Flow Through Super-elliptic Ducts Filled with a Darcy–Brinkman Medium

The fully developed flow and constant flux heat transfer in super-elliptic ducts filled with a porous (Darcy–Brinkman) medium are studied. Super-elliptic ducts resemble rectangular ducts with rounded corners. An efficient Ritz method is used to determine the velocity and temperature fields. Extensive tables for friction factor–Reynolds number product and Nusselt number are given.     

Response to Comments on the Article Titled ‘The Starting Flow in Ducts Filled with a Darcy–Brinkman Medium’

The fully developed flow and constant flux heat transfer in super-elliptic ducts filled with a porous (Darcy–Brinkman) medium are studied. Super-elliptic ducts resemble rectangular ducts with rounded corners. An efficient Ritz method is used to determine the velocity and temperature fields. Extensive tables for friction factor–Reynolds number product and Nusselt number are given.     

Pulsating flow in a heat exchanger

The problem of heat transfer in industrial processes, heat exchangers, and combustion chambers is formulated for a case where flow inside the chamber consists of a periodic motion imposed on a fully developed turbulent flow. It is shown that the velocity pulsations induce harmonic oscillations in temperature, thus breaking the temperature field into a steady mean part and a harmonic part. The interaction between the velocity and temperature oscillations introduces an extra term into the energy equation which reflects the effect of pulsations in producing higher heat transfer rates. The analysis shows that when the mean temperature is fully developed with constant heat flux at the wall, there is no effect of the velocity pulsations on the total heat transfer rate along the chamber. For the case where the mean temperature profile is not fully developed, analytical solutions are obtained for asymptotic values of the pulsations frequency. The results show the temperature gradient and its dependence on the frequency. These results are used to evaluate the feasibility of pulsating the flow in a heat exchanger for obtaining higher rates of heat transfer.     

An experimental study of nonrheometric flows of visco-elastic fluids: I. flow into a re-entrant tube

A laser anemometer has been used to study the developing flow both upstream and downstream from the entry plane in a re-entrant tube geometry. A 0.75% polyacrylamide/water solution was used and Reynolds numbers (based on wall conditions in the fully developed downstream flow) in the range 100–500 were obtained in 1.82-cm and 2.40-cm-diameter tubes.The shear stress-shear rate relationship for the fluid was measured using a cone and plate geometry in conjunction with a Weissenberg rheogoniometer. Theoretical fully developed velocity profiles were calculated numerically from these measurements. The measured fully developed velocity profiles were found to be in excellent agreement with those calculated.Velocity profiles measured at the tube entry plane showed the pronounced wall region distortion typically predicted by recent numerical solutions of the flow of purely-viscous fluids through an abrupt tube contraction.It was found that the major velocity rearrangements were achieved within only a few diameters (both upstream and downstream) of the entry plane. In particular, the velocity distribution near the tube wall varied negligibly over the relatively longer distance (many diameters) that it took for the centreline velocity to achieve its fully developed value. Entry lengths were found to be only about half those for purely-viscous fluids.Calculation of the time of flight along the central streamline confirmed that the major rearrangements of velocity suffered by the fluid occurred over a relatively short time period. This indicates that hereditary integral constitutive equations may have to be used in theoretical analyses of this type of flow situation.     

Heat transfer in vertical annular laminar flow of two immiscible liquids

The paper investigates heat transfer in annular laminar undisturbed flow of two immiscible liquids, with constant heat-flux generated at the wall of the tube. It presents an analytical solution for the fully developed temperature field. This is used to obtain a more general solution from a model, describing the temperature field as a superposition of the fully developed and the developing fields. This superposition model is solved by an orthogonal collocation method. An asymptotic model for short entry lengths is also described. Calculations for a kerosene-water system, show that the superposition solution converges to the entrance solution below 100 diameters and converges asymptotically to the solution of the fully developed temperature field beyond 5000 diameters. The effect of the wavy interface is assessed experimentally for annular kerosene-water flow, by comparing predicted and measured temperature profiles. It is found that experimental profiles are considerably flatter and measured Nusselt numbers for the kerosene phase are accordingly higher by 40–320% as compared to the undisturbed flow analyses.     

The influence of thermal radiation on MHD station-point flow past a stretching sheet with heat generation

This letter is concerned with the plane and axisymmetric stagnation-point flows and heat transfer of an electrically-conducting fluid past a stretching sheet in the presence of the thermal radiation and heat generation or absorption. The analytical solutions for the velocity distribution and dimensionless temperature profiles are obtained for the various values of the ratio of free stream velocity and stretching velocity, heat source parameter, Prandtl number, thermal radiation parameter, the suction and injection velocity parameter and magnetic parameter and dimensionality index in the series form with the help of homotopy analysis method (HAM). Convergence of the series is explicitly discussed. In addition, shear stress and heat flux at the surface are calculated.     

