Non-similar Boundary Layer Equations Research Articles

Convection boundary layers with transpiration: Non-similar solutions and scaling

We present a new formulation and novel scaling relations for natural convection boundary layers on horizontal surfaces with weak transpiration velocities Vi, for intermediate Schmidt numbers (or Prandtl numbers Pr) 1<Sc<20. Using characteristic scales derived from integral boundary layer equations, we define a new similarity variable η=(y/x)(Rex/Grx)1/4/Sc1/5 that has Vi within it, where Rex and Grx are the local Reynolds and Grashof numbers based on Vi and the concentration difference ΔC, respectively. Using η, and the dimensionless transpiration velocity ξ=(Rex/Grx1/5)5/4 as a non-similarity variable, we obtain novel, reduced, non-similar boundary layer equations. Numerical solution of these reduced boundary layer equations using a local non-similarity approach gives us the profiles of horizontal velocity and concentration, boundary layer thicknesses, wall shear stress, and mass flux as functions of Sc, for various ξ in the range ξ<1. Using normalizing functions, we then obtain scaling relations for these parameters; we observe extended similarity where the non-similarity parameter ξ itself appears in the obtained similarity scaling relations.

MAGNETOHYDRODYNAMIC FREE CONVECTION FLOW OF FLUID OF A VERTICAL PLANE WITH VISCOSITY AND THERMAL CONDUCTIVITY DEPENDING ON TEMPERATURE

A two-dimensional natural convection flow of a viscous incompressible and electrically conducting fluid with both the viscosity and the thermal conductivity depending on temperature past a vertical impermeable flat plate is considered in presence of a uniform transverse magnetic field. The governing equations are reduced to non-similar boundary layer equations by introducing coordinate transformations appropriate to the cases (i) near the leading edge (ii) in the region far away from the leading edge and (iii) for the entire regime from leading edge to down stream. The governing equations for the flow in the upstream regime are investigated by perturbation method for smaller values of x, the stream-wise distributed magnetic field parameter. The equations governing the flow for large x and for all x, have been investigated by employing the implicit finite difference method with Keller box scheme. The effects of the viscosity variation parameter, e and the thermal conductivity variation parameter g, on the skin friction as well as the rate of heat transfer for the fluid for low Prandtl number are shown graphically.

Keller-Box Scheme to Mixed Convection Flow Over a Solid Sphere with the Effect of MHD

Mixed convection is the combination of a free convection caused by the buoyancy forces due to the different density and a forced convection due to external forces that increase the heat exchange rate. This means that, in free convection, the effect of external forces is significant besides buoyancy forces. In this study the fluid type with viscoelastic effect is non-Newtonian. The viscoelastic fluids that pass over a surface of a sphere form a thin layer, which due to their dominant viscosity is called by the border layer. The obtained limiting layer is analyzed with the thickness of the boundary layer- near the lower stagnating point, then obtained dimensional boundary layer equations, continuity, momentum, and energy equations. These dimensional boundary layer equations are then transformed into non-dimensional boundary layer equations by using non-dimensional variables. Further, the non-dimensional boundary layer equations are transformed into ordinary differential equations by using stream function, so that obtained the non-similar boundary layer equations. These non-similar boundary layer equations are solved numerically by using finite difference method of Keller-Box. The discretization results are non-linear and it should be linearized using newton linearization technique. The numerical solutions are analyzed the effect of Prandtl number, viscoelastic, mixed convection, and MHD parameters towards velocity profile, temperature profile, and wall temperature.

Effect of pressure work on free convection flow from an isothermal truncated cone with temperature dependent viscosity

The object of this article is to analyze the impact of pressure work on a steady free flow of convection over an isothermal truncated cone with viscosity dependent on temperature. The numerical results of the converted non-similar boundary layer equations are found in this paper by a finite differential method with quasi-linearization scheme. The effects of several physical parameters are represented graphically for changed values of the pressure work (ϵ) and the parameter of viscosity dependent on temperature (e). Results indicate that both pressure function parameter and temperature dependent viscosity are strongly impacted by skin friction and transferring of heat. It is perceived that the skin friction coefficient decreases and the transferring of heat coefficient rises, increasing the pressure work (ϵ).

