Temperature distribution in a laminar plane wall jet due to variably heated wall

The heat transfer in a laminar incompressible plane wall jet due to a variably heated wall has been studied. It is assumed that the difference of temperatures between the wall and the issuing jet is inversely proportional to an arbitrary exponent of the distance from the slit. A similar solution of the energy equation is possible. The solutions, for arbitrary values of the Prandtl number and of the exponent are obtained. It is found that in some cases the heat transfer at the wall may become zero or negative.     

Heat transfer of a generalized stretching/shrinking wall problem with convective boundary conditions

In this paper, we investigate the heat transfer of a viscous fluid flow over a stretching/shrinking sheet with a convective boundary condition. Based on the exact solutions of the momentum equations, which are valid for the whole Navier–Stokes equations, the energy equation ignoring viscous dissipation is solved exactly and the effects of the mass transfer parameter, the Prandtl number, and the wall stretching/shrinking parameter on the temperature profiles and wall heat flux are presented and discussed. The solution is given as an incomplete Gamma function. It is found the convective boundary conditions results in temperature slip at the wall and this temperature slip is greatly affected by the mass transfer parameter, the Prandtl number, and the wall stretching/shrinking parameters. The temperature profiles in the fluid are also quite different from the prescribed wall temperature cases.     

Homotopy analysis method for the heat transfer in a asymmetric porous channel with an expanding or contracting wall

The flow of a viscous incompressible fluid between two parallel plates due to the normal motion of the porous upper plate is investigated and an analysis is made to determine the heat and mass transfer. The unsteady Navier–Stokes equations are reduced to a generalization of the Proudman–Johnson equation retaining the effect of wall motion using a suitable similarity transformation. The analytical solution for stream function and heat transfer characteristics are obtained by employing homotopy analysis method. The effects of various physical parameters like expansion ratio, Prandtl number, Reynolds number on various momentum and heat transfer characteristics are discussed in detail.     

Heat transfer enhancement for boundary layer flow over a wedge by the use of electric fields

The effect of an electric field is investigated for heat transfer properties in a laminar, incompressible, non-isothermal boundary layer gas flow over a wedge. The governing boundary layer equations are reduced to an ordinary differential equation system using similarity transformations. The reduced equations are solved numerically for different values of electric and flow field parameters characterizing the ratio of electric force to fluid inertia force, Joule heating and ion kinetic work. For specific electric field function forms, leading to similarity solutions, velocity boundary layers are observed to become thinner and heat transfer properties are shown to be enhanced near the wall. The level of enhancement is controlled by the electric body force with additional effects of Joule heating and ion kinetic work on the bulk flow. The effects of low and high Prandtl numbers are also demonstrated. Heat transfer enhancement is observed to increase with increasing Prandtl number.     

A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition

This paper considers the classical problem of hydrodynamic and thermal boundary layers over a flat plate in a uniform stream of fluid. It is well known that similarity solutions of the energy equation are possible for the boundary conditions of constant surface temperature and constant heat flux. However, no such solution has been attempted for the convective surface boundary condition. The paper demonstrates that a similarity solution is possible if the convective heat transfer associated with the hot fluid on the lower surface of the plate is proportional to x^?1/2. Numerical solutions of the resulting similarity energy equation are provided for representative Prandtl numbers of 0.1, 0.72, and 10 and a range of values of the parameter characterizing the hot fluid convection process. For the case of constant heat transfer coefficient, the same data provide local similarity solutions.     

Long-wave oscillatory Marangoni instability in a horizontal liquid layer

The long-wave instability in the problem of thermocapillary convection in a horizontal layer with a free deformable boundary and a solid bottom is investigated. The transcendental equation for the main asymptotic term of the spectral parameter is written in explicit form. The main attention is paid to investigating oscillatory instability. For the frequency of neutral oscillations, simple transcendental equations are obtained that contain the Prandtl and Biot numbers. In a number of cases, exact solutions are indicated. Explicit formulae are given for the main asymptotic term of the Marangoni number. In the case of a non-heat-conducting solid wall, the relation between the critical values of the parameters for inverse Prandtl numbers is found. It is shown that, for different Prandtl numbers, the asymptotic values are in good agreement with the numerical values.     

