Scaling Transformations for Boundary Layer Flow near the Stagnation-Point on a Heated Permeable Stretching Surface in a Porous Medium Saturated with a Nanofluid and Heat Generation/Absorption Effects

In this article, a similarity solution of the steady boundary layer flow near the stagnation-point flow on a permeable stretching sheet in a porous medium saturated with a nanofluid and in the presence of internal heat generation/absorption is theoretically studied. The governing partial differential equations with the corresponding boundary conditions are reduced to a set of ordinary differential equations with the appropriate boundary conditions via Lie-group analysis. Copper (Cu) with water as its base fluid has been considered and representative results have been obtained for the nanoparticle volume fraction parameter f{\phi} in the range 0 ￡ f ￡ 0.2{0\leq \phi \leq 0.2} with the Prandtl number of Pr = 6.8 for the water working fluid. Velocity and temperature profiles as well as the skin friction coefficient and the local Nusselt number are determined numerically. The influence of pertinent parameters such as nanofluid volume fraction parameter, the ratio of free stream velocity and stretching velocity parameter, the permeability parameter, suction/blowing parameter, and heat source/sink parameter on the flow and heat transfer characteristics is discussed. Comparisons with published results are also presented. It is shown that the inclusion of a nanoparticle into the base fluid of this problem is capable to change the flow pattern.     

Effect of Viscous Dissipation on the Boundary Layer Flow and Heat Transfer Past a Permeable Stretching Surface Embedded in a Porous Medium with a Second-Order Slip Using Chebyshev Finite Difference Method

This article investigates a theoretical and numerical study for the effect of viscous dissipation on the steady flow with heat transfer of Newtonian fluid toward a permeable stretching surface embedded in a porous medium with a second-order slip and thermal slip. The governing nonlinear partial differential equations are converted into nonlinear ordinary differential equations (ODEs) using similarity variables. The resulting ODEs are successfully solved numerically with the help of Chebyshev finite difference method. Graphically results are shown for non-dimensional velocities and temperature. The effects of the porous parameter, the suction (injection) parameter, Eckert number, first- and second-order velocity slip parameter, the thermal slip parameter and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and Nusselt numbers are presented. A comparison of numerical results is made with the earlier published results under limiting cases.     

Three-dimensional flow of a magnetohydrodynamic Casson fluid over an unsteady stretching sheet embedded into a porous medium

A three-dimensional flow of a magnetohydrodynamic Casson fluid over an unsteady stretching surface placed into a porous medium is examined. Similarity transformations are used to convert time-dependent partial differential equations into nonlinear ordinary differential equations. The transformed equations are then solved analytically by the homotopy analysis method and numerically by the shooting technique combined with the Runge–Kutta–Fehlberg method. The results obtained by both methods are compared with available reported data. The effects of the Casson fluid parameter, magnetic field parameter, and unsteadiness parameter on the velocity and local skin friction coefficients are discussed in detail.     

Melting effect and Cattaneo-Christov heat flux in fourth-grade material flow through a Darcy-Forchheimer porous medium

The melting phenomenon in two-dimensional (2D) flow of fourth-grade material over a stretching surface is explored. The flow is created via a stretching surface. A Darcy-Forchheimer (D-F) porous medium is considered in the flow field. The heat transport is examined with the existence of the Cattaneo-Christov (C-C) heat flux. The fourth-grade material is electrically conducting subject to an applied magnetic field. The governing partial differential equations (PDEs) are reduced into ordinary differential equations (ODEs) by appropriate transformations. The solutions are constructed analytically through the optimal homotopy analysis method (OHAM). The fluid velocity, temperature, and skin friction are examined under the effects of various involved parameters. The fluid velocity increases with higher material parameters and velocity ratio parameter while decreases with higher magnetic parameter, porosity parameter, and Forchheimer number. The fluid temperature is reduced with higher melting parameter while boosts against higher Prandtl number, magnetic parameter, and thermal relaxation parameter. Furthermore, the skin friction coefficient decreases against higher melting and velocity ratio parameters while increases against higher material parameters, thermal relaxation parameter, and Forchheimer number.