On the Multidomain Bivariate Spectral Local Linearisation Method for Solving Systems of Nonsimilar Boundary Layer Partial Differential Equations

In this work, a novel approach for solving systems of nonsimilar boundary layer equations over a large time domain is presented. The method is a multidomain bivariate spectral local linearisation method (MD-BSLLM), Legendre-Gauss-Lobatto grid points, a local linearisation technique, and the spectral collocation method to approximate functions defined as bivariate Lagrange interpolation. The method is developed for a general system of n nonlinear partial differential equations. The use of the MD-BSLLM is demonstrated by solving a system of nonlinear partial differential equations that describe a class of nonsimilar boundary layer equations. Numerical experiments are conducted to show applicability and accuracy of the method. Grid independence tests establish the accuracy, convergence, and validity of the method. The solution for the limiting case is used to validate the accuracy of the MD-BSLLM. The proposed numerical method performs better than some existing numerical methods for solving a class of nonsimilar boundary layer equations over large time domains since it converges faster and uses few grid points to achieve accurate results. The proposed method uses minimal computation time and its accuracy does not deteriorate with an increase in time.

Effects of Temperature Dependent Thermal Conductivity and Viscous Dissipation on Free Convective Flow along a Vertical Flat Plate with Asymptotic Solution

This paper reports us the surface temperature distribution effects of free convective flow along a vertical flat plate with temperature dependent thermal conductivity and viscous dissipation with asymptotic solution. The governing equations with associated boundary conditions reduce to local non-similarity boundary layer equations for this phenomenon are converted to dimensionless forms using a suitable transformation. The transformed non-linear equations are then solved using the implicit finite difference method together with Keller-box technique. Numerical results of the velocity and temperature profiles, skin friction and surface temperature profiles for different values of the thermal conductivity variation parameter, the Prandtl number and the viscous dissipation parameters are presented graphically. Also we considered the asymptotic solution for the effect of the thermal conductivity variation parameter, the Prandtl number and the viscous dissipation parameters in skin friction and surface temperature profiles. Detailed discussion is given for the aforementioned parameters. A good similarity is found in small and large value solution with all value solution for the thermal conductivity variation parameter, the Prandtl number and the viscous dissipation parameter for skin friction and surface temperature. Divergence is found near one (1).GANIT J. Bangladesh Math. Soc.Vol. 36 (2016) 33-46

Unsteady MHD Free Convective Flow Past an Inclined Parabolic Accelerated Plate with Hall Current, Radiation Effects and Variable Temperature in a Porous Medium

We analyze the effects of hall current and radiation on unsteady hydromagnetic free convection flow of a viscous incompressible electrically conducting fluid past an inclined parabolic accelerated plate in a porous medium by applying a transverse magnetic field which makes an angle α to the inclined plate.The fluid is assumed to be viscous, incompressible and electrically conducting with a strong magnetic field. Using the modified Ohm’s law and the Bossinesq approximation the governing equations of the problem are reduced to local non-similarity boundary layer equations using suitable transformation.The dimensionless governing equations of flow field are solved numerically by Crank-Nicolson finite difference method for different values of governing flow parameters. The velocity and temperature profiles are shown through graphs.

Effects of Conduction Variation on MHD Natural Convection Flow along a Vertical Flat Plate with Thermal Conductivity

This paper reports the effect of conduction variation of free convection flow along a vertical flat plate on magnetohydrodynamic (MHD) with thermal conductivity. The governing equations with associated boundary conditions reduce to local non-similarity boundary layer equations for this phenomenon are converted to dimensionless forms using a suitable transformation. The transformed non-linear equations are then solved using the implicit finite difference method together with Keller-box technique. Numerical results of the velocity and temperature profiles, skin friction and surface temperature profiles for different values of the magnetic parameter, the thermal conductivity variation parameter, the Prandtl number and the conduction variation parameters are presented graphically. Detailed discussion is given for the effect of the aforementioned parameters. Opposite scenario is found in skin friction and surface temperature for the thermal conductivity variation parameter. Significant effect is found in skin friction and surface temperature for conduction variation parameter.

Effects of Conduction Variation on MHD Natural Convection Flow Along a vertical Flat Plate

This paper reports free convection flow along a vertical flat plate with conduction variation on magneto hydrodynamic (MHD) effects. The governing equations with associated boundary conditions reduce to local non-similarity boundary layer equations for this phenomenon are converted to dimensionless forms using a suitable transformation. The transformed non-linear equations are then solved using the implicit finite difference method together with Keller-box technique. Numerical results of the velocity and temperature profiles, skin friction and surface temperature profiles for different values of the magnetic parameter, the Prandtl number and the conduction variation parameters are presented graphically. Detailed discussion is given for the effect of the aforementioned parameters.GANIT J. Bangladesh Math. Soc.Vol. 34 (2014) 63-73

Effect of thermal radiation on free convection flow and heat transfer over a truncated cone in the presence of pressure work and heat generation/absorption

Effect of heat generation or absorption and thermal radiation on free convection flow and heat transfer over a truncated cone in the presence of pressure work is considered. The governing boundary layer equations are reduced to non-similarity boundary layer equations and solved numerically by using Mathematica technique. Comparisons with previously published work on special cases of the problem are performed and the results are found to be in excellent agreement. The solutions are presented in terms of local skin friction, local Nusselt number, velocity and temperature profiles for values of Prandtl number, pressure work parameter, radiation parameter and heat generation or absorption parameter.