Homotopy analysis method for the asymmetric laminar flow and heat transfer of viscous fluid between contracting rotating disks

The present work investigates the effects of the disks contracting, rotation, heat transfer and different permeability on the viscous fluids and temperature distribution between two heated contracting rotating disks. Two cases are considered. For the first case, we neglect the viscous dissipation effects in the energy equation and reduce the Navier-Stokes equations and energy equation into nonlinear coupled ODEs by introducing the Von Kármán type similarity transformations. The effects of various physical parameters like expansion ratio, Prandtl number, Reynolds number and rotation ratio on the velocity and temperature are discussed in detail. The second and more general case is that we consider the viscous dissipation in the energy equation. Under this assumption, the energy equation is reduced to a ordinary differential equation including the Eckert number, whose solution also is solved by HAM.     

Momentum and heat transfer of the Falkner–Skan flow with algebraic decay: An analytical solution

In this paper, an analytical solution of the Falkner–Skan equation with mass transfer and wall movement is obtained for a special case, namely a velocity power index of ?1/3, with an algebraically decaying velocity profile. The solution is given in a closed form. Under different values of the mass transfer parameter, the wall can be fixed, moving in the same direction as the free stream, or opposite to the free stream (reversal flow near the wall). The thermal boundary layer solution is also presented with a closed form for a prescribed power-law wall temperature, which is expressed by the confluent hypergeometric function of the second kind. The temperature profiles and the wall temperature gradients are plotted. Interesting but complicated variation trends for certain combinations of controlling parameters are observed. Under certain parameter combinations, there exist singular points or poles for the wall temperature gradients, namely wall heat flux. With poles, the temperature profiles can cross the zero temperature line and become negative. From the results, it is also found empirically that under certain given values of the Prandtl number (Pr) and flow controlling parameter (b), the number of times for the temperature profiles crossing the zero line is the same as the number of poles when the wall temperature power index varies between zero and a given value n. The current result provides a new analytical solution for the Falkner–Skan equation with algebraic decay and greatly enriches the understanding of this important equation as well as the heat transfer characteristics for this flow.     

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.     

Stationary statistical properties of Rayleigh‐Bénard convection at large Prandtl number

This is the third in a series of our study of Rayleigh‐Bénard convection at large Prandtl number. Here we investigate whether stationary statistical properties of the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number are related to those of the infinite Prandtl number model for convection that is formally derived from the Boussinesq system via setting the Prandtl number to infinity. We study asymptotic behavior of stationary statistical solutions, or invariant measures, to the Boussinesq system for Rayleigh‐Bénard convection at large Prandtl number. In particular, we show that the invariant measures of the Boussinesq system for Rayleigh‐Bénard convection converge to those of the infinite Prandtl number model for convection as the Prandtl number approaches infinity. We also show that the Nusselt number for the Boussinesq system (a specific statistical property of the system) is asymptotically bounded by the Nusselt number of the infinite Prandtl number model for convection at large Prandtl number. We discover that the Nusselt numbers are saturated by ergodic invariant measures. Moreover, we derive a new upper bound on the Nusselt number for the Boussinesq system at large Prandtl number of the form which asymptotically agrees with the (optimal) upper bound on Nusselt number for the infinite Prandtl number model for convection. © 2007 Wiley Periodicals, Inc.     

