entropy generation number, and Bejan number are shown in graphs and analyzed. The results reveal that the entropy generation number increases with the increase of wedge angle parameter, while it decreases with the increase of velocity ratio parameter. Also, in order to validate the obtained numerical results of the present work, comparisons are made with the available results in the literature as special cases, and the results are found to be in a very good agreement.

Aggregation effects on water base Al2O3—nanofluid over permeable wedge in mixed convection

AbstractTwo‐dimensional heat transfer flow of a nanofluid in the neighborhood of a stagnation point flow in the presence of mixed convection is investigated. The mathematical model consists of continuity and the momentum equations, while a new model is proposed to see the effects aggregations on water base Al2O3 nanofluid over permeable wedge. The variable wall temperature is taken into account. The Mathematica package based on homotopy analysis method is used to solve this problem. Several aspects of the aggregation parameters on velocity and temperature profiles of nanofluid are investigated and shown graphically with respect to the physical parameters involved in it. The tabular results are demonstrated for heat transfer rate and skin friction coefficient. © 2015 Curtin University of Technology and John Wiley & Sons, Ltd.

Non-similar Boundary Layers over a Wedge due to Thermophoresis and Viscosity Effects

Background/Objectives: The objective of the present work is to investigate the effects of the thermophoresis and the fluid viscosity on a forced convection heat and mass transfer on non-similar flow past a permeable wedge embedded in a porous medium. Methods of Analysis: The analysis is conducted by using a local non-similarity scheme of the second level truncation in the presence of thermophoresis and variable viscosity. The governing equations of the non-similar convection are presented in the dimensionless form by applying Falkner-Skan variable to obtain the set of ordinary differential equations. Runge-Kutta-Gill in conjunction with shooting method is applied to find a skin friction, rate of heat and mass transfer. The initial value problems are solved numerically using Runge Kutta-Fehlberg methods and presented graphically. Findings: The thermophoresis has significant effect to speed up the process for decreasing concentration significantly, while the effects of viscosity is to increase or decrease the velocity of fluid depend on the value of viscosity parameter, positive or negative, respectively. Validation is conducted by comparing the current works with previous works and found that the results are in excellent agreement. The conclusion is drawn that the non-similar velocity and the concentration of fluid are significantly affected by thermophoresis and viscosity. Application/Improvements: The results of an analysis on the effects of thermophoresis and viscosity on boundary layer flow over solid surfaces can be applied to design an instrument to remove pollutants from environment

Similarity Solutions for Heat and Mass Transfer of MHD Flow Over a Permeable Stretching Wedge with Variable Fluid Properties and Heat Generation/Absorption

The problem of steady boundary layer flow for Newtonian fluid with variable viscosity and thermal conductivity over stretching wedge in the presence of magnetic field and heat generation (absorption) is studied. The system of governing partial differential equations with the boundary conditions is reduced to a system of ordinary differential equations with appropriate boundary conditions using Lie group. The reduced ordinary differential equations along with the boundary conditions are solved numerically using the fourth order Runge-Kutta method algorithm with the shooting technique. The effects of various parameters on the velocity, the temperature and the concentration profiles are discussed. Finally, the numerical values of the physical quantities, such as local skin-friction coefficient, the local Nusselt number and the local Sherwood number are illustrated graphically.

Convective slip flow of rarefied fluids over a permeable wedge plate embedded in a Darcy-Forchheimer porous medium

This paper is devoted to investigate the influences of thermal radiation and temperature-dependent fluid properties on convective slip flow of slightly rarefied fluids over a porous wedge plate embedded in a Darcy-Forchheimer porous medium. Using the similarity transformation, the governing system of non-linear partial differential equations is transformed into similarity non-linear ordinary differential equations which are solved by employing a numerical shooting technique with a fourth-order Runge-Kutta integration scheme. Numerical results are analyzed for the effect of different pertinent parameters on the flow and heat transfer characteristics.

Hydromagnetic flow of a Cu–water nanofluid past a moving wedge with viscous dissipation

A numerical study is performed to investigate the flow and heat transfer at the surface of a permeable wedge immersed in a copper (Cu)–water-based nanofluid in the presence of magnetic field and viscous dissipation using a nanofluid model proposed by Tiwari and Das (Tiwari I K and Das M K 2007 Int. J. Heat Mass Transfer 50 2002). A similarity solution for the transformed governing equation is obtained, and those equations are solved by employing a numerical shooting technique with a fourth-order Runge–Kutta integration scheme. A comparison with previously published work is carried out and shows that they are in good agreement with each other. The effects of velocity ratio parameter λ, solid volume fraction φ, magnetic field M, viscous dissipation Ec, and suction parameter Fw on the fluid flow and heat transfer characteristics are discussed. The unique and dual solutions for self-similar equations of the flow and heat transfer are analyzed numerically. Moreover, the range of the velocity ratio parameter for which the solution exists increases in the presence of magnetic field and suction parameter.

Mixed Convective Heat Transfer for MHD Viscoelastic Fluid Flow over a Porous Wedge with Thermal Radiation

The main concern of the present paper is to study the MHD mixed convective heat transfer for an incompressible, laminar, and electrically conducting viscoelastic fluid flow past a permeable wedge with thermal radiation via a semianalytical/numerical method, called Homotopy Analysis Method (HAM). The boundary-layer governing partial differential equations (PDEs) are transformed into highly nonlinear coupled ordinary differential equations (ODEs) consisting of the momentum and energy equations using similarity solution. The current HAM solution demonstrates very good agreement with previously published studies for some special cases. The effects of different physical flow parameters such as wedge angle (β), magnetic field ( M), viscoelastic ( k1), suction/injection ( fw), thermal radiation ( Nr), and Prandtl number (Pr) on the fluid velocity component ( f′( η)) and temperature distribution ( θ( η)) are illustrated graphically and discussed in detail.

