Analysis of slip flow heat transfer between two unsymmetrically heated parallel plates with viscous dissipation

This paper presents an analytical investigation to study the heat transfer and fluid flow characteristics in the slip flow region for hydrodynamically and thermally fully developed flow between parallel plates. Both upper and lower plates are subjected to asymmetric heat flux boundary conditions. The effect of first order velocity slip, temperature jump, asymmetric heat flux ratio and viscous dissipation on the heat transfer performance is analyzed. Closed form expressions are obtained for the temperature distribution and Nusselt number. Present predictions are verified for the cases that neglect the viscous heating and microscale effects. The effect of asymmetric heat flux ratio with and without viscous dissipation on Nusselt number for both macroscale and microscale is highlighted. The heat transfer characteristics are found to depend on various modeling parameters, namely, modified Brinkman number, Knudsen number and heat flux ratio.     

Slip flow and heat transfer in rectangular and circular microchannels using hybrid FE/FV method

Rarefied gas flows typically encountered in MEMS systems are numerically investigated in this study. Fluid flow and heat transfer in rectangular and circular microchannels within the slip flow regime are studied in detail by our recently developed implicit, incompressible, hybrid (finite element/finite volume) flow solver. The hybrid flow solver methodology is based on the pressure correction or projection method, which involves a fractional step approach to obtain an intermediate velocity field by solving the original momentum equations with the matrix‐free, implicit, cell‐centered finite volume method. The Poisson equation resulting from the fractional step approach is then solved by node based Galerkin finite element method for an auxiliary variable, which is closely related to pressure and is used to update the velocity field and pressure field. The hybrid flow solver has been extended for applications in MEMS by incorporating first order slip flow boundary conditions. Extended inlet boundary conditions are used for rectangular microchannels, whereas classical inlet boundary conditions are used for circular microchannels to emphasize on the entrance region singularity. In this study, rarefaction effects characterized by Knudsen number (Kn) in the range of 0 ⩽ Kn ⩽ 0.1 are numerically investigated for rectangular and circular microchannels with constant wall temperature. Extensive validations of our hybrid code are performed with available analytical solutions and experimental data for fully developed velocity profiles, friction factors, and Nusselt numbers. The influence of rarefaction on rectangular microchannels with aspect ratios between 0 and 1 is thoroughly investigated. Friction coefficients are found to be decreasing with increasing Knudsen number for both rectangular and circular microchannels. The reduction in the friction coefficients is more pronounced for rectangular microchannels with smaller aspect ratios. Effects of rarefaction and gas‐wall surface interaction parameter on heat transfer are analyzed for rectangular and circular microchannels. For most engineering applications, heat transfer is decreased with rarefaction. However, for fluids with very large Prandtl numbers, velocity slip dominates the temperature jump resulting in an increase in heat transfer with rarefaction. Depending on the gas‐wall surface interaction properties, extreme reductions in the Nusselt number can occur. Present results confirm the existence of a transition point below and above wherein heat transfer enhancement and reduction can occur. Copyright © 2011 John Wiley & Sons, Ltd.     

水平管外沸腾强化换热的数值模拟与场协同分析

应用FLUENT软件对制冷剂R134a在光管和横纹槽管水平管外沸腾传热进行三维数值模拟,得到其饱和泡状沸腾过程中体积含气率的分布规律,并比较它们的换热系数。结果表明横纹槽管外侧能够很好地强化沸腾传热。此外,还通过改变边界条件分析质量流量、热流密度的变化对横纹槽管管外沸腾换热系数的影响。最后应用场协同理论,从局部换热角度分析其强化机制。研究表明,横纹槽管水平管外沸腾换热得到强化的原因是其凹槽前后的速度场与温度梯度场之间夹角较小,协同程度更好。     

Effects of tangential and radial velocity on fluid flow and heat transfer for flow through a pipe with twisted tape insert—laminar flow

The present study reports the numerical analysis of fluid flow and heat transfer in a pipe with full length twisted tape insert. The investigation is carried out for five different twist ratios of 4, 5, 6, 8 and 10 at 100 ≤ Re ≤ 1000. The velocity field in terms of streamwise, tangential and radial velocity and temperature field are studied as a function of Reynolds number and twist ratio. The variation of friction factor and Nusselt number with Reynolds number for different twist ratios is also presented. The heat transfer enhancement due to insertion of twisted tape mainly comes from the tangential and radial components of velocities, which are regarded as secondary fluid motion. It is evident from the results that with increase in Reynolds number the axial convection increases. However, with the decrease in the twist ratio, the tangential and radial convection increases, leading to increased heat transfer. The secondary flow affects the thermal boundary layer inside the tube and increases the cross-flow mixing, which increases the heat transfer. The correlations for prediction of friction factor and Nusselt number based on the numerical data are also proposed.     