Heat transfer in a Walter’s liquid B fluid over an impermeable stretching sheet with non-uniform heat source/sink and elastic deformation

In this present article an analysis is carried out to study the boundary layer flow behavior and heat transfer characteristics in Walter’s liquid B fluid flow. The stretching sheet is assumed to be impermeable, the effects of viscous dissipation, non-uniform heat source/sink in the presence and in the absence of elastic deformation (which was escaped from attention of researchers while formulating the viscoelastic boundary layer flow problems)on heat transfer are addressed. The basic boundary layer equations for momentum and heat transfer, which are non-linear partial differential equations, are converted into non-linear ordinary differential equations by means of similarity transformation. Analytical solutions are obtained for the resulting boundary value problems. The effects of viscous dissipation, Prandtl number, Eckert number and non-uniform heat source/sink on heat transfer (in the presence and in the absence of elastic deformation) are shown in several plots and discussed. Analytical expressions for the wall frictional drag coefficient, non-dimensional wall temperature gradient and non-dimensional wall temperature are obtained and are tabulated for various values of the governing parameters. The present study reveals that, the presence of work done by deformation in the energy equation yields an augment in the fluid’s temperature.     

A note on the similarity equations arising in free convection boundary layers with blowing and suction

The similarity equations for the free convection boundary-layer flow on a vertical plate with prescribed wall temperature and transpiration velocity are considered. The range of existence of solution is discussed first. For blowing this is seen to be independent of the transpiration parameter , depending only on the Prandtl number. For suction this range of existence of solutions is seen to depend on as well. Asymptotic solutions for strong suction and strong blowing are derived and compared with numerical solutions of the similarity equations.     

Hiemenz flow and heat transfer of a third grade fluid

The laminar flow and heat transfer of an incompressible, third grade, electrically conducting fluid impinging normal to a plane in the presence of a uniform magnetic field is investigated. The heat transfer analysis has been carried out for two heating processes, namely, (i) with prescribed surface temperature (PST-case) and (ii) prescribed surface heat flux (PHF-case). By means of the similarity transformation, the governing non-linear partial differential equations are reduced to a system of non-linear ordinary differential equations and are solved by a second-order numerical technique. Effects of various non-Newtonian fluid parameters, magnetic parameter, Prandtl number on the velocity and temperature fields have been investigated in detail and shown graphically. It is found that the velocity gradient at the wall decreases as the third grade fluid parameter increases.     

An extended similarity theory applied to heated flows in complex geometries

In the traditional similarity theory the influence of temperature- and pressure-dependent fluid properties on the flow field and heat transfer is not described by the basic dimensionless parameters, i.e. Prandtl, Reynolds, Rayleigh, . . . number. We present an extended similarity theory that not only takes into account the variable material properties but also can handle small variations in other parameters of the physical model like small changes in the (reference) Prandtl number. The method has general applicability that is suitable for a wide variety of fluid dynamic and heat transfer situations in which variable properties with a strong dependence on temperature and pressure play a significant role. It is especially useful in predicting the behaviour of a certain fluid based on the results for a different one. As an example the Nu?elt number of a lid driven heated cavity is determined with fluid properties being temperature dependent.     

The effect of variable viscosity on MHD viscoelastic fluid flow and heat transfer over a stretching sheet

An analysis has been carried out to study the momentum and heat transfer characteristics in an incompressible electrically conducting non-Newtonian boundary layer flow of a viscoelastic fluid over a stretching sheet. The partial differential equations governing the flow and heat transfer characteristics are converted into highly non-linear coupled ordinary differential equations by similarity transformations. The effect of variable fluid viscosity, Magnetic parameter, Prandtl number, variable thermal conductivity, heat source/sink parameter and thermal radiation parameter are analyzed for velocity, temperature fields, and wall temperature gradient. The resultant coupled highly non-linear ordinary differential equations are solved numerically by employing a shooting technique with fourth order Runge–Kutta integration scheme. The fluid viscosity and thermal conductivity, respectively, assumed to vary as an inverse and linear function of temperature. The analysis reveals that the wall temperature profile decreases significantly due to increase in magnetic field parameter. Further, it is noticed that the skin friction of the sheet decreases due to increase in the Magnetic parameter of the flow characteristics.     