An analysis on forced convection in a channel filled with a Brinkman-Darcy porous medium: Exact and approximate solutions

An analysis was made to investigate non-Darcian fully developed flow and heat transfer in a porous channel bounded by two parallel walls subjected to uniform heat flux. The Brinkmanextended Darcy model was employed to study the effect of the boundary viscous frictional drag on hydrodynamic and heat transfer characteristics. An exact expression has been derived for the Nusselt number under the uniform wall heat flux condition. Approximate results were also obtained by exploiting a momentum integral relation and an auxiliary relation implicit in the Brinkmanextended Darcy model. Excellent agreement was confirmed between the approximate and exact solutions even in details of velocity and temperature profiles.     

Influence of hall effect and ion slip on velocity and temperature fields in an MHD channel

The combined influence of viscosity, Hall effect and ion slip on hydrodynamic fields and on heat transfer is investigated. The exact solutions for velocity, induced magnetic field and temperature are derived for the laminar MHD flow in a flat channel assuming a small magnetic Reynolds number, finely segmented electrodes, fully developed flow and uniform heat flux at channel walls. The internal generation of heat is not considered. The Kantorowitsch method of variational calculus is employed to approximate the complicated velocity distribution.     

A Local Thermal Non-Equilibrium Analysis of Fully Developed Forced Convective Flow in a Tube Filled with a Porous Medium

A local thermal non-equilibrium model has been considered for the case of thermally fully developed flow within a constant heat flux tube filled with a porous medium. Exact temperature profiles for the fluid and solid phases are found after combining the two individual energy equations and then transforming them into a single ordinary differential equation with respect to the temperature difference between the solid phase and the wall subject to constant heat flux. The exact solutions for the case of metal-foam and air combination reveal that the local thermal equilibrium assumption may fail for the case of constant heat flux wall. The Nusselt number is presented as a function of the Peclet number, which shows a significant increase due to both high stagnant thermal conductivity and thermal dispersion resulting from the presence of the metal-foam.     

Melting heat transfer in steady laminar flow over a moving surface

The steady laminar boundary layer flow and heat transfer from a warm, laminar liquid flow to a melting surface moving parallel to a constant free stream is studied in this paper. The continuity, momentum and energy equations, which are coupled nonlinear partial differential equations are reduced to a set of two nonlinear ordinary differential equations, before being solved numerically using the Runge–Kutta–Fehlberg method. Results for the skin friction coefficient, local Nusselt number, velocity profiles as well as temperature profiles are presented for different values of the governing parameters. Effects of the melting parameter, moving parameter and Prandtl number on the flow and heat transfer characteristics are thoroughly examined. It is found that the problem admits dual solutions.     

Thermally developing flow in curved square ducts with internal fins

The laminar incompressible hydrodynamically fully developed and thermally developing flow is studied in a curved square duct with four longitudinal fins. The duct is successively subjected to constant wall temperature, to circumferentially uniform temperature and axially linearly or exponentially varying temperature. The local and fully developed Nusselt numbers are examined for various values of the Dean number and it is found that the heat transfer rate increases for high fins. The parameters that affect the entry length are studied and the fluctuations of the local Nu that appear in the entrance region are investigated. Temperature contour plots are presented for the visualization of the temperature field and functional relations for the Nusselt number are proposed in terms of the Dean and Prandtl numbers.     

Developing laminar forced convection in eccentric annuli

The paper presents a boundary-layer model describing the laminar forced convection heat transfer in the entry region of eccentric annuli. A finite-difference numerical algorithm is developed for solving this model. Numerical results are presented for the developing velocity profiles and the pressure drop in annuli of radius ratio 0.5 and 0.9 over a dimensionless eccentricity ranged from 0.1 to 0.8. Heat transfer parameters are presented for a fluid of Pr=0.7 under the conditions of an isothermally heated inner wall while the outer wall is kept at the inlet fluid temperature. Received on 18 March 1997     

Analytical Investigation of the Effect of Viscous Dissipation on Couette Flow in a Channel Partially Filled with a Porous Medium

An analytical study of viscous dissipation effect on the fully developed forced convection Couette flow through a parallel plate channel partially filled with porous medium is presented. A uniform heat flux is imposed at the moving plate while the fixed plate is insulated. In the fluid-only region the flow field is governed by Navier–Stokes equation while the Brinkman-extended Darcy law relationship is considered in the fully saturated porous medium. The interface conditions are formulated with an empirical constant β due to the stress jump boundary condition. Fluid properties are assumed to be constant and the longitudinal heat conduction is neglected. A closed-form solution for the velocity and temperature distributions and also the Nusselt number in the channel are obtained and the viscous dissipation effect on these profiles is briefly investigated.     

