Free Convection Boundary Layer Research Articles

Soret and Dufour effects on free convection heat and mass transfer from an arbitrarily inclined plate in a porous medium with constant wall temperature and concentration

The free convection boundary layer flow over an arbitrarily inclined heated plate in a porous medium with Soret and Dufour effects is studied by transforming the governing equations into a universal form. The generalized equations can be used to derive the similarity solutions for limiting cases of horizontal and vertical plates and to calculate the heat and mass transfer characteristics between these two limiting cases. The heat and mass transfer characteristics are presented as functions of Soret parameter, Dufour parameter, inclination variable, Lewis number, and buoyancy ratio. Results show that an increase in the Dufour parameter tends to decrease the local heat transfer rate, and an increase in the Soret parameter tends to decrease the local mass transfer rate. As the inclination variable increases, the local Nusselt number and the local Sherwood number decrease from their respective values for horizontal plates, reach their respective minima, and then increase to their respective values for vertical plates. The minima are where the tangential and normal components of buoyancy force are comparable.

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 | fw|. 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.

Natural convection boundary layer of a non-Newtonian fluid about a permeable vertical cone embedded in a porous medium saturated with a nanofluid

An analysis was performed to study the effect of uniform transpiration velocity on free convection boundary-layer flow of a non-Newtonian fluid over a permeable vertical cone embedded in a porous medium saturated with a nanofluid. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The governing partial differential equations are transformed into a set of non-similar equations and solved numerically by an efficient implicit, iterative, finite-difference method. Comparisons with previously published work are performed and excellent agreement is obtained. A parametric study of the physical parameters is conducted and a representative set of numerical results for the velocity, temperature, and volume fraction profiles as well as the local Nusselt and Sherwood numbers is illustrated graphically to show interesting features of the solutions.

Influence of yield stress on free convective boundary-layer flow of a non-Newtonian nanofluid past a vertical plate in a porous medium

The effect of yield stress on the free convective heat transfer of dilute liquid suspensions of nanofluids flowing on a vertical plate saturated in porous medium under laminar conditions is investigated considering the nanofluid obeys the mathematical model of power-law. The model used for non-Newtonian nanofluid incorporates the effects of Brownian motion and thermophoresis. The governing boundary-layer equations are cast into dimensionless system which is solved numerically using a deferred correction technique and Newton iteration. This solution depends on yield stress parameter Ω, a power-law index n, Lewis number Le, a buoyancy-ratio number Nr, a Brownian motion number Nb, and a thermophoresis number Nt. Analyses of the results found that the reduced Nusselt and Sherwood numbers are decreasing functions of the higher yield stress parameter for each dimensionless numbers, n and Le, except the reduced Sherwood number is an increasing function of higher Nb for different values of yield stress parameter.

Solving of boundary-layer equations with transpiration effects, governance on a vertical permeable cylinder using modified differential transform method

The suction and injection effects on the free convection boundary-layer flow over a vertical cylinder are studied. The main stream velocity and wall temperature are proportional to the axial distance along the surface of the cylinder. Both analytic and numerical solutions of the arising mathematical problem are obtained. An analytic solution is derived by a new analytical method (DTM-Pade) and numerical solutions have been performed by using a fourth-order Runge–Kutta and shooting methods. Velocity and temperature profiles are shown graphically. It is shown that the differential transform method (DTM) solutions are only valid for small values of the independent variable but the obtained results by DTM-Pade are valid for the whole solution domain with high accuracy. These methods can be easily extended to other linear and nonlinear equations and so can be found widely applicable in engineering and sciences. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.20366

Free Convection Boundary Layer Flow Past a Horizontal Flat Plate Embedded in a Porous Medium Filled With a Nanofluid

A similarity solution is presented for the steady free convection boundary layer flow past a horizontal flat plate embedded in a porous medium filled with nanofluids. The model used for the nanofluid is the one which incorporates the effects of Brownian motion and thermophoresis. For the porous medium the Darcy-Boussinesq model is employed. This solution depends on a Lewis number Le, a buoyancy-ratio parameter Nr, a Brownian motion parameter Nb, and a thermophoresis parameter Nt. The effects of these parameters on the velocity, temperature and nanoparticle fraction profiles are discussed. The dependency of the local Nusselt and Sherwood numbers on these four parameters is also investigated.

