An extended study of the hydrodynamics of gravity-driven film flow of power-law fluids

A new similarity transformation has been devised for extensive studies of accelerating non-Newtonian film flow. The partial differential equations governing the hydrodynamics of the flow of a power-law fluid down along an inclined plane surface are transformed into a set of two ordinary differential equations by means of the dimensionless velocity component approach. Although the analysis is applicable for any angle of inclination (0<π/2), the resulting one-parameter problem involves only the power-law index n. Nevertheless, physically essential quantities, like the velocity components and the skin-friction coefficient, do depend on and relevant relationships are deduced between the vertical and inclined cases. Accurate numerical similarity solutions are provided for n in the range from 0.1 to 2.0. The present method enables solutions to be obtained also for highly pseudo-plastic films, i.e. for n below 0.5. The mass flow rate entrained into the momentum boundary layer from the inviscid freestream is expressed in terms of a dimensionless mass flux parameter Φ, which depends on the dimensionless boundary layer thickness and the velocity components at the edge of the viscous boundary layer. Φ, which is thus an integral part of the similarity solution, turns out to decrease monotonically with n. This parameter is of particular relevance in the determination of the streamwise position at which the entire freestream has been entrained and viscous stresses prevail all the way to the free surface of the film. A short-cut method to facilitate rapid and yet accurate estimates of the mass flux parameter is developed to this end.     

Mixed convection boundary layer flow along vertical moving thin needles with variable heat flux

The problem of steady laminar mixed convection boundary layer flow of an incompressible viscous fluid along vertical moving thin needles with variable heat flux for both assisting and opposing flow cases is theoretically considered in this paper. The governing boundary layer equations are first transformed into non-dimensional forms. The curvature effects are incorporated into the analysis whereas the pressure variation in the axial direction has been neglected. These equations are then transformed into similarity equations using the similarity variables, which are solved numerically using an implicit finite-difference scheme known as the Keller-box method. The solutions are obtained for a blunt-nosed needle (m = 0). Numerical calculations are carried out for various values of the dimensionless parameters of the problem, which include the mixed convection parameter λ, the Prandtl number Pr and the parameter a representing the needle size. It is shown from the numerical results that the skin friction coefficient, the surface (wall) temperature and the velocity and temperature profiles are significantly influenced by these parameters. The results are presented in graphical form and are discussed in detail.     

Natural convection flow of a couple stress fluid between two vertical parallel plates with Hall and ion-slip effects

The Hall and ion-slip effects on fully developed electrically conducting couple stress fluid flow between vertical parallel plates in the presence of a temperature dependent heat source are investigated. The governing non-linear partial differential equations are transformed into a system of ordinary differential equations using similarity transformations. The resulting equations are then solved using the homotopy analysis method (HAM). The effects of the magnetic parameter, Hall parameter, ion-slip parameter and couple stress fluid parameter on velocity and temperature are discussed and shown graphically.     

Non-darcy double-diffusive mixed convection from heated vertical and horizontal plates in saturated porous media

The non-darcy mixed convection flows from heated vertical and horizontal plates in saturated porous media have been considered using boundary layer approximations. The flows are considered to be driven by multiple buoyancy forces. The similarity solutions for both vertical and horizontal plates have been obtained. The governing equations have been solved numerically using a shooting method. The heat transfer, mass transfer and skin friction are reduced due to inertial forces. Also, they increase with the buoyancy parameter for aiding flow and decrease for the opposing flow. For aiding flow, the heat and mass transfer coefficients are found to approach asymptotically the forced or free convection values as the buoyancy parameter approaches zero or infinity.     

Chemical reaction effects on unsteady MHD flow past semi-infinite vertical porous plate with viscous dissipation

The chemical reaction effect on an unsteady magnetohydrodynamic (MHD) flow past a semi-infinite vertical porous plate with viscous dissipation is analyzed. The governing equations of motion, energy, and species are transformed into ordinary differential equations (ODEs) using the time dependent similarity parameter. The resultant ODEs are then solved numerically by a finite element method. The effects of various parameters on the velocity, temperature, and concentration profiles are presented graphically, and the values of the skin-friction, Nusselt number, and Sherwood number for various values of physical parameters are presented through tables.     

MHD free convective flow past semi-infinite vertical permeable wall

In this paper,the basic equations governing the flow and heat transfer of an incompressible viscous and electrically conducting fluid past a semi-infinite vertical permeable plate in the form of partial differential equations are reduced to a set of non-linear ordinary differential equations by applying a suitable similarity transformation.Approximate solutions of the transformed equations are obtained by employing the perturbation method for two cases,i.e.,small and large values of the suction parameter.From the numerical evaluations of the solution,it can be seen that the velocity field at any point decreases as the values of the magnetic and suction parameters increase.The effect of the magnetic parameter is to increase the thermal boundary layer.It is also found that the velocity and temperature fields decrease with the increase in the sink parameter.     