MHD mixed convection flow over a permeable non-isothermal wedge

Numerical solutions are obtained for the hydro-magnetic mixed convection boundary layer flow of an electrically conducting fluid over a non-isothermal wedge in the presence of variable thermal conductivity. The effects due to viscous dissipation, internal heat generation/absorption, thermal radiation, Joule heating and stress work are included. The governing partial differential equations of the problem, subjected to the appropriate boundary conditions are solved numerically by an efficient finite difference scheme. Numerical calculations are carried out for several sets of values of the dimensionless parameters and a careful study of the results obtained reveal that the flow field is influenced appreciably by the applied magnetic field in addition to the other parameters. Numerical results for the velocity and temperature fields, the local skin-friction coefficient and the local Nusselt number are presented graphically and discussed. To validate the numerical method, comparisons are made with the available results in the literature as special cases and the results are found to be in good agreement. The results obtained reveal many interesting behaviors that warrant further study of the flow and heat transfer characteristics over the permeable wedge.

MHD mixed convective heat transfer over a permeable stretching wedge with thermal radiation and ohmic heating

In this paper, a new analytical method named DTM-BF is proposed and applied to study the MHD mixed convective flow and heat transfer of an incompressible, viscous and electrically conducting fluid over a stretching permeable wedge in the presence of thermal radiation and ohmic heating. The boundary layer governing equations of PDES are transformed into highly nonlinear coupled ODES and the approximate solutions are derived by the new proposed analytical technique—the differential transformation and basis functions method (DTM-BF) on unbounded domains. The effects of the velocity ratio parameter, magnetic field parameter, the wedge angle parameter, the mixed convection parameter, suction/injection parameter, thermal radiation parameter, Eckert number and Prandtl number are presented. Moreover, the reliability and effectiveness of the DTM-BF have been verified by comparing analytical results with numerical solutions. Some new characteristics of the mixed convective flow and heat transfer over a wedge are investigated and discussed.

Analytical Solution of Slip Flow of Nanofluids over a Permeable Wedge in the Presence of Magnetic field

In this paper, a boundary layer analysis is presented for the slip flow of three types of incompressible viscous nanofluids past a permeable wedge in the presence of a magnetic field. Due to the appearance of a slip boundary condition at the surface, local similarity solution of the reduced nonlinear ordinary differential equation is obtained by the HAM coupled with minimizing the square residual error. The effects of pertinent parameters, such as the magnetic parameter, the solid volume fraction of nanoparticles, the slip parameter and the type of nanofluid on the flow, are analyzed and studied in details. It is found that Ag-water has the highest skin friction coefficient at the surface compared with the others.

MHD Effects on Non-Darcy Forced Convection Boundary Layer Flow past a Permeable Wedge in a Porous Medium with Uniform Heat Flux

The steady two-dimensional laminar forced flow and heat transfer of a viscous incompressible electrically conducting and heat-generating fluid past a permeable wedge embedded in non-Darcy high-porosity ambient medium with uniform surface heat flux has been studied. The governing equations are derived using the usual boundary layer and Bossinesq approximations and accounting for the applied magnetic filed, permeability of porous medium, variable porosity, inertia and heat generation effects. These equations and boundary conditions are non-dimenstionalized and transformed using non-similarity transformation. The resulting non-linear partial differential equations are then solved numerically subject to the transformed boundary conditions by a finite difference method. Comparisons with previously published works are performed and the results are found to be in excellent agreement. Numerical and graphical results for the velocity and temperature profiles as well as the skin friction and Nusselt number are presented and discussed for various parametric conditions.

An iterative approach to scattering by edges and wedges

A new approach to the calculation of the scattering of waves by the edge of a thin plate or the edge of a wedge of any angle is presented. It has a number of advantages relative to previous methods. It is intuitively easy to grasp, being close in spirit to ray optics methods for solving problems. It deals with the whole range of configurations in the one way, rather than the patchwork of ways that has evolved in the past. It provides extremely good approximate solutions with minimal effort. The mathematics involved is reasonably simple. It promises to be applicable to a range of problems which have yet to be solved by other methods. In particular, we show how it can be applied in principle to obtain successive approximations to the solution of the problem of diffraction by a permeable wedge, where its relationship to the ray optics method is very clear.

Flow of a viscous incompressible fluid of temperature dependent viscosity past a permeable wedge with uniform surface heat flux

The effect of uniform suction on the steady two-dimensional laminar forced flow of a viscous incompressible fluid of temperature dependent viscosity past a wedge with uniform surface heat flux is considered. The governing equations for the flow are obtained by using suitable transformations and are solved by using an implicit finite difference method. Perturbation solutions are also obtained near the leading edge and in the downstream regime. The results are obtained in terms of the local skin friction coefficient and the rate of heat transfer for various values of the pertinent parameters, such as the Prandtl number, Pr, the velocity gradient parameter, m, the local suction parameter, ξ, and the viscosity variation parameter, ɛ. Perturbation solutions are compared with the finite difference solutions and are found to be in excellent agreement. The effect of ξ, m and ɛ on the dimensionless velocity profiles and viscosity distribution are also presented graphically for Pr = 0.7 and 7.0, which are the appropriate values for gases and water respectively.