Free convection stagnation-point boundary-layer flow in a porous medium with a density maximum

The steady free convection boundary-layer flow near a stagnation point in a fluid-saturated porous medium is considered when the convecting fluid is close to its maximum density. Three forms for the wall boundary condition are treated, a prescribed wall temperature, prescribed wall heat flux and Newtonian heating. In each case the flow and heat transfer characteristics are determined by a dimensionless parameter δ that measures the difference between the ambient temperature and the temperature at which the fluid attains its density maximum. We find that solutions are possible for δ ≥ 0 for each case. For δ < 0 there is a critical value δ c of δ, the value of which depends on the boundary conditions applied, with solutions possible only for δ ≥ δ c . The nature of this critical value, as well as other limiting asymptotic forms is discussed.

A comparison of three quantitative schlieren techniques

We compare the results of three quantitative schlieren techniques applied to the measurement and visualization of a two-dimensional laminar free-convection boundary layer. The techniques applied are Schardin's “calibrated” schlieren technique, in which a weak lens in the field-of-view provides a calibration of light deflection angle to facilitate quantitative measurements, “rainbow schlieren”, in which the magnitude of schlieren deflection is coded by hue in the image, and “background-oriented schlieren” (BOS), in which quantitative schlieren-like results are had from measuring the distortion of a background pattern using digital-image-correlation software. In each case computers and software are applied to process the data, thus streamlining and modernizing the quantitative application of schlieren optics. (BOS, in particular, is only possible with digital-image-correlation software.) Very good results are had with the lens-calibrated standard schlieren method in the flow tested here. BOS likewise produces good results and requires less expensive apparatus than the other methods, but lacks the simplification of parallel light that they feature. Rainbow schlieren suffers some unique drawbacks, including the production of the required rainbow cutoff filter, and provides little significant benefit over the calibrated schlieren technique.

Soret and Dufour effects on free convection boundary layers over an inclined wavy surface in a porous medium

The problem of free convection boundary layer flow near an inclined wavy surface in a fluid saturated porous medium with Soret and Dufour effects has been solved using boundary layer approximations. A suitable coordinate transformation is used to transform the complex wavy surface to a smooth surface, and the obtained boundary layer equations are solved by the cubic spline collocation method. Effects of Soret parameter, Dufour parameter, angle of inclination, Lewis number, buoyancy ratio, and wavy geometry on the heat and mass transfer characteristics are studied. Results show that increasing the Soret parameter leads to a greater fluctuation of the local Nusselt number with the streamwise coordinate and a smaller fluctuation of the local Sherwood number with the streamwise coordinate. Moreover, as the Dufour parameter increases, the fluctuation of the local Sherwood number with the streamwise coordinate is enhanced, while the amplitude of fluctuation of the local Nusselt number with the streamwise coordinate is reduced. An increase in the Dufour parameter tends to decrease the dimensionless total heat transfer rate while an increase in the Soret parameter tends to decrease the dimensionless total mass transfer rate.

Radiation effect on chemically reacting magnetohydrodynamics (MHD) boundary layer flow of heat and mass transfer through a porous vertical flat plate

A mathematical model is presented for a two-dimensional, steady, viscous, incompressible, electrically conducting and laminar free convection boundary layer flow with radiation from a flat plate in a chemically reactive medium in the presence of a transverse magnetic field. The basic equations governing the flow are in the form of partial differential equations and have been reduced to a set of non-linear ordinary differential equations by applying suitable similarity transformations. The problem is tackled numerically using shooting techniques with the fourth order Runge-Kutta integration scheme. Pertinent results with respect to embedded parameters are displayed graphically for the velocity, temperature and concentration profiles and discussed quantitatively. Key words: Heat transfer, thermal radiation, magnetic field, chemical species, mixed convection.