Boundary integral equation for tangential derivative of flux in Laplace and Helmholtz equations

In this paper, the boundary integral equations (BIEs) for the tangential derivative of flux in Laplace and Helmholtz equations are presented. These integral representations can be used in order to solve several problems in the boundary element method (BEM): cubic solutions including degrees of freedom in flux's tangential derivative value (Hermitian interpolation), nodal sensitivity, analytic gradients in optimization problems, or tangential derivative evaluation in problems that require the computation of such variable (elasticity problems in BEM). The analysis has been developed for 2D formulation. Kernels for tangential derivative of flux lead to high‐order singularities (O(1/r³)). The limit to the boundary analysis has been carried out. Based on this analysis, regularization formulae have been obtained in order to use such BIE in numerical codes. A set of numerical benchmarks have been carried out in order to validate theoretical and practical aspects, by considering known analytic solutions for the test problems. The results show that the tangential BIEs have been properly developed and implemented. Copyright © 2005 John Wiley & Sons, Ltd.     

Three dimensional boundary formulations for nonlinear thermal shape sensitivities

Implicit differentiation of the discretized boundary integral equations governing the conduction of heat in three dimensional (3D) solid objects, subjected to nonlinear boundary conditions, and with temperature dependent material properties, is shown to generate an accurate and economical approach for the computation of shape sensitivities. The theoretical formulation for primary response (surface temperature and normal heat flux) sensitivities and secondary response (surface tangential heat flux components and internal temperature and heat flux components) sensitivities is given. Iterative strategies are described for the solution of the resulting sets of nonlinear equations and computational performances examined. Multi-zone analysis and zone condensation strategies are demonstrated to provide substantial computational economies in this process for models with either localized nonlinear boundary conditions or regions of geometric insensitivity to design variables. A series of nonlinear sensitivity example problems are presented that have closed form solutions. Sensitivities computed using the boundary formulation are shown to be in excellent agreement with these exact expressions.     

Effect of surface slip on Stokes flow past a spherical particle in infinite fluid and near a plane wall

The motion of a spherical particle in infinite linear flow and near a plane wall, subject to the slip boundary condition on both the particle surface and the wall, is studied in the limit of zero Reynolds number. In the case of infinite flow, an exact solution is derived using the singularity representation, and analytical expressions for the force, torque, and stresslet are derived in terms of slip coefficients generalizing the Stokes–Basset–Einstein law. The slip velocity reduces the drag force, torque, and the effective viscosity of a dilute suspension. In the case of wall-bounded flow, advantage is taken of the axial symmetry of the boundaries of the flow with respect to the axis that is normal to the wall and passes through the particle center to formulate the problem in terms of a system of one-dimensional integral equations for the first sine and cosine Fourier coefficients of the unknown traction and velocity along the boundary contour in a meridional plane. Numerical solutions furnish accurate predictions for (a) the force and torque exerted on a particle translating parallel to the wall in a quiescent fluid, (b) the force and torque exerted on a particle rotating about an axis that is parallel to the wall in a quiescent fluid, and (c) the translational and angular velocities of a freely suspended particle in simple shear flow parallel to the wall. For certain combinations of the wall and particle slip coefficients, a particle moving under the influence of a tangential force translates parallel to the wall without rotation, and a particle moving under the influence of a tangential torque rotates about an axis that is parallel to the wall without translation. For a particle convected in simple shear flow, minimum translational velocity is observed for no-slip surfaces. However, allowing for slip may either increase or decrease the particle angular velocity, and the dependence on the wall and particle slip coefficients is not necessarily monotonic.     