Analytic study on angular fields of HRR solution

By using the constructing function method, a systematic and strict analysis is carried out on the angular distribution field near a crack tip in a power-law hardening material and the analytic solution is provided for HRR problem. In addition, the equivalence of H equation and RR equation is proved. The present analytic solutions for HRR problem can reduce to (he well-known Williams solution in the limit case ofN→1 (orn→1) and Prandtl solution in the limit case ofN→0 (orn→∞). It is particularly interesting that from the deformation theory of plasticity one obtains the Prandtl solution based on the increatment theory of plasticity. Project supported by the National Natural Science Foundation of China (Grant No. 19132022).     

Natural convection on a vertical stretching cylinder

The flow and natural (or mixed) convection due to a vertical stretching cylinder is studied. Using similarity transforms, the Navier-Stokes and energy equations reduce to a set of nonlinear ordinary differential equations. Asymptotic analysis for large Reynolds numbers shows the relation between axisymmetric flow and two-dimensional flow. Due to the algebraic decay of the similarity functions, numerical integration is performed using a compressed coordinate. The axial velocity is composed of forced convection due to stretching and natural convection from the heated cylinder. The heat transfer increases with both the Reynolds number and the Prandtl number. The result is also a rare similarity solution of the free convection and Navier-Stokes equations.     

Mixed convection heat transfer in the boundary layers on an exponentially stretching surface with magnetic field

An analysis has been carried out to describe mixed convection heat transfer in the boundary layers on an exponentially stretching continuous surface with an exponential temperature distribution in the presence of magnetic field, viscous dissipation and internal heat generation/absorption. Approximate analytical similarity solutions of the highly non-linear momentum and energy equations are obtained. The present results are found to be in excellent agreement with previously published work on various special cases of the problem. Numerical results for temperature distribution and the local Nusselt number have been obtained for different values of the governing parameters. The numerical solutions are obtained by considering an exponential dependent stretching velocity and prescribed boundary temperature on the flow directional coordinate. The effects of various physical parameters like Prandtl number, Hartman number, Grashof number on dimensionless heat transfer characteristics are discussed in detail. In particular, it has been found that increase in Prandtl number decreases the skin-friction coefficient at the stretching surface, while increase in the strength of the magnetic field leads to increase in the local Nusselt number.     

Laminar plane jet in a gas medium

The effects of compressibility on a laminar plane jet are investigated. First, assuming that the viscosity is directly proportional to the enthalpy it is found that the similarity solutions both for momentum and energy equations exist. By a proper transformation of the similarity variable, the energy equation, in the case of viscous heating, is reduced to a hypergeometric equation and solutions for arbitrary Prandtl number are obtained. Secondly, taking the Prandtl number of the fluid as unity and a power law for the dependence of viscosity on enthalpy the velocity distribution is obtained with the help of Crocco's integral.

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Zusammenfassung Der Einfluss der Kompressibilität auf einer ebenen laminaren Strahl wird untersucht. Zunächst wird unter der Annahme, dass Zähigkeit und Enthalpie einander proportional sind, gefunden, dass sowohl die Impulsgleichung als auch die Energiegleichung Aehnlichkeitslösungen haben. Im Falle der Erwärmung durch Dissipation wird die Energiegleichung durch eine geeignete Transformation der Aehnlichkeitsvariablen auf eine hypergeometrische Differentialgleichung reduziert, und Lösungen für beliebige Prandtl-Zahlen werden bestimmt. Im Falle, dass zwischen Zähigkeit und Enthalpie ein Potenzgesetz herrscht, wird für die Prandtl-Zahl1 die Geschwindigkeitsverteilung mit Hilfe von Crocco's Integral erhalten.

     

Bessel方程的边值问题解的相似结构

将微分方程的边值问题的解式整理成具有相似结构的形式,使其更有利于进一步地分析解的内在规律;并提出了解式的相似性,与相对应的数和形的相似性统一起来,完善了数学上的三位一体和谐统一的理论.