Unsteady mixed convection of a micropolar fluid past a circular cylinder due to time-dependent free stream velocity and temperature

Unsteady mixed convection boundary-layer flow of an electrically conducting micropolar fluid past a circular cylinder is investigated taking into account the effect of thermal radiation and heat generation or absorption. The reduced non-similar boundary-layer equations are solved using the finite difference method. It is found that the magnitude of the friction factor and the couple stress significantly increases due to the increase of the mixed convection parameter, the conduction-radiation parameter, the surface temperature parameter, the heat absorption parameter and the frequency parameter. However the magnitude of the heat transfer rate decreases with these parameters. The converse characteristics are observed for the Prandtl number. The magnitude of the couple stress and the heat transfer rate is seen to decrease whereas the magnitude of the skin factor increases with increasing the vortex viscosity parameter. The magnetic field parameter reduces the skin factor, couple stress and heat transfer rate. The amplitude of oscillation of the transient skin factor and couple stress gradually increases owing to an increase of $\xi$. But the transient heat transfer rate is found to be oscillating with almost the same amplitude for any value of $\xi$. The amplitude of oscillation of the transient skin factor and couple stress increases with an increase of $S$ and $\xi$ while the amplitude of the transient heat transfer rate increases with increasing Pr and $S$.

EFFECTS OF THERMAL CONDUCTIVITY OF FLUID ON FREE CONVECTION FLOW ALONG A VERTICAL FLAT PLATE WITH TRANSVERSE CONDUCTION

This paper reports the effects of transverse conduction variation with thermal conductivity on freeconvection flow along a vertical flat plate. The governing equations with associated boundary conditions reduceto local non-similarity boundary layer equations for this phenomenon are converted to dimensionless forms usinga suitable transformation. The transformed non-linear equations are then solved using the implicit finitedifference method together with Keller-box technique. Numerical results of the velocity and temperature profiles,skin friction and surface temperature profiles for different values of the thermal conductivity parameter, thePrandtl number and the transverse conduction variation parameters are presented graphically. Detaileddiscussion is given for the effect of the aforementioned parameters. Mentionable effect is found in skin friction andsurface temperature for the transverse conduction variation parameter.

Effects of radiation--conduction interaction on mixed convection from a vertical cone embedded in a porous media with high porosity

The problem of steady laminar mixed convection heat transfer about a vertical cone embedded in a porous medium with high porosity was studied numerically, taking into account the radiation--conduction effect. The fluid was assumed to be incompressible and dense. The nonlinear coupled parabolic partial differential equations governing the flow were transformed into nonsimilar boundary layer equations, which were then solved numerically using the Keller box method. The effects of the exponent in the power law variation of the free stream velocity m, the mixed convection parameter Ri, the radiation--conduction parameter Rd, the surface temperature parameter qw, and Forchheimer parameter g on the velocity and temperature profiles, as well as on the local skin friction and local heat transfer, are presented and analyzed. The validity of the methodology and analysis was checked by comparing the results obtained for some specific cases with those available in the literature.

Heat Transfer Flow for Magnetohydrodynamic and Heat Generation Effects Along a Vertical Flat Plate

Free convection heat transfer flow of an electrically conducting fluid along a vertical flat plate with heat generation and Magnetohydrodynamic (MHD) combined effects are studied in this paper. The governing equations with associated boundary conditions reduce to local non-similarity boundary layer equations for this phenomenon are converted to dimensionless forms using a suitable transformation. The transformed non-linear equations are then solved using the implicit finite difference method. Numerical results of the velocity and temperature profiles for the magnetic and heat generation parameter, skin friction coefficient and local rate of heat transfer for different values of the magnetic parameter, Prandtl number and heat generation parameters are presented graphically. Detailed discussion is given for the effect of the aforementioned parameters. Significant effects are found in velocity, temperature, skin friction and local rate of heat transfer profiles for heat generation parameter.

Hall current effects on magnetohydrodynamic natural convection flow with strong cross magnetic field

A boundary layer analysis has been presented to study the effect of Hall current on the MHD natural convection flow of a viscous incompressible fluid along a semi-infinite uniformly heated vertical plate in the presence of a strong cross magnetic field. The governing equations are expressed in terms of local non-similar boundary layer equations. Solutions are obtained for small and large values of local Hartmann parameter, ξ. For small ξ solutions are obtained through power series method and for large ξ analytical solutions are obtained via matched asymptotic expansion method. Numerical solutions are also obtained by implicit finite difference method together with Thompson's algorithm, over the entire range of ξ. The boundary layer structure is investigated in terms of shear stress, τw, rate of heat transfer, qw, and rate of mass transfer, mw, for low Prandtl number (i.e. Pr << 1). Comprehensive interpretation of energy distribution is also given in terms of heatlines.