HOMOTOPY ANALYSIS OF FREE CONVECTION BOUNDARY LAYER FLOW WITH HEAT AND MASS TRANSFER

A steady two-dimensional laminar boundary layer flow of an incompressible viscous fluid over a semi-infinite surface is considered to investigate the accuracy of the homotopy analysis method. The governing coupled nonlinear system of differential equations is solved by means of the HAM approach. Explicit analytical series solutions are obtained and compared with numerical solutions. Good agreement is observed between the numerical results and HAM analytical solutions.

Soret and Dufour effects on free convection boundary layers of non-Newtonian power law fluids with yield stress in porous media over a vertical plate with variable wall heat and mass fluxes

This work studies the Soret and Dufour effects on the free convection boundary layers over a vertical plate with variable wall heat and mass fluxes in a porous medium saturated with a non-Newtonian power law fluid with yield stress. The governing equations are transformed into a dimensionless form by the similarity transformation and then solved by a cubic spline collocation method. Results are presented for the local surface temperature and concentration for various parameters of the power law fluid with yield stress in porous media. An increase in the power law exponent decreases the local surface temperature and concentration, thus increasing the local Nusselt and Sherwood numbers. An increase in the Soret parameter tends to increase the local surface concentration, thus decreasing the local Sherwood number. Moreover, increasing the Dufour number increases the surface temperature and thus decreases the local Nusselt number.

Variable viscosity effect on free convection of a non-Newtonian power-law fluid over a vertical cone in a porous medium with variable heat flux

The effect of variable viscosity on the free-convection boundary layer over a permeable vertical cone in a porous medium saturated with non-Newtonian power-law fluid has been studied numerically. The governing equations describing the problem are transformed into a system of nonlinear ordinary differential equations by using a similarity transformation, which is solved numerically using the Chebyshev spectral method. The effects of the power-law index, blowing/suction parameter and the viscosity parameter on the temperature profiles and the local Nusselt number are discussed.

An Exponential Series Method for the Solution of Free Convection Boundary Layer Flow in a Saturated Porous Medium

Third order nonlinear ordinary differential equation, subject to appropriate boundary conditions, arising in fluid mechanics is solved exactly using more suggestive schemes- Dirichlet series and method of stretching variables. These methods have advantages over pure numerical methods in obtaining derived quantities accurately for various values of the parameters involved at a stretch and are valid in a much larger domain compared with classical numerical schemes.

Numerical Analysis of Couple Stress Fluid Past an Infinite Vertical Cylinder

:Numerical analysis is performed to study the transient free convective boundary layer flow of a couple stress fluid past an infinite vertical cylinder, in the absence of body forces and body couples. A set of nondimensional governing equations, namely, the continuity, momentum and energy equations are derived, which are unsteady, non-linear and coupled. The couple stress fluid flow model introduces the length dependent effect based on the material constant and dynamic viscosity. Also, it introduces the biharmonic operator in the Navier-Stokes equations, which is not present in the case of Newtonian fluids. As there are no analytical or direct numerical method available to solve these unsteady, non-linear and coupled equations, they are solved using the CFD techniques. An unconditionally stable Crank-Nicolson type of implicit finite difference scheme is employed to obtain the discretized forms of the governing equations. The discretized equations are solved using the Thomas and pentadiagonal algorithm. The results concerning the velocity and temperature profiles across the boundary layer are illustrated graphically and discussed for different values of Prandtl number. Transient effects of velocity and temperature are analyzed and compared with those of the Newtonian fluids. The heat transfer characteristics are analyzed and compared with those of Newtonian fluids with the help of average skin-friction and Nusselt number and are shown graphically. It is observed that the deviation of transient velocity and temperature profiles from the hot wall of a couple stress fluid is more in comparison with that of Newtonian fluids.

