Application of a variational method to flow over a flat plate in the entrance region with variable physical properties

A variational method has been used to solve the flow over a flat plate in the entrance region at constant wall temperature. The physical properties, i.e. Thermal conductivity and viscosity, were assumed to be linear functions of temperature in the study. Two coupled equations were derived from the variational formulation and then solved by the analog/hybrid computer. Consequently, momentum boundary layer thickness,thermal boundary layer thickness local Nusselt number and local friction factor were found for the flow. For the constant properties case a comparison was made between the exact solution, and results obtained using the solution approach suggested in this paper.     

理想流体对流传热问题的理论解

研究理想流体受迫对流传热和自然对流传热问题的理论解。采用流体无垂直于壁面法线方向运动(即无穿透)的条件取代黏性流体在壁面无滑移条件，解决了流体在边界上有滑移时计算对流传热系数的困难，给出了理想流体与平壁受迫对流传热、理想流体与竖直壁面自然对流传热和理想流体在管内受迫对流传热的理论解。结果表明：理想流体的对流传热与黏性流体同样存在着热边界层。在外部流动的情况下，无论受迫对流传热还是自然对流传热，对流传热系数都与流体的导热系数、密度和比热三乘积的二分之一次方成正比。在管内受迫对流的情况下，当无因次长度大于0．05时，局部Nu和界面无因次温度分布都不再变化，对于恒热流边界条件，Nu等于8，截面无因次平均温度等于2；对于恒壁温边界条件，Nu等于5．782，截面无因次平均温度等于2．316。     

Effects of viscous dissipation on the heat transfer in a forced pipe flow. Part 2: Thermally developing flow

In this part of the study, consideration is given to thermally developing laminar forced convection in a pipe including viscous dissipation. The axial heat conduction in the fluid is neglected. Two different thermal boundary conditions are considered: the constant heat flux (CHF) and the constant wall temperature (CWT). Both the wall heating (the fluid is heated) case and the wall cooling (the fluid is cooled) case are considered. The distributions for the developing temperature and local Nusselt number in the entrance region are obtained. Results show that the temperature profiles and local Nusselt number are influenced by the Brinkman number (Br) and the thermal boundary condition used for the wall. Significant viscous dissipation effects have been observed for large Br.     

Entry flow in heated curved pipes

The flow in the entrance region of heated curved pipes is analysed. Two cases of heating—a constant temperature at the wall, and a constant flux of heat at the wall—are considered. Using boundary layer approximations and the method of matched asymptotic expansions, the combined effects of curvature, entrance region and the buoyancy is studied. It is found that buoyancy disturbs the symmetric secondary motion induced by curvature, the deviation depending on the type of thermal input at the wall. It is also found that the oscillatory nature of the Nusselt number in the constant temperature case decreases as the Peclet number is increased.     

Possible transformations for natural convection on a vertical flat plate embedded in porous media with prescribed wall heat flux

This study derived the transformation of boundary layer equations for two-dimensional steady natural convection on a vertical wall embedded in porous media. Three different kinds of thermal boundary conditions are prescribed for wall heat flux: uniform distribution, power law variation, and exponential variation. The flow pattern contains three subregions based on the distance along the flat plate. When inertia resistance is ignored, similarity solution exists in case wall heat flux is in linear distribution. The known relationships of uniform wall temperature relative to wall heat flux variation and uniform wall heat flux relative to wall temperature variation in both cases of the pure fluid flow and the pure porous flow can all be obtained in the present transformation.     

Asymptotic solutions for turbulent mass transfer augmented by a first order chemical reaction

We present asymptotic solutions for turbulent mass transfer in a smooth conduit at high Schmidt number in the presence of a first-order chemical reaction in the fluid. Far-field solutions are derived for a case dominated by (1) mass transfer at the wall and (2) the first-order chemical reaction. An approximate solution is derived for the regime where both are important. The solutions are in good agreement with numerical solutions and with the literature. At high Damköhler numbers the system is governed by a reaction–diffusion equation and the observed increase in mass transfer coefficient is caused by thinning of the mass transfer boundary layer due to the fast chemical reaction in the fluid. We present closed-form far-field solutions for Dirichlet, Neumann and Robin boundary conditions and comment on grid resolution requirements to accurately resolve the mass transfer boundary layer. The solution strategy presented can be straightforwardly extended to non-linear wall- and bulk-reactions.     