     

Boundary layer flow and heat transfer over an unsteady stretching vertical surface

The solution to the unsteady mixed convection boundary layer flow and heat transfer problem due to a stretching vertical surface is presented in this paper. The unsteadiness in the flow and temperature fields is caused by the time-dependent of the stretching velocity and the surface temperature. The governing partial differential equations with three independent variables are first transformed into ordinary differential equations, before they are solved numerically by a finite-difference scheme. The effects of the unsteadiness parameter, buoyancy parameter and Prandtl number on the flow and heat transfer characteristics are thoroughly examined. Both assisting and opposing buoyant flows are considered. It is observed that for assisting flow, the solutions exist for all values of buoyancy parameter, whereas for opposing flow, they exist only if the magnitude of the buoyancy parameter is small. Comparison with known results for steady-state flow is excellent.     

Differential transformation method for studying flow and heat transfer due to stretching sheet embedded in porous medium with variable thickness,variable thermal conductivity,and thermal radiation

This article presents a numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet embedded in a porous medium with the power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The governing partial differential equations （PDEs） are transformed into a system of coupled non-linear ordinary differential equations （ODEs） with appropriate boundary conditions for various physical parameters. The remaining system of ODEs is solved numerically using a differential transformation method （DTM）. The effects of the porous parameter, the wall thickness parameter, the radiation parameter, the thermal conductivity parameter, and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and the Nusselt numbers are presented. Comparison of the obtained numerical results is made with previously published results in some special cases, with good agreement. The results obtained in this paper confirm the idea that DTM is a powerful mathematical tool and can be applied to a large class of linear and non-linear problems in different fields of science and engineering.     

Effects of variable fluid viscosity on flow past a heated stretching sheet embedded in a porous medium in presence of heat source/sink

The boundary layer flow and heat transfer of a fluid through a porous medium towards a stretching sheet in presence of heat generation or absorption is considered in this analysis. Fluid viscosity is assumed to vary as a linear function of temperature. The symmetry groups admitted by the corresponding boundary value problem are obtained by using a special form of Lie group transformations viz. scaling group of transformations. These transformations are used to convert the partial differential equations corresponding to the momentum and the energy equations into highly non-linear ordinary differential equations. Numerical solutions of these equations are obtained by shooting method. It is found that the horizontal velocity decreases with increasing temperature-dependent fluid viscosity parameter up to the crossing-over point but increases after that point and the temperature decreases in this case. With the increase of permeability parameter of the porous medium the fluid velocity decreases but the temperature increases at a particular point of the sheet. Effects of Prandtl number on the velocity boundary layer and on the thermal boundary layer are studied and plotted.     

Free Convection Boundary Layer Flow Past a Vertical Surface in a Porous Medium with Temperature-Dependent Properties

Numerical solutions for the free convection heat transfer in a viscous fluid at a permeable surface embedded in a saturated porous medium, in the presence of viscous dissipation with temperature-dependent variable fluid properties, are obtained. The governing equations for the problem are derived using the Darcy model and the Boussinesq approximation (with nonlinear density temperature variation in the buoyancy force term). The coupled non-linearities arising from the temperature-dependent density, viscosity, thermal conductivity, and viscous dissipation are included. The partial differential equations of the model are reduced to ordinary differential equations by a similarity transformation and the resulting coupled, nonlinear ordinary differential equations are solved numerically by a second order finite difference scheme for several sets of values of the parameters. Also, asymptotic results are obtained for large values of | f _w|. Moreover, the numerical results for the velocity, the temperature, and the wall-temperature gradient are presented through graphs and tables, and are discussed. It is observed that by increasing the fluid variable viscosity parameter, one could reduce the velocity and thermal boundary layer thickness. However, quite the opposite is true with the non-linear density temperature variation parameter.     

On the numerical solution for the flow and heat transfer in a thin liquid film over an unsteady stretching sheet in a saturated porous medium in the presence of thermal radiation

The problem of the flow and heat transfer over an unsteady stretching sheet embedded in a porous medium in the presence of thermal radiation is studied theoretically and numerically. The continuity, momentum, and energy equations, which are coupled nonlinear partial differential equations, are reduced to a set of two nonlinear ordinary differential equations. Special attention is given to study the convergence of the proposed method. The error estimation is also given. The effects of various parameters, such as the Darcy parameter, the radiation parameter, and the Prandtl number, on the flow and temperature profiles, as well as on the local skin-friction coefficient and the local Nusselt number are presented and discussed. The results obtained agree very well with the data obtained by the Runge-Kutta method coupled with the shooting technique.     