Temperature distribution and heat flow around a crack of arbitrary orientation in a functionally graded medium

This paper investigates the heat transfer problem of an infinite functionally graded medium containing an arbitrarily oriented crack under uniform remote heat flux. In the mathematical treatment the crack is approximated as a perfectly insulating cut. By using Fourier transformation, the mixed boundary value problem is reduced to a Cauchy-type singular integral equation for an unknown density function. The singular integral equation is then solved by representing the density function with a Chebyshev polynomial-based series and solving the resulting linear equation using a collocation technique. The temperature field in the vicinity of the crack and the crack-tip heat flux intensity factor are presented to quantify the effect of crack orientation and grading inhomogeneity on the heat flow around the crack.     

Effect of magnetic fields on heat convection inside a concentric annulus filled with Al2O3–water nanofluid

In the current study, forced convective heat transfer of an MHD fully developed laminar nanofluid between two concentric horizontal cylinders is investigated in the presence of a radial magnetic field. In contrast to a conventional no-slip condition at the surfaces, the Navier’s slip condition is considered at the surface to represent the non-equilibrium region near the surfaces. Employing the modified Buongiorno model, the conservative partial differential equations have been collapsed to two-point ordinary boundary value differential equations before being numerically solved. To consider the effects of thermal boundary condition on nanoparticle migration, two distinctive cases including constant heat flux at the outer wall and adiabatic inner wall (Case A) and constant heat flux at the inner wall with adiabatic outer wall (Case B) have been considered. Our results indicate that due to thermophoresis force, the distribution of nanoparticles was denser at the adiabatic wall for the case A which affects the local and the universal fluid flow and heat transfer characteristics. Moreover, inducing a radial magnetic field on the system, heat transfer rate was increased for the case A which had a decreasing effect on the case B. Finally, slip velocity at the walls enhances heat transfer rate for both cases.     

Galerkin boundary integral method for evaluating surface derivatives

A Galerkin boundary integral procedure for evaluating the complete derivative, e.g., potential gradient or stress tensor, is presented. The expressions for these boundary derivatives involve hypersingular kernels, and the advantage of the Galerkin approach is that the integrals exist when a continuous surface interpolation is employed. As a consequence, nodal derivative values, at smooth surface points or at corners, can be obtained directly. This method is applied to the problem of electromigration-driven void dynamics in thin film aluminum interconnects. In this application, the tangential component of the electric field on the boundary is required to compute the flux of atoms at the void surface.     

Viscous flow due to an exponentially shrinking permeable sheet in nanofluid in presence of slip

Classical problem of steady boundary layer flow of nanofluid over an exponentially permeable shrinking sheet in presence of slip is investigated. The model used for nanofluid includes Brownian motion and thermophoresis effects. The governing equations for momentum, energy, and nanofluid solid volume fraction are transformed to ordinary differential equations with the help of similarity transformations and then solved numerically using fourth order Runge–Kutta method with shooting technique. It is found that the governing parameters, viz. the suction/blowing parameter, velocity slip, thermal and mass slip parameters, Brownian motion parameter, thermophoresis parameter, Prandtl number, and Lewis number significantly affect the flow field, heat, and mass transfer. The results obtained indicate that the dual solutions exist for certain values of the mass suction parameter. Velocity increases whereas the temperature and nanoparticle volume fraction decrease due to suction through the porous sheet. It is noted that with the increase in velocity slip fluid velocity increases whereas temperature and concentration decrease. Due to increase in thermal slip and mass slip both temperature and concentration decrease.     

Analytical approximations to the viscous glass-flow problem in the mould-plunger pressing process, including an investigation of boundary conditions