Boundary layer theory approach to the concentration layer adjacent to a ceiling wall of a hydrogen leakage: Far region

The field of the hydrogen leakage in partially open space can be divided into two main regions according to the importance of the hydrogen concentration distribution and the flow behavior. These two regions are the jet region and the boundary layer region which are adjacent to the ceiling wall of the space, resulting from impinging the hydrogen jet to the wall. The boundary layer region in turn can be divided into two regions, according to the modeling of the flow. These regions are the stagnation-point boundary layer region and the far boundary layer region. Previously, we studied the region of stagnation-point flow (Hiemenz flow) [El-Amin MF, Kanayama H. Boundary layer theory approach to the concentration layer adjacent to a ceiling wall at impinging region of a hydrogen leakage. Int J Hydrogen Energy, in press.]. The current paper is devoted to analyze the far region of the boundary layer adjacent to the ceiling wall using the boundary layer theory. Also, an experiment has been conducted on the hydrogen leakage in partially open space to estimate the concentration distribution vertically at the center of the domain under the ceiling wall. In order to verify the boundary layer theory approach, a comparison between the measurements and the boundary layer theory approximations is investigated and the results showed a good agreement. The wall shear stress, the local friction factor, the friction drag and the non-dimensional drag coefficient of the ceiling wall are calculated. Also, both momentum and concentration boundary layer thicknesses are estimated.     

Heat and fluid flow characteristics of gases in micropipes

In this study, laminar forced convective heat transfer of a Newtonian fluid in a micropipe is analyzed by taking the viscous dissipation effect, the velocity slip and the temperature jump at the wall into account. Hydrodynamically and thermally fully developed flow case is examined. Two different thermal boundary conditions are considered: the constant heat flux (CHF) and the constant wall temperature (CWT). Either wall heating (the fluid is heated) case or wall cooling (the fluid is cooled) case is examined. The Nusselt numbers are analytically determined as a function of the Brinkman number and the Knudsen number. Different definitions of the Brinkman number based on the definition of the dimensionless temperature are discussed. It is disclosed that for the cases studied here, singularities for the Brinkman number-dependence of the Nusselt number are observed and they are discussed in view of the energy balance.     

Analysis of temperature gradient bifurcation in porous media – An exact solution

The phenomenon of temperature gradient bifurcation in a porous medium is analyzed by studying the convective heat transfer process within a channel filled with a porous medium, with internal heat generation. A local thermal non-equilibrium (LTNE) model is used to represent the energy transport within the porous medium. Exact solutions are derived for both the fluid and solid temperature distributions for two primary approaches (Models A and B) for the constant wall heat flux boundary condition. The Nusselt number for the fluid at the channel wall is also obtained. The effects of the pertinent parameters such as fluid and solid internal heat generations, Biot number and fluid to solid thermal conductivity ratio are discussed. It is shown that the internal heat generation in the solid phase is significant for the heat transfer characteristics. The validity of the one equation model is investigated by comparing the Nusselt number obtained from the LTNE model with that from the local thermal equilibrium (LTE) model. The results demonstrate the importance of utilizing the LTNE model in the present study. The phenomenon of temperature gradient bifurcation for the fluid and solid phases at the wall for Model A is established and demonstrated. In addition, the temperature distributions for Models A and B are compared. A numerical study for the constant temperature boundary condition was also carried out. It was established that the phenomenon of temperature gradient bifurcation for the fluid and solid phases for the constant temperature boundary condition can occur over a given axial region.     

Entropy generation in a binary gas mixture in the presence of thermal and solutal mixed convection

Steady-state, laminar, fully-developed mixed convection of a binary non-reacting gas mixture flowing upwards in a vertical parallel-plate channel has been investigated from the point of view of the second law of thermodynamics. Analytical expressions are derived for the entropy generation rate for two combinations of boundary conditions: uniform wall temperature with uniform wall concentration (UWT-UWC) and uniform wall heat flux with uniform wall concentration (UHF-UWC). These expressions include three sources of irreversibility: heat conduction, fluid friction and species diffusion. In the UWT-UWC case, the entropy generation rate depends on the thermal and solutal boundary conditions and the mean velocity while in the UHF-UWC case it also depends on the streamwise coordinate. For humid air, the contribution of fluid friction is negligible for both cases while heat conduction and species diffusion effects appear to be of comparable orders of magnitude.     

Experimental study and theoretical analysis of local heat transfer distribution between smooth flat surface and impinging air jet from a circular straight pipe nozzle

An experimental investigation is performed to study the effect of jet-to-plate spacing and Reynolds number on the local heat transfer distribution to normally impinging submerged circular air jet on a smooth and flat surface. A single jet from a straight circular nozzle of length-to-diameter ratio (l/d) of 83 is tested. Reynolds number based on nozzle exit condition is varied between 12,000 and 28,000 and jet-to-plate spacing between 0.5 and 8 nozzle diameters. The local heat transfer characteristics are estimated using thermal images obtained by infrared thermal imaging technique. Measurements for the static wall pressure distribution due to impinging jet at different jet-to-plate spacing are made. The local heat transfer distributions are analyzed based on theoretical predictions and experimental results of the fluid flow characteristics in the various regions of jet impingement. The heat transfer at the stagnation point is analyzed from the static wall pressure distribution. Semi-analytical solution for heat transfer in the stagnation region is obtained assuming an axisymmetric laminar boundary layer with favourable pressure gradient. The heat transfer in the wall jet region is studied considering fluid flow over a flat plate of constant heat flux. However, heat transfers in the transition region are explained from reported fluid dynamic behaviour in this region. Correlations for the local Nusselt numbers in different regions are obtained and compared with experimental results.     