On the Stability of a Laminar Wall Jet with Heat Transfer

The hydrodynamic stability of a low speed, plane, non-isothermal laminar wall jet at a constant temperature boundary condition was investigated theoretically and experimentally. The mean velocity and temperature profiles used in the stability analysis were obtained by implementing the Illingworth–Stewartson transformation that allows one to extend the classical Glauert solution to a thermally non-uniform flow. The stability calculations showed that the two unstable eigenmodes coexisting at moderate Reynolds numbers are significantly affected by the heat transfer. Heating is destabilizing the flow while cooling is stabilizing it. However, the large-scale instabilities associated with the inflection point of the velocity profile still amplify in spite of the high level of the stabilizing temperature difference. The calculated stability characteristics of the wall jet with heat transfer were compared with experimental data. The comparison showed excellent agreement for small amplitudes of the imposed perturbations. The agreement is less good for the phase velocities of the sub-harmonic wave and this is attributed to experimental difficulties and to nonlinear effects.     

Wärmeübergang bei turbulenter Ringspaltströmung und konstanter thermischer Randbedingung zweiter Art

An analysis has been carried out to determine the heat transfer characteristics for fully developed turbulent flow in concentric annular ducts with constant thermal boundary conditions of second kind. The energy equation has been solved analytically by separation of variables. Results for heat transfer in the thermal entrance region, in the thermal fully developed flow and thermal entrance-length results are presented over a wide range of Reynolds number (10⁴?Re?10⁶) and Prandtl-number (0?Pr?100) and compared to available empirical data.     

Flow and heat transfer over a generalized stretching/shrinking wall problem—Exact solutions of the Navier-Stokes equations

In this paper, we investigate the steady momentum and heat transfer of a viscous fluid flow over a stretching/shrinking sheet. Exact solutions are presented for the Navier-Stokes equations. The new solutions provide a more general formulation including the linearly stretching and shrinking wall problems as well as the asymptotic suction velocity profiles over a moving plate. Interesting non-linear phenomena are observed in the current results including both exponentially decaying solution and algebraically decaying solution, multiple solutions with infinite number of solutions for the flow field, and velocity overshoot. 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 strength on the temperature profiles and wall heat flux are also presented and discussed. The exact solution of this general flow configuration is a rare case for the Navier-Stokes equation.     

Laminar forced convection with viscous dissipation in a concentric annular ductConvection laminaire forcée avec dissipation visqueuse dans un conduit annulaire concentrique

Forced convection heat transfer in fully developed flows of viscous dissipating fluids in concentric annular ducts is analyzed analytically. Special attention has been paid to the effect of the viscous dissipation. Two different cases of the thermal boundary conditions are considered: uniform heat flux at the outer wall and adiabatic inner wall (Case A) and uniform heat flux at the inner wall and adiabatic outer wall (Case B). Solutions for the velocity and temperature distributions and the Nusselt number are obtained for different values of the aspect ratio and the Brinkman number. The present analytical results for the case without the viscous dissipation effect are compared with those available in the literature and an excellent agreement is observed. To cite this article: M. Avc?, O. Ayd?n, C. R. Mecanique 334 (2006).     

具有热源项的驻点流动和传热问题的近似解析解

研究持续拉伸变形表面上二维平面和轴对称驻点流的动量和热量传输问题。利用同伦分析方法获得速度分布和温度分布的级数解,讨论了级数的敛散性。通过图形分析主流速度与拉伸速度的比率参数,普朗特参数,热源参数和流动类型指标对速度边界层和温度边界层的影响。结果表明,这些参数对二维平面驻点流动和传热有较大的影响。     

Horizontal boundary-layer natural convection in a porous medium saturated with a gas

Boundary-layer analysis is performed for free convection flow over a hot horizontal surface embedded in a porous medium saturated with a gas of variable properties. The variable gas properties are accounted for via the assumption that thermal conductivity and dynamic viscosity are proportional to temperature. A similarity solution is shown to exist for the case of constant surface temperature. Numerical results for the stream function, horizontal velocity, and temperature profiles within the boundary layer as well as for the mass of entrained gas, surface slip velocity, and heat transfer rate at different values of the wall-temperature parameter are presented. Asymptotic solutions for large heating are also available to support the numerical work.