Industrial glass is produced at temperatures above 600°C, where glass becomes a highly viscous incompressible fluid, usually considered as Newtonian. In the production two phases may be distinguished, namely the pressing phase and the blowing phase. This study will be concerned with glass flow in the pressing phase, which is called thus because a blob of fluid glass (called a gob) is pressed in a mould by a plunger, such that the glass flows between mould and plunger, in order to obtain the preform of a bottle or jar, called a parison. In the blowing phase (not considered here) the parison is subsequently blown into the final shape of the product. By application of the slender geometry of mould and plunger and a cylindrical symmetry, a form of Reynolds's lubrication flow equations is obtained. These equations are solved by utilizing the incompressibility of the glass, by which the flux at any axial cross section is determined for prescribed plunger velocity, leading to analytical results in closed form for velocity field and pressure gradient. The glass level is implicitly defined by the integral over the varying volume which is to remain constant. The pressure may then be determined by integration. Special attention is given to the required boundary conditions. It is known that, depending on several problem parameters like temperature, pressure, and smoothness of the wall, the glass flow slips, to some extent, along the wall. Therefore, this study includes a general formulation of the boundary condition of partial slip in the form of a linear relation between shear stress and slip velocity, also known as Navier's slip condition. The coefficient of this relation, a positive number, may vary in our solution with axial position, but depends on the problem and is to be obtained from (for example) experiment. Two special cases, which seem to be relevant in practice, are considered as examples: (i) no slip on both plunger and mould; (ii) no slip on the mould and full slip (zero friction) on the plunger. The results are compared with fully numerical (FEM) solutions of a Stokes-flow model, and the agreement is good or excellent. Since in any practical situation it is not the plunger velocity which is prescribed, but (within practical limits) the force applied by the plunger, the problem of a prescribed plunger force has also been investigated.     

Numerical Modeling of the Tangential Stress Effects on Convective Fluid Flows in an Open Cavity

The study of convective processes caused by impact of various forces on the fluid and gas media is actual nowadays. The increased interest to these problems is determined by preparation of the new experiments on the International Space Station in the frame of the scientific project CIMEX of the European Space Agency. They are the experiments to investigate the convective flows of the fluids with a thermocapillary interface between liquid and gas phases. In the case, when a co-current gas flux generates the tangential stresses on a free boundary the additional characteristics of the convective flows should be detected. In this paper the exact solutions of a stationary problems of convection and of gas flow in the horizontal layers with a free thermocapillary interface are constructed. An evaporation through the fluid-gas flow interface is modeled qualitatively with the help of a heat exchange condition. It is determined that the velocity on the fluid-gas interface can be positive and negative. The equal-zero condition for the interface velocity is found, as well. In the experiments an open horizontal fluid layer is studied so that the closed flux condition is not needed. Although the closed flux requirement is not set explicitly a parameters relation is found, when this condition is fulfilled. The paper presents the velocity and temperature profiles in the conditions, which correspond qualitatively to the CIMEX experiment.     

Mixed convective flow and heat transfer in a vertical helicoidal pipe with finite pitch

Fully developed laminar mixed convective flow and heat transfer in a vertical helicoidal pipe with finite pitch is studied numerically in this paper. The centrifugal, buoyancy, and torsion forces created by the pitch of the helix are considered. The thermal boundary conditions are uniform heat flux axially and uniform wall temperature peripherally. The velocity, the temperature profiles, the friction factor, and the Nusselt number are obtained. The results indicate that torsion can significantly change the axial and secondary flow patterns and temperature distributions in the helicoidal pipe with a fixed pressure gradient. Torsion can also remarkably affect the flow rate, the friction factor, and the Nusselt number. However, at a fixed Dean number, there is only slight difference in the Nusselt numbers of the upward and downward flow.The results presented in this paper were obtained in the course of research sponsored by the National Science Foundation (NSF) under Grant No. CTS-9017732     

A molecular dynamics simulation of TIP4P and Lennard-Jones water in nanochannel

Summary. The flow characteristics of the plane Poiseuille flow in the nanochannel driven by a constant external force are studied by the Lennard-Jones and TIP4P potentials. The problem is investigated by the leap-frog method in the field of molecular dynamics. In this work, the wall boundary condition is considered to be the situation that the water is absorbed on the metal wall and is then formed to be flat ice. Both global effect (effective channel width) and local effect (wall boundary types) are examined to demonstrate the features of the distributions of velocity and its gradient in the system. When the effective channel width is less than a critical value, the numerical results show that the Navier-Stokes theory would fail to predict the velocity distribution. Furthermore, the velocity profile at a virtual slip plane presents the slip condition. Finally, we can reason that the surface effect exists and will affect the shear stress in the nanochannel.     

Theoretical analysis on mixed convection boundary layer flow over a wedge with uniform suction or injection

Summary Forced and free mixed convection boundary layer flow over a wedge with uniform suction or injection is theoretically investigated. Nonsimilar partial differential equations are transformed into ordinary differential equations by means of difference-differential method. The solutions of the resulting equations are obtained in integral forms and are calculated by iterative numerical procedures. The results were given for velocity profiles, temperature profiles, friction and heat transfer parameters for various values of suction/injection parameter, pressure gradient parameter and buoyancy parameter.     