Hydromagnetic thin film flow of Casson fluid in non-Darcy porous medium with Joule dissipation and Navier's partial slip

In this paper, the effects of viscous and Ohmic heating and heat generation/absorption on magnetohydrodynamic flow of an electrically conducting Casson thin film fluid over an unsteady horizontal stretching sheet in a non-Darcy porous medium are investigated. The fluid is assumed to slip along the boundary of the sheet. Similarity transformation is used to translate the governing partial differential equations into ordinary differential equations. A shooting technique in conjunction with the 4 th order Runge-Kutta method is used to solve the transformed equations. Computations are carried out for velocity and temperature of the fluid thin film along with local skin friction coefficient and local Nusselt number for a range of values of pertinent flow parameters. It is observed that the Casson parameter has the ability to enhance free surface velocity and film thickness, whereas the Forchheimer parameter, which is responsible for the inertial drag has an adverse effect on the fluid velocity inside the film. The velocity slip along the boundary tends to decrease the fluid velocity. This investigation has various applications in engineering and in practical problems such as very large scale integration(VLSI) of electronic chips and film coating.     

Radiation effect on mixed convection from a horizontal surface in a porous medium

The effect of thermal radiation on the non-Darcy mixed convection flow over a non-isothermal horizontal surface immersed in a saturated porous medium has been studied. The wall temperature is assumed to have a power-law variation with the distance measured from the leading edge of the plate. The non-linear coupled parabolic partial differential equations governing the flow have been solved numerically using a finite-difference scheme. For some particular cases, the self-similar solution has also been obtained. The heat transfer is found to be strongly influenced by the radiative flux number, buoyancy parameter, variation of wall temperature, non-Darcy parameter and the nature of the free stream velocity.     

The influence of thermal radiation on hydromagnetic Darcy-Forchheimer mixed convection flow past a stretching sheet embedded in a porous medium

A boundary layer analysis is performed to study the influence of thermal radiation and buoyancy force on two-dimensional magnetohydrodynamic flow of an incompressible viscous and electrically conducting fluid over a vertical stretching sheet embedded in a porous medium in the presence of inertia effect. The governing system of partial differential equations is first transformed into system of ordinary differential equations using self-similarity transformation. A special form for magnetic field is chosen to obtain the similarity solution. The transformed boundary layer equations are solved numerically for some important values of the physical parameters. The present results are compared with the previously published papers and the results are found to be in excellent agreement. The important features of the flow, heat and mass transfer characteristics for different values of thermal radiation, porous permeability, magnetic field and buoyancy parameters are analyzed and discussed. The effects of various physical parameters on the skin friction coefficient, local Nusselt number and local Sherwood number are also presented. It is found that increase in the value of thermal radiation parameter R ₁ increases the skin friction coefficient and Sherwood number whereas reverse trend is seen for the local Nusselt number.     

Stretching surface in rotating viscoelastic fluid

The boundary layer flow over a stretching surface in a rotating viscoelastic fluid is considered. By applying a similarity transformation, the governing partial differ- ential equations are converted into a system of nonlinear ordinary differential equations before being solved numerically by the Keller-box method. The effects of the viscoelastic and rotation parameters on the skin friction coefficients and the velocity profiles are thor- oughly examined. The analysis reveals that the skin friction coefficients and the velocity in the x-direction increase as the viscoelastic parameter and the rotation parameter in- crease. Moreover, the velocity in the y-direction decreases as the viscoelastic parameter and the rotation parameter increase.     

Computational modeling of heat transfer over an unsteady stretching surface embedded in a porous medium

The present paper deals with the study of heat transfer characteristics in the laminar boundary layer flow of an incompressible viscous fluid over an unsteady stretching sheet which is placed in a porous medium in the presence of viscous dissipation and internal absorption or generation. Similarity transformations are used to convert the governing time dependent nonlinear boundary layer equations into a system of non-linear ordinary differential equations containing Prandtl number, Eckert number, heat source/sink parameter, porous parameter and unsteadiness parameter with appropriate boundary conditions. These equations are solved numerically by applying shooting method using Runge-Kutta-Fehlberg method. Comparison of numerical results is made with the earlier published results under limiting cases. The effects of the parameters which determine the velocity and temperature fields are discussed in detail.     