MHD Couette two-fluid flow and heat transfer in presence of uniform inclined magnetic field

The MHD Couette flow of two immiscible fluids in a parallel plate channel in the presence of an applied electric and inclined magnetic field is investigated in the paper. One of the fluids is assumed to be electrically conducting, while the other fluid and the channel plates are assumed to be electrically insulating. Separate solutions with appropriate boundary conditions for each fluid are obtained and these solutions are matched at the interface using suitable matching conditions. The partial differential equations governing the flow and heat transfer are transformed to ordinary differential equations and closed-form solutions are obtained in both fluid regions of the channel. The results for various values of the Hartmann number, the angle of magnetic field inclination, the loading parameter and the ratio of the heights of the fluids are presented graphically to show their effect on the flow and heat transfer characteristics.     

Numerical Study of a Magneto-fluid Motion Through a Porous Medium Between Two Wavy Plates with Constant Suction

In this note, the problem of an incompressible viscous fluid moving through a porous medium (Brinkman model) between two wavy plates under the effects of a constant inclined magnetic field that makes an angle with the vertical axis and constant suction, are studied numerically by a method related to the method of Takabatake and Ayukawa in 1982. The present approach is not restricted by any of the parameters appearing in the problem such as Reynolds number, magnetic parameter, suction parameter, the wave number and amplitude ratio. The variations in velocity, flow rate and pressure gradient with the above governing parameters are presented. Moreover, the effect of varying the porous medium and the inclined angle is also studied.     

Similarity solutions for hydromagnetic simultaneous heat and mass transfer by natural convection from an inclined plate with internal heat generation or absorption

 The problem of coupled heat and mass transfer by natural convection from a semi-infinite inclined flat plate in the presence of an external magnetic field and internal heat generation or absorption effects is formulated. The plate surface has a power-law variation of both wall temperature and concentration and is permeable to allow for possible fluid wall suction or blowing. The resulting governing equations are transformed using a similarity transformation and then solved numerically by an implicit, iterative, finite-difference scheme. Comparisons with previously published work are performed and good agreement is obtained. A parametric study of all involved parameters is conducted and a representative set of numerical results for the velocity and temperature profiles as well as the skin-friction parameter, average Nusselt number, and the average Sherwood number is illustrated graphically to show typical trends of the solutions. Received on 26 October 1998     

Fully Developed Mixed Convection Flow Between Inclined Parallel Plates Filled with a Porous Medium

A fully developed mixed convection flow between inclined parallel flat plates filled with a porous medium is considered through which there is a constant flow rate and with heat being supplied to the fluid by the same uniform heat flux on each plate. The equations governing this flow are made non-dimensional and are seen to depend on two dimensionless parameters, a mixed convection parameter λ and the Péclet number Pe, as well as the inclination γ of the plates to the horizontal. The velocity and temperature profiles are obtained in terms of λ, Pe and γ when the channel is inclined in an upwards direction as well as for horizontal channels. The limiting cases of small and large λ and small Pe are considered with boundary-layer structures being seen to develop on the plates for large values of λ.     

Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet with prescribed surface heat flux

The similarity solution for the problem of mixed convection boundary layer flow adjacent to a stretching vertical sheet in an incompressible electrically conducting fluid in the presence of a transverse magnetic field is presented. It is assumed that the sheet is stretched with a power-law velocity and is subjected to a variable surface heat flux. The governing partial differential equations are first transformed into a system of non-linear ordinary differential equations, before being solved numerically by the Keller-box method. The numerical results obtained are then compared with previously reported cases available in the literature as well as the series solution for certain values of parameters, to support their validity. The effects of the governing parameters on the flow field and heat transfer characteristics are obtained and discussed.     

Double-Diffusive Free Convection Flow Past an Inclined Plate Embedded in a Non-Darcy Porous Medium Saturated with a Nanofluid

In this investigation, we intend to present the influence of the prominent Soret effect on double-diffusive free convection heat and mass transfer in the boundary layer region of a semi-infinite inclined flat plate in a nanofluid saturated non-Darcy porous medium. The transformed boundary layer ordinary differential equations are solved numerically using the shooting and matching technique. Consideration of the nanofluid and the coupled convective process enhanced the number of non-dimensional parameters considerably thereby increasing the complexity of the present problem. A wide range of parameter values are chosen to bring out the effect of Soret parameter on the free convection process with varying angle of inclinations making the wall geometry from vertical to horizontal plate. The effects of angle of inclination and Soret parameter on the flow, heat and mass transfer coefficients are analyzed. The numerical results obtained for the velocity, temperature, volume fraction, and concentration profiles, local wall temperature, local nanoparticle concentration, and local wall concentration reveal interesting phenomenon, and some of these qualitative results are presented through the plots.     

MHD non-Darcian flow through a non-isothermal vertical surface embedded in a porous medium with radiation

The effect of MHD on steady two-dimensional laminar mixed flow about a vertical porous surface is numerically analyzed. Also the effects of radiation and heat generation and absorption are considered. A power law variation of temperature along the vertical wall is assumed. The nonlinear boundary-layer equations were transformed and the resulting differential equations were solved by an implicit finite difference scheme (Keller box method). Numerical results for the velocity distribution and the temperature distribution are presented for various values of Prandtl number Pr, magnetic parameter, porous medium parameter and internal heat generation or absorption coefficient. Further validation with previous works is carried out.     