Flow due to a porous stretching/shrinking sheet with thermal radiation and mass transpiration

This study investigates the behavior of carbon nanotubes (CNT) approaching an unsteady flow of a Newtonian fluid over a stagnation point on a stretching surface employing porous media. It flows when the liquid begins to move with the progression of time. Heat exchange with the environment has an impact on the flow. The implicitly limited component technique is used to solve the nondimensional partial differential equation with an associated boundary layer, which is an unstable system. Analytically, the solutions, as well as the required boundary conditions, are obtained. The effects of mass transpiration, volume fraction, and heat radiation on Newtonian fluid flow through porous media are explored. Single- and multi-walled CNTs are used as well as water, as base fluids in the experiment. The impact of thermal radiation and heat source/sink is shown in the energy equation, which is solved under four different cases: uniform heat flux case, constant wall temperature case, general power-law wall heat flux case, and general power-law wall temperature case. By supplying distinct physical characteristics, a theoretical analysis of the existence and nonexistence of unique and dual solutions may be explored. These physical parameters determine the velocity distribution and temperature distribution. Prescribed surface temperature (PST) and prescribed wall heat flux (PHF) heat transfer solutions can be written using confluent hypergeometric equations, and generic power-law PST and PHF situations can also be expressed using confluent hypergeometric equations. The graphical representations assist in the discussion of the current study's findings.     

Natural convection heat and mass transfer from a sphere in micropolar fluids with constant wall temperature and concentration

This work examines the natural convection heat and mass transfer near a sphere with constant wall temperature and concentration in a micropolar fluid. A coordinate transformation is used to transform the governing equations into nondimensional nonsimilar boundary layer equations and the obtained boundary layer equations are then solved by the cubic spline collocation method. Results for the local Nusselt number and the local Sherwood number are presented as functions of the vortex viscosity parameter, Schmidt number, buoyancy ratio, and Prandtl number. For micropolar fluids, higher viscosity tends to retard the flow and thus decreases the natural convection heat and mass transfer rates from the sphere with constant wall temperature and concentration. Moreover, the natural convection heat and mass transfer rates from a sphere in Newtonian fluids are higher than those in micropolar fluids.     

Boundary layer theory approach to the concentration layer adjacent to the ceiling wall of a hydrogen leakage: Axisymmetric impinging and far regions

As hydrogen leaks into a partially open space with a ceiling wall, a boundary layer of hydrogen can be constructed under that wall due to the impingement on the wall and the buoyancy force. The resulting boundary layer can be divided into two regions, namely the stagnation-point region and the far region. When the geometry of the source of the hydrogen leak is circular, such as a pinhole or an o-ring, the behavior of leakage flow will be axisymmetric due to the resulting radial jet. In contrast, when the geometry of the source of the hydrogen leak is planar, such as a crack, the behavior of leakage flow will be planar due to the resulting planar jet. Previously, we studied the planar case in the context of both the stagnation-point flow region [El-Amin MF, Kanayama H. Boundary layer theory approach to the concentration layer adjacent to a ceiling wall at impinging region of a hydrogen leakage. Int J Hydrogen Energy 2008; 33(21): 6393–00] and the far region [El-Amin MF, Inoue M, Kanayama H. Boundary layer theory approach to the concentration layer adjacent to a ceiling wall of a hydrogen leakage: far region. Int J Hydrogen Energy 2008; 33(24):7642–7]. This paper is concerned with both the stagnation-point flow region and the far region of the axisymmetric concentration boundary layer adjacent to a ceiling wall. Flow in the stagnation-point region is treated as Hiemenz flow, while it is treated as Blasius flow in the far region. The current results are compared with the planar cases [El-Amin MF, Kanayama H. Boundary layer theory approach to the concentration layer adjacent to a ceiling wall at impinging region of a hydrogen leakage. Int J Hydrogen Energy 2008; 33(21): 6393–00; El-Amin MF, Inoue M, Kanayama H. Boundary layer theory approach to the concentration layer adjacent to a ceiling wall of a hydrogen leakage: far region. Int J Hydrogen Energy 2008; 33(24):7642–7] for both stagnation-point flow and far regions. Both momentum and concentration boundary layer thicknesses are estimated, as well as the local friction factor.     

