Solutions of thermal boundary layer equations when temperature gradient at the moving flat plate in parabolic flow is prescribed

Solutions of steady as well as unsteady three-dimensional incompressible thermal boundary layer equations are studied when the temperature gradient at the moving flat plate in parabolic flow is prescribed. A general analysis is made and different cases are studied by giving values to β and Cx which determine the gradient and curvature of the outer flow steam lines. The components of velocity in boundary layer are discussed by Sarma and Gupta and those results are used to analyse the thermal boundary layer equations. The response of temperature of the plate are studied for large and small times and curves are drawn representing the variation of temperature with time for various cases. The limiting time is also calculated.     

Similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition

The steady laminar boundary layer flow over a permeable flat plate in a uniform free stream, with the bottom surface of the plate is heated by convection from a hot fluid is considered. Similarity solutions for the flow and thermal fields are possible if the mass transpiration rate at the surface and the convective heat transfer from the hot fluid on the lower surface of the plate vary like x^−1/2, where x is the distance from the leading edge of the solid surface. The governing partial differential equations are first transformed into ordinary differential equations, before being solved numerically. The effects of the governing parameters on the flow and thermal fields are thoroughly examined and discussed.     

Parabolic flow over a flat plate with wave disturbance in the main stream

In this paper the effects of unsteady flow past a flat plate, which is at rest, are studied when the steady parabolic flow outside the boundary layer is in a wave disturbance. Solutions are developed for large and small times. The behaviour of the amplitudes and phase leads of skin friction components along chordwise and spanwise directions are studied in detail. Two cases, one when the main stream lines are straight and yawed, and the other when the main stream lines are parabolic and concave with respect to the points on the chordwise direction are studied separately and the results are interpreted graphically.     

Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivity,non-uniform heat source and radiation

An analysis has been carried out to study the magnetohydrodynamic boundary layer flow and heat transfer characteristics of a non-Newtonian viscoelastic fluid over a flat sheet with a linear velocity in the presence of thermal radiation and non-uniform heat source. The thermal conductivity is assumed to vary as a linear function of temperature. The basic equations governing the flow and heat transfer are in the form of partial differential equations, the same have been reduced to a set of non-linear ordinary differential equations by applying suitable similarity transformation. The transformed equations are solved analytically by regular perturbation method. Numerical solution of the problem is also obtained by the efficient shooting method, which agrees well with the analytical solution. The effects of various physical parameters such as viscoelastic parameter, Chandrasekhar number, Prandtl number, variable thermal conductivity parameter, Eckert number, thermal radiation parameter and non-uniform heat source/sink parameters which determine the temperature profiles are shown in several plots and the heat transfer coefficient is tabulated for a range of values of said parameters. Some important findings reported in this work reveals that combined effect of variable thermal conductivity, radiation and non-uniform heat source have significant impact in controlling the rate of heat transfer in the boundary layer region.     

Influence of radiation on MHD free convection from a vertical flat plate embedded in porous media with thermophoretic deposition of particles

Magneto-hydrodynamics and thermal radiation effects on heat and mass transfer in steady laminar boundary layer flow of a Newtonian, viscous fluid over a vertical flat plate embedded in a fluid saturated porous media in the presence of the thermophoresis particle deposition effect is studied in this paper. The governing equations are transformed by special transformations. Brownian motion of particles and thermophoretic transport are considered in the flow equations. The magnetic field is considered to be applied. Rosseland approximation is used to describe the radiative heat flux in the energy equation. The resulting similarity equations are solved numerically by the fourth-order Runge–Kutta method with shooting technique. Many results are obtained and representative set is displayed graphically to illustrate the influence of the various parameters on the wall thermophoretic deposition velocity, concentration, temperature and velocity profiles.     

A Mixed Analytical/Numerical Method for Velocity and Heat Transfer of Laminar Power-Law Fluids

This paper presents a relatively simple numerical method to investigate the flow and heat transfer of laminar power-law fluids over a semi-infinite plate in the presence of viscous dissipation and anisotropy radiation. On one hand, unlike most classical works, the effects of power-law viscosity on velocity and temperature fields are taken into account when both the dynamic viscosity and the thermal diffusivity vary as a power-law function. On the other hand, boundary layer equations are derived by Taylor expansion, and a mixed analytical/numerical method (a pseudo-similarity method) is proposed to effectively solve the boundary layer equations. This method has been justified by comparing its results with those of the original governing equations obtained by a finite element method. These results agree very well especially when the Reynolds number is large. We also observe that the robustness and accuracy of the algorithm are better when thermal boundary layer is thinner than velocity boundary layer.     