Flow and heat transfer of a slightly rarefied gas over a stretching surface

This work deals with the study of the boundary layer flow and mass transfer of a visco-elastic fluid immersed in a porous medium over a stretching surface in the presence of surface slip, chemical reaction and variable viscosity. The partial differential equations governing the flow have been transformed by similarity transformation into a system of coupled nonlinear ordinary differential equations which is solved numerically by means of the fourth order Runge-Kutta integration scheme coupled with the shooting technique. The effects of various involved interesting parameters on the velocity fields and concentration fields are shown graphically and investigated. In addition, tabulated results for the local skin-friction coefficient and the local Sherwood number are presented and discussed.     

Steady nonlinear hydromagnetic flow and heat transfer over a stretching surface of variable temperature

The steady nonlinear hydromagnetic flow of an incompressible, viscous and electrically conducting fluid with heat transfer over a surface of variable temperature stretching with a power-law velocity in the presence of variable transverse magnetic field is analysed. Utilizing similarity transformation, governing nonlinear partial differential equations are transformed to nonlinear ordinary differential equations and they are numerically solved using fourth-order Runge–Kutta shooting method. Numerical solutions are illustrated graphically by means of graphs. The effects of magnetic field, stretching parameter and Prandtl number on velocity, skin friction, temperature distribution and rate of heat transfer are discussed.     

Chemical reaction and variable viscosity effects on flow and mass transfer of a non-Newtonian visco-elastic fluid past a stretching surface embedded in a porous medium

This work deals with the study of the boundary layer flow and mass transfer of a visco-elastic fluid immersed in a porous medium over a stretching surface in the presence of surface slip, chemical reaction and variable viscosity. The partial differential equations governing the flow have been transformed by similarity transformation into a system of coupled nonlinear ordinary differential equations which is solved numerically by means of the fourth order Runge-Kutta integration scheme coupled with the shooting technique. The effects of various involved interesting parameters on the velocity fields and concentration fields are shown graphically and investigated. In addition, tabulated results for the local skin-friction coefficient and the local Sherwood number are presented and discussed.     

Effects of thermal stratification on flow and heat transfer past a porous vertical stretching surface

The aim of this paper is to present the boundary layer flow of viscous incompressible fluid due to a porous vertical stretching surface with a power-law stretching velocity in a thermally stratified medium. Using a special form of Lie group transformations viz. scaling group of transformations, similarity solutions for this problem are obtained. The equations are then solved numerically. With increasing values of the stratification parameter, the velocity as well as temperature decreases. At a particular point of the porous stretching sheet, the velocity decreases with the increasing suction parameter. The dimensionless temperature at a point of the sheet decreases due to suction but increases due to injection. The findings of this study reveal that stratification and suction can be used as means of cooling the boundary layer flow region.     

Soret and dufour effects on MHD free convective heat and mass transfer with thermophoresis and chemical reaction over a porous stretching surface: Group theory transformation

The group theoretic method is applied for solving the problem of the combined influence of the thermal diffusion and diffusion thermoeffect on magnetohydrodynamic free convective heat and mass transfer over a porous stretching surface in the presence of thermophoresis particle deposition with variable stream conditions. The application of one-parameter groups reduces the number of independent variables by one; consequently, the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The equations along with the boundary conditions are solved numerically by using the Runge-Kutta-Gill integration scheme with the shooting technique. The impact of the Soret and Dufour effects in the presence of thermophoresis particle deposition with a chemical reaction plays an important role on the flow field.     

Boundary-layer flow of Reiner–Philippoff fluids past a stretching wedge

This paper presents a numerical analysis of the steady boundary-layer flow of a Reiner–Philippoff fluid induced by a 90° stretching wedge in a variable free stream. The governing partial differential equations are converted into a set of two ordinary differential equations by the use of a similarity transformation. The flow is therefore governed by a stretching velocity parameter λ and two non-Newtonian fluid parameters γ and μ₀. The variation of the skin friction, as well as other flow characteristics, as a function of the governing parameters is presented graphically and tabulated. A stability analysis has also been performed for this self-similar flow based on linear disturbances to the steady similarity solutions. The results presented in this paper reveal that there are no multiple (dual) solutions for the present problem and the unique solution is stable.