A transient natural convection of micropolar fluids over a vertical cylinder

A transient free convective boundary layer flow of micropolar fluids past a semi-infinite cylinder is analysed in the present study. The transformed dimensionless governing equations for the flow, microrotation and heat transfer are solved by using the implicit scheme. For the validation of the current numerical method heat transfer results for a Newtonian fluid case where the vortex viscosity is zero are compared with those available in the existing literature, and an excellent agreement is obtained. The obtained results concerning velocity, microrotation and temperature across the boundary layer are illustrated graphically for different values of various parameters and the dependence of the flow and temperature fields on these parameters is discussed. An increase in the vortex viscosity tends to increase the magnitude of microrotation and thus decreases the peak velocity of fluid flow. An increase in the vortex viscosity in micropolar fluids is shown to decrease the heat transfer rate.

Soret and Dufour effects on free convection boundary layer over a vertical cylinder in a saturated porous medium

This paper deals with an analysis of the Soret and Dufour effects on the boundary layer flow due to free convection heat and mass transfer over a vertical cylinder in a porous medium saturated with Newtonian fluids with constant wall temperature and concentration. A suitable coordination transformation is used to derive the similar governing boundary-layer equations, and the cubic spline collocation method is then employed to solve the similar governing boundary-layer equations. The variation of the Nusselt number and the Sherwood number with the Dufour parameter and the Soret parameter for various Lewis numbers and buoyancy ratios have been presented in this work. Results show that an increase in the Soret number leads to a decrease in the local Sherwood number and an increase in the local Nusselt number. The local Nusselt number tends to decrease as the Dufour parameter is increased. Moreover, an increase in the Lewis number enhances the effect of the Dufour parameter on the local Nusselt number.

Consequences of the Translation Invariance on the Darcy Free Convection Flow Past a Vertical Surface

It is shown that the governing equation for the stream function of the Darcy free convection boundary layer flows past a vertical surface is invariant under arbitrary translations of the transverse coordinate y. The consequences of this basic symmetry property on the solutions corresponding to a prescribed surface temperature distribution T w (x) are investigated. It is found that starting with a “primary solution” which describes the temperature boundary layer on an impermeable surface, infinitely many “translated solutions” can be generated which form a continuous group, the “translation group” of the given primary solution. The elements of this group describe free convection boundary layer flows from permeable counterparts of the original surface with a transformed temperature distribution $${\tilde {T}_w \left( x \right)}$$ , when simultaneously a suitable lateral suction/injection of the fluid is applied. It turns out in this way that several exact solutions discovered during the latter few decades are in fact not basically new solutions, but translated counterparts of some formerly reported primary solutions. A few specific examples are discussed in detail.

Parameter-dependent systems on the half-line and free convection boundary-layers in porous media

In this paper we use a recently elaborated abstract method, for parameter-dependent ODE systems over the interval (0, ∞), to obtain existence results for the problem of self-similar solutions in boundary-layer free convection in porous media. Using a generalization of the method to exponentially decaying solutions, we are able to recover some known results, and to obtain a new branch of solutions in the case of the so-called backward boundary-layer. The arguments involve the derivation of suitable a priori estimates for the solutions of the problem.

Modeling of Free Convection Boundary Layer Flow on a Solid Sphere with Newtonian Heating

In this paper, the mathematical model of free convection boundary layer flow on a solid sphere with Newtonian heating, in which the heat transfer from the surface is proportional to the local surface temperature, is considered. The transformed boundary layer equations are solved numerically using an efficient numerical scheme known as the Keller-box method. Numerical solutions are obtained for the local wall temperature, the local skin friction coefficient, as well as the velocity and temperature profiles. The features of the flow and heat transfer characteristics for different values of the Prandtl number Pr and conjugate parameter ? are analyzed and discussed.