MHD mixed convection of a viscous dissipating fluid about a vertical slender cylinder

The problem of steady laminar magnetohydrodynamic (MHD) mixed convection heat transfer about a heated/cooled vertical slender cylinder is studied numerically, taking into account the effects of transverse curvature and viscous dissipation. A uniform magnetic field is applied perpendicular to the cylinder. The resulting governing equations are transformed into the nonsimilar boundary layer equations and solved using the Keller box method. The velocity and temperature profiles as well as the local skin friction and the local heat transfer parameters are determined for different values of the governing parameters, mainly the magnetic parameter, the Richardson number, and the Eckert number. For some specific values of the governing parameters, the results agree very well with those available in the literature. Generally, it is determined that the local skin friction coefficient and the local heat transfer coefficient increase, increasing the Richardson number and the magnetic parameter Mn or decreasing the Eckert number.

Transient mixed convection flow arising due to thermal and mass diffusion over porous sensor surface inside squeezing horizontal channel

The double diffusion effect on the mixed convection flow over a horizontal porous sensor surface placed inside a horizontal channel is analyzed. With the appropriate transformations, the unsteady equations governing the flow are reduced to non-similar boundary layer equations which are solved numerically for the time-dependent mixed convection parameter. The asymptotic solutions are obtained for small and large values of the time-dependent mixed convection parameter. The results are discussed in terms of the skin friction, the heat transfer coefficient, the mass transfer coefficient, and the velocity, temperature, and concentration profiles for different values of the Prandtl number, the Schmidt number, the squeezing index, and the mixed convection parameter.

Nonsimilar solutions for mixed convection of water at 4°C over a vertical surface with a convection boundary condition in a porous medium

A boundary layer analysis has been presented for the mixed convection of water at 4°C over a vertical plate embedded in a porous medium. The Robin or convective boundary condition at the surface has been considered where the heat lost from the surface is the product of a heat transfer coefficient and the temperature difference between the surface and the free stream. The governing non-similar boundary layer equations for both the forced and free convection dominated regimes were solved numerically by means of an implicit finite difference method. The friction factor and dimensionless heat transfer rate (Nusselt number) are presented for several values of the dimensionless heat transfer coefficient and buoyancy parameter. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21022

Unsteady Mixed Convection Flow over a Cylinder of Elliptic Cross Section near Forward Stagnation Point

The unsteady mixed convection flow over a cylinder of elliptic cross section when the major axis is horizontal (blunt orientation) or vertical (slender orientation) near forward stagnation point is studied. The unsteadiness is due to an impulsive motion on the free stream. The governing boundary layer equations are first reduced into a non-dimensional form, and then transformed into a set of non-similar boundary layer equations, which are solved numerically using an efficient implicit finite-difference method known as Keller-Box method. The numerical results are obtained for various values of the Prandtl number, Pr, the mixed convection parameter, and parameter for blunt and slender orientation, !. The effects of these parameters on velocity profiles, temperature profiles as well as Nusselt number are presented through graphs and tables. The results show that as the mixed convection parameter, is increased, the velocity of the fluid is increased but the temperature of the fluid is decreased, for both blunt and slender orientations. Furthermore, the slender orientation gives earlier separation times for opposing flow while there is no separation of flow detected for the assisting flow. Keywords: unsteady flow; mixed convection; cylinder of elliptic cross section; forward stagnation point. 2010 Mathematics Subject Classification: 76B99; 76R99.

Effects of Conjugate Heat Transfer on Steady MHD Mixed Convective Heat Transfer Flow over a Thin Vertical Plate Embedded in a Porous Medium with High Porosity

This study investigates mixed convection heat transfer about a thin vertical plate in the presence of magneto and conjugate heat transfer effects in the porous medium with high porosity. The fluid is assumed to be incompressible and dense. The nonlinear coupled parabolic partial differential equations governing the flow are transformed into the nonsimilar boundary layer equations, which are then solved numerically using the Keller box method. The effects of the conjugate heat transfer parameter p, the porous medium parameter k1, the Forchheimer parameter F*, the mixed convection parameter Ri, the magnetic parameter Mn, and the electric field parameter E1 on the velocity and temperature profiles as well as on the local skin friction and local heat transfer are presented and analyzed. The validity of the methodology and analysis is checked by comparing the results obtained for some specific cases with those available in the literature.