Hall电流对表面热通量均匀的竖直可渗透平板上MHD自然对流的影响

在横向磁场作用下,研究Hall电流对竖直可渗透平板上MHD自然对流的影响,平板具有均匀的热通量.和外部磁场相比,假设感应磁场可以忽略不计.利用自由变量公式化(FVF)和流函数公式化(SFF),将边界层方程简化为适当的形式.对局部蒸发系数ζ的整个取值范围,由FVF得到的抛物型方程,用简明的有限差分法进行数值积分;另一方面,由SFF得到的非相似方程,采用局部非相似法求解.有些区域,如局部蒸发系数ζ值足够大或足够小时,用正规的摄动法求解.对低值Prandtl数Pr,例如Pr=0.005,0.01,0.05时,用图形表示磁场参数M和Hall参数m,对局部表面摩擦因数和局部Nusselt数的影响.最后对不同的局部蒸发系数ζ值,给出流体的速度和温度分布.     

A new family of unsteady boundary layers over a stretching surface

In this paper, a new family of unsteady boundary layers over a stretching flat surface was proposed and studied. This new class of unsteady boundary layers involves the flows over a constant speed stretching surface from a slot, and the slot is moving at a certain speed. Depending on the slot moving parameter, the flow can be treated as a stretching sheet problem or a shrinking sheet problem. Both the momentum and thermal boundary layers were studied. Under special conditions, the solutions reduce to the unsteady Rayleigh problem and the steady Sakiadis stretching sheet problem. Solutions only exist for a certain range of the slot moving parameter, α. Two solutions are found for −53.55° < α < −45°. There are also two solution branches for the thermal boundary layers at any given Prandtl number in this range. Compared with the upper solution branch, the lower solution branch leads to simultaneous reduction in wall drag and heat transfer rate. The results also show that the motion of the slot greatly affects the wall drag and heat transfer characteristics near the wall and the temperature and velocity distributions in the fluids.     

Stagnation point flow and heat transfer over a non-linearly moving flat plate in a parallel free stream with slip

An analysis is presented for the steady boundary layer flow and heat transfer of a viscous and incompressible fluid in the stagnation point towards a non-linearly moving flat plate in a parallel free stream with a partial slip velocity. The governing partial differential equations are converted into nonlinear ordinary differential equations by a similarity transformation, which are then solved numerically using the function bvp4c from Matlab for different values of the governing parameters. Dual (upper and lower branch) solutions are found to exist for certain parameters. Particular attention is given to deriving numerical results for the critical/turning points which determine the range of existence of the dual solutions. A stability analysis has been also performed to show that the upper branch solutions are stable and physically realizable, while the lower branch solutions are not stable and, therefore, not physically possible.     

Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition

In this paper the boundary layer flow over a flat plat with slip flow and constant heat flux surface condition is studied. Because the plate surface temperature varies along the x direction, the momentum and energy equations are coupled due to the presence of the temperature gradient along the plate surface. This coupling, which is due to the presence of the thermal jump term in Maxwell slip condition, renders the momentum and energy equations non-similar. As a preliminary study, this paper ignores this coupling due to thermal jump condition so that the self-similar nature of the equations is preserved. Even this fundamental problem for the case of a constant heat flux boundary condition has remained unexplored in the literature. It was therefore chosen for study in this paper. For the hydrodynamic boundary layer, velocity and shear stress distributions are presented for a range of values of the parameter characterizing the slip flow. This slip parameter is a function of the local Reynolds number, the local Knudsen number, and the tangential momentum accommodation coefficient representing the fraction of the molecules reflected diffusively at the surface. As the slip parameter increases, the slip velocity increases and the wall shear stress decreases. These results confirm the conclusions reached in other recent studies. The energy equation is solved to determine the temperature distribution in the thermal boundary layer for a range of values for both the slip parameter as well as the fluid Prandtl number. The increase in Prandtl number and/or the slip parameter reduces the dimensionless surface temperature. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow properties, and the applied heat flux.     

Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with thermal radiation and non-uniform heat source/sink

An analysis has been carried out to study the flow and heat transfer characteristics for MHD viscoelastic boundary layer flow over an impermeable stretching sheet with space and temperature dependent internal heat generation/absorption (non-uniform heat source/sink), viscous dissipation, thermal radiation and magnetic field due to frictional heating. The flow is generated due to linear stretching of the sheet and influenced by uniform magnetic field, which is applied vertically in the flow region. The governing partial differential equations for the flow and heat transfer are transformed into ordinary differential equations by a suitable similarity transformation. The governing equations with the appropriate conditions are solved exactly. The effects of viscoelastic parameter and magnetic parameter on skin friction and the effects of viscous dissipation, non-uniform heat source/sink and the thermal radiation on heat transfer characteristics for two general cases namely, the prescribed surface temperature (PST) case and the prescribed wall heat flux (PHF) case are presented graphically and discussed. The numerical results for the wall temperature gradient (the Nusselt number) are presented in tables and are discussed.     

Combined effects of non-uniform heat source/sink and thermal radiation on heat transfer over an unsteady stretching permeable surface

The present paper is concerned with the study of flow and heat transfer characteristics in the unsteady laminar boundary layer flow of an incompressible viscous fluid over continuously stretching permeable surface in the presence of a non-uniform heat source/sink and thermal radiation. The unsteadiness in the flow and temperature fields is because of the time-dependent stretching velocity and surface temperature. Similarity transformations are used to convert the governing time-dependent nonlinear boundary layer equations for momentum and thermal energy are reduced to a system of nonlinear ordinary differential equations containing Prandtl number, non-uniform heat source/sink parameter, thermal radiation 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 unsteadiness parameter, thermal radiation, suction/injection parameter, non-uniform heat source/sink parameter on flow and heat transfer characteristics as well as on the local Nusselt number are shown graphically.     

Group method analysis of melting effect on MHD mixed convection flow from radiate vertical plate embedded in a saturated porous media

The linear transformation group approach is developed to simulate problem of hydromagnetic heat transfer by mixed convection along vertical plate in a liquid saturated porous medium in the presence of melting and thermal radiation effects for opposing external flow. The application of a one-parameter transformation group reduces the number of independent variables by one so that the governing partial differential equations with the boundary conditions reduce to an ordinary differential equations with appropriate corresponding conditions. The Runge-Kutta shooting method is used to solve the determining equations of the set of nonlinear ordinary differential equations. are presented in the form of the temperature and flow fields in the melting region within the boundary layer for different parameters entering into the analysis. Also the effects of the pertinent parameters on the rate of the heat transfer in terms of the local Nusselt number at the solid–liquid interface are also discussed.     

Spatial Stability for Mixed Convection Boundary Layer over a Heated Horizontal Plate

The spatial stability properties of a mixed convection boundary layer developing over a heated horizontal plate is studied here under linear and quasi-parallel flow assumption. The main aim of the present work is to find out if there is a critical buoyancy parameter that would indicate the importance of heat transfer in destabilizing mixed convection boundary layers, when the buoyancy effect is given by Boussinesq approximation. The undisturbed flow used here is that given by the similarity solution of [ 1 ] that implies the wall temperature to vary as the inverse square root of the distance from the leading edge of the plate. The stability of this flow has been investigated by using the compound matrix method (CMM)—that allows finding all the modes in the chosen range in the complex wave number plane for spatial stability analysis. Presented neutral curves for mixed convection boundary layer show the existence of two types of disturbances present simultaneously, for large buoyancy parameter. One notices very unstable high-frequency mode when the buoyancy parameter exceeds the above-mentioned critical value. This unstable thermal mode is in addition to the hydrodynamic mode of isothermal flow given by corresponding similarity profile. The calculated critical buoyancy parameter is shown to qualitatively match with experimental results.     