Heat‐flux‐specified boundary treatment for gas flow and heat transfer in microchannel using direct simulation Monte Carlo method

Direct simulation Monte Carlo (DSMC) method has been widely used to study gaseous flow and heat transfer in micro‐fluidic devices. For flows associated with microelectromechanical systems (MEMS), the heat‐flux‐specified (HFS) boundary condition broadly exists. However, problems with HFS boundary have not been realized in the simulation of microchannel flows with DSMC method. To overcome this problem, a new technique named as inverse temperature sampling (ITS) is developed. This technique provides an approach to calculate the molecular reflective characteristic temperature from the specified heat flux at the wall boundary. Coupling with DSMC method, the ITS technique can treat the HFS boundary condition in DSMC method for both simple gas and gas mixtures. For validation, heat flux obtained from two‐dimensional Poiseuille flows with wall‐temperature‐specified (WTS) boundary condition is employed as the initial thermal boundary condition of our new method. Sampled wall temperature by the ITS method agrees well with the expected value. Pressure, velocity and temperature distributions under these two thermal boundary conditions (WTS and HFS) are compared. Effects of molecule collision model and gas–surface interaction model are also investigated. Results show that the proposed ITS method could accurately simulate gaseous flow and heat transfer in MEMS. Copyright © 2007 John Wiley & Sons, Ltd.     

Equivalent histories,states and a variational problem for a rigid linear heat conductor with memory

The model of a rigid linear heat conductor which exhibits a constitutive equation with memory for the heat flux can be characterized by processes and states. The equivalence between histories is introduced in order to consider minimal states. The inversion of the constitutive equation, in which the heat flux appears as a linear functional of the history of the temperature gradient, allows to consider states expressed in terms of the heat flux vector instead of the temperature gradient. A variational formulation of an evolution problem with mixed initial boundary conditions is also given.     

An adjoint method for the inverse design of solidification processes with natural convection

This paper presents a finite element algorithm based on the adjoint method for the design of a certain class of solidification processes. In particular, the paper addresses the design of directional solidification processes for pure materials such that a desired freezing front heat flux and growth velocity are achieved. This is the first time that an infinite-dimensional continuum adjoint formulation is obtained and implemented for the solution of such inverse/design problems with moving boundaries and Boussinesq incompressible flow. The present design problem belongs to a category of inverse problems in which one is looking for the unknown conditions in part of the boundary, while overspecified boundary conditions are supplied in another part of the boundary (here the freezing interface). The solidification design problem is mathematically posed as a whole time-domain optimization problem. The gradient of the cost functional is calculated using the solution of an appropriately defined continuous adjoint problem. The minimization process is realized by the conjugate gradient method via the solutions of the direct, adjoint and sensitivity sub-problems. The proposed methodology is demonstrated with the solidification of an initially superheated liquid aluminum confined in a square mold. The non-uniformity in the casting product in the direction of gravity due to the existence of natural convection in the melt is emphasized. The inverse design problem is then posed as finding the appropriate spatial-temporal variations of the boundary heat flux on the vertical mold walls that can eliminate or reduce the effects of convection on the freezing interface heat fluxes and growth velocity. The numerical example demonstrates the accuracy and convergence of the adjoint formulation. Finally, open related research design problems are discussed. © 1998 John Wiley & Sons, Ltd.     

Experimental measurements and modeling of transient heat transfer in forced flow of He II at high velocities

An experiment has been built to study heat transfer in forced flow of He II at flow velocities up to 22 m/s. The main part of this experiment is a 10 mm ID, 0.86 m long straight test section instrumented with a heater, thermometers and pressure transducers. The high flow velocities allow clear observation of the effects of the forced convection, counterflow heat transfer and the Joule–Thomson effect. A numerical model based on the He II energy conservation equation and including pressure effects has been developed to compare with the experimental results. The model works well for low flow velocities where the heat flux is primarily driven by the temperature gradient and for high flow velocities where the heat flux is primarily driven by the pressure gradients. In the intermediate velocity region, discrepancies between the model and experiment may result from an inappropriate representation of the heat flux by counterflow when the temperature and pressure gradients have an effect of similar magnitude on the heat flux.