Heat transfer in a viscoelastic boundary layer flow over a stretching sheet

Studies are made on the viscoelastic fluid flow and heat transfer characteristics over a stretching sheet with frictional heating and internal heat generation or absorption. The heat transfer analysis has been carried out for the cases of prescribed surface temperature (PST) and prescribed surface heat flux (PHF). The momentum equation is decoupled from the energy equation for the present incompressible boundary layer flow problem with constant physical parameters. Exact solution for the velocity field and the skin-friction are obtained. Also, the solutions for the temperature and heat transfer characteristics are obtained in terms of Kummer’s function. The work due to deformation in energy equation, which is essential while formulating the viscoelastic boundary layer flow problems, is considered. This paper examines the effect of viscoelastic parameter, Eckert number, Prandtl number and non-uniform heat source/sink parameter on temperature distribution, wall temperature gradient for PST-case and wall temperature for PHF-case.     

Effects of interface wettability on microscale flow by molecular dynamics simulation

Non-equilibrium molecular dynamic simulations have been carried out to study the effect of the interface wettability on the pressure driven flow of a Lennard-Jones (LJ) fluid in a nanochannel. The results show that the hydrodynamic boundary condition at the solid-liquid interface depends on both the interface wettability and the magnitude of the driving force. For a LJ fluid in a nanochannel with hydrophilic surfaces, the velocity profiles have the traditional parabolic shape. The no-slip boundary condition may break down when the driving force exceeds a critical value that overcomes the interfacial resistance. In such a case, the MD results show a pattern of an adsorbing layer sliding along the solid wall. For a LJ fluid in a nanochannel with hydrophobic interfaces, the results show that a gap exists between the liquid and the surface, resulting in almost frictionless resistance; the velocity shows a plug flow profile and the slip length is not constant but depends on the driving force. Furthermore, it is found that the non-uniform temperature and pressure profiles near the solid walls are owing to the effect of interface wettability.     

Effect of chemical reaction and thermal radiation on heat and mass transfer flow of MHD micropolar fluid in a rotating frame of reference

This paper studies the effect of first order chemical reaction and thermal radiation on hydromagnetic free convection heat and mass transfer flow of a micropolar fluid via a porous medium bounded by a semi-infinite porous plate with constant heat source in a rotating frame of reference. The plate is assumed to oscillate in time with constant frequency so that the solutions of the boundary layer are the same oscillatory type. The dimensionless governing equations for this investigation are solved analytically using small perturbation approximation. The effect of the various dimensionless parameters entering into the problem on the velocity, temperature and concentration profiles across the boundary layer are investigated through graphs. Also the results of the skin friction coefficient, couple stress coefficient, the rate of heat and mass transfer at the wall are prepared with various values of the parameters.     

Mixed convection in the stagnation point flow adjacent to a vertical surface in a viscoelastic fluid

An analytic technique, namely the homotopy analysis method (HAM), is applied to study the steady mixed convection in two-dimensional stagnation flows of a viscoelastic fluid around heated surfaces for the case when the temperature of the wall varies linearly with the distance from the stagnation point. The two-dimensional boundary layer equations governing the flow and thermal fields are reduced by appropriate transformations to a set of two ordinary differential equations. These equations are solved analytically using the HAM in the buoyancy assisting and opposing regions. It is found that, as for the Newtonian flow case, a reverse flow region develops in the buoyancy opposing flow region, and dual solutions are found to exist in that flow regime for a certain ranges of the buoyancy and viscoelastic parameters.     

The effect of viscous dissipation and rarefaction on rectangular microchannel convective heat transfer

The effect of viscous dissipation and rarefaction on rectangular microchannel convective heat transfer rates, as given by the Nusselt number, is numerically evaluated subject to constant wall heat flux (H2) and constant wall temperature (T) thermal boundary conditions. Numerical results are obtained using a continuum based, three-dimensional, compressible, unsteady computational fluid dynamics algorithm with slip velocity and temperature jump boundary conditions applied to the momentum and energy equations, respectively. For the limiting case of parallel plate channels, analytic solutions for the thermally and hydrodynamically fully developed momentum and energy equations are derived, subject to both first- and second-order slip velocity and temperature jump boundary conditions, from which analytic Nusselt number solutions are then obtained. Excellent agreement between the analytical and numerical results verifies the accuracy of the numerical algorithm, which is then employed to obtain three-dimensional rectangular channel and thermally/hydrodynamically developing Nusselt numbers. Nusselt number data are presented as functions of Knudsen number, Brinkman number, Peclet number, momentum and thermal accommodation coefficients, and aspect ratio. Rarefaction and viscous dissipation effects are shown to significantly affect the convective heat transfer rate in the slip flow regime.