Numerical investigation of transient heat transfer to hydromagnetic channel flow with radiative heat and convective cooling

This present study consists of a numerical investigation of transient heat transfer in channel flow of an electrically conducting variable viscosity Boussinesq fluid in the presence of a magnetic field and thermal radiation. The temperature dependent nature of viscosity is assumed to follow an exponentially model and the system exchanges heat with the ambient following Newton’s law of cooling. The governing nonlinear equations of momentum and energy transport are solved numerically using a semi-implicit finite difference method. Solutions are presented in graphical form and given in terms of fluid velocity, fluid temperature, skin friction and heat transfer rate for various parametric values. Our results reveal that combined effect of thermal radiation, magnetic field, viscosity variation and convective cooling have significant impact in controlling the rate of heat transfer in the boundary layer region.     

Magnetohydrodynamic non-Darcy mixed convection heat transfer from a vertical heated plate embedded in a porous medium with variable porosity

A numerical model is developed to study magnetohydrodynamics (MHD) mixed convection from a heated vertical plate embedded in a Newtonian fluid saturated sparsely packed porous medium by considering the variation of permeability, porosity and thermal conductivity. The boundary layer flow in the porous medium is governed by Forchheimer–Brinkman extended Darcy model. The conservation equations that govern the problem are reduced to a system of non-linear ordinary differential equations by using similarity transformations. Because of non-linearity, the governing equations are solved numerically. The effects of magnetic field on velocity and temperature distributions are studied in detail by considering uniform permeability (UP) and variable permeability (VP) of the porous medium and the results are discussed graphically. Besides, skin friction and Nusselt number are also computed for various physical parameters governing the problem under consideration. It is found that the inertial parameter has a significant influence in increasing the flow field and the rate of heat transfer for variable permeability case. The important finding of the present work is that the magnetic field has considerable effects on the boundary layer velocity and on the rate of heat transfer for variable permeability of the porous medium. Further, the results obtained under the limiting conditions were found to be in good agreement with the existing ones.     

Minimum error solutions of boundary layer equations

The minimum error solutions of boundary layer equations in the least square sense have been studied by employing the Euler-Lagrange equations. To test the method a class of problems,i.e., boundary layer on a flat plate, Hiemenz flow, boundary layer on a moving sheet and boundary layer in non-Newtonian fluids have been studied. The comparison of the results with approximate methods, like Karman-Pohlhuasen, local potential and other variational methods, shows that the present predictions are invariably better.     

Einfluss eines Verbrennungsvorganges auf den Wärme- und Stoffaustausch in einer laminaren Grenzschicht

Summary The influence of combustion on heat and mass transfer is investigated on the following model. A mixture of an inert with a combustible gas (air) flows in steady, laminar flow over a flat plate. A mass flux of gaseous fuel away from the plate surface is produced by some means. Combustion is assumed to occur with very fast reaction rate so that the process is purely controlled by diffusion and the equilibrium is assumed as very close to complete combustion. It is studied under which conditions the combustion occurs at the surface or when the flame is displaced into the boundary layer. The influence of combustion on the heat transfer from a hot gas to the plate surface is calculated, for the condition that combustion occurs at the surface.      

Analytical approximate solution of the cooling problem by Adomian decomposition method

The Adomian decomposition method (ADM) can provide analytical approximation or approximated solution to a rather wide class of nonlinear (and stochastic) equations without linearization, perturbation, closure approximation, or discretization methods. In the present work, ADM is employed to solve the momentum and energy equations for laminar boundary layer flow over flat plate at zero incidences with neglecting the frictional heating. A trial and error strategy has been used to obtain the constant coefficient in the approximated solution. ADM provides an analytical solution in the form of an infinite power series. The effect of Adomian polynomial terms is considered and shows that the accuracy of results is increased with the increasing of Adomian polynomial terms. The velocity and thermal profiles on the boundary layer are calculated. Also the effect of the Prandtl number on the thermal boundary layer is obtained. Results show ADM can solve the nonlinear differential equations with negligible error compared to the exact solution.     

Nonlinear convection at a porous flat plate with application to heat transfer from a dike

Solutions for a class of coupled nonlinear differential equations, arising in free convection flow at a vertical flat plate embedded in a saturated porous medium at high Reynolds numbers in the presence of heat sources (or sinks) and with nonlinear density temperature variation, are obtained. Further, using the Schauder theory and numerical results, existence and analyticity results are established.