Radiation effect on Darcy free convection flow along an inclined surface placed in porous media

The paper investigates the effect of radiation on Darcy's buoyancy induced flow of an optically dense viscous incompressible fluid along a heated inclined flat surface maintained at uniform temperature placed in a saturated porous medium with Rosseland diffusion approximation employing the implicit finite difference method together with Keller box elimination technique. Both the streamwise and normal components of the buoyancy force are retained in the momentum equations. The numerical results show that as the buoyancy parameter, ξ, increases the local Nusselt number increases. The results for the locally nonsimilar solutions are compared with the locally similar solutions for small angle of inclination and approximate similar solutions along vertical surface. The effect of the conduction-radiation parameter, R _d , and the surface temperature excess ration, θ _w , on the local Nusselt number, the tangential velocity distribution and the temperature distribution are also shown graphically.     

MHD mixed convection–radiation interaction along a permeable surface immersed in a porous medium in the presence of Soret and Dufour’s Effects

This work is focused on the numerical modeling of steady, laminar, heat and mass transfer by MHD mixed convection from a semi-infinite, isothermal, vertical and permeable surface immersed in a uniform porous medium in the presence of thermal radiation and Dufour and Soret effects. A mixed convection parameter for the entire range of free-forced-mixed convection is employed and the governing equations are transformed into non-similar equations. These equations are solved numerically by an efficient, implicit, iterative, finite-difference scheme. The obtained results are checked against previously published work on special cases of the problem and are found to be in excellent agreement. A parametric study illustrating the influence of the thermal radiation coefficient, magnetic field, porous medium inertia parameter, concentration to thermal buoyancy ratio, and the Dufour and Soret numbers on the fluid velocity, temperature and concentration as well as the local Nusselt and the Sherwood numbers is conducted. The obtained results are shown graphically and the physical aspects of the problem are discussed.     

Non-uniform heat generation effect on heat transfer of a non-Newtonian power-law fluid over a non-linearly stretching sheet

The effects of non-uniform heat generation/absorption and viscous dissipation on heat transfer of a non-Newtonian power-law fluid on a non-linearly stretching surface have been examined. The governing nonlinear partial differential equations describing the problem are transformed to a system of non-linear ordinary differential equations by using suitable similarity transformation. The transformed system of ordinary differential equations is solved numerically using fourth order Runge-Kutta method with the shooting technique. Graphical solutions for the dimensionless temperature are presented and discussed for various values of the power-law index parameter, the Prandtl number, the heat generation/absorption parameter and the Eckert number. The results show that the local Nusselt number is reduced with increasing the Eckert number or the heat generation parameter, whereas the heat absorption parameter has the effect of enhancing the local Nusselt number.     

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.     

Two fluid flow and heat transfer in an inclined channel containing porous and fluid layer

Convective flow and heat transfer in an inclined channel bounded by two rigid plates held at constant different temperatures with one region filled with porous matrix saturated with a viscous fluid and another region with a clear viscous fluid different from the fluid in first region is studied analytically. The coupled nonlinear governing equations are solved using regular perturbation method. It is found that the presence of porous matrix in one of the region reduces the velocity and temperature. Results have been presented for a wide range of governing parameters such as Grashof number, porous parameter, angle of inclination, ratio of heights of the two layers and also the ratio of viscosities.     

Effects of variable thermal conductivity,viscous dissipation on steady MHD natural convection flow of low Prandtl fluid on an inclined porous plate with Ohmic heating

Abstract Aim of the paper is to investigate the effects of linearly varying thermal conductivity, viscous dissipation and Ohmic heating on steady free convection flow of a viscous incompressible electrically conducting liquid having low Prandtl number along an inclined isothermal non-conducting porous plate in the presence of transverse magnetic field. The governing equations of continuity, momentum and energy are transformed into ordinary differential equations using similarity transformation. The resulting coupled and non-linear ordinary differential equations are solved using Runge-Kutta fourth order method and shooting technique. The velocity and temperature distributions are discussed numerically and presented through graphs. Skin-friction coefficient and Nusselt number at the plate are derived, discussed and their numerical values for various values of physical parameters are presented through tables.     

On stagnation point flow of Sisko fluid over a stretching sheet

The steady two-dimensional stagnation-point flow, represented by Sisko fluid constitutive model, over a stretching sheet is investigated theoretically. Using suitable similarity transformations, the governing boundary-layer equations are transformed into the self-similar non-linear ordinary differential equation. The transformed equation is then solved using a very efficient analytic technique namely the homotopy analysis method (HAM) and the HAM solutions are validated by the exact analytic solutions obtain in certain special cases. The influence of the power-law index (n), the material parameter (A) and the velocity ratio parameter (d/c) on the flow characteristics is studied and presented through several graphs. In addition, the local skin friction coefficient for several values of these parameters is tabulated and examined. The similarity solutions for both the Newtonian and the power-law fluids are presented as special cases of the analysis. The results obtained reveal that, in comparison with the Newtonian and the power-law fluids, the velocity profiles of the Sisko fluid are much faster (slower), for d/c<1 (d/c>1), respectively.