SLIP FLOW IN RECTANGULAR MICROTUBES

Microscale fluid dynamics has received intensive interest due to the extraordinary advances in electronic device miniaturization, where the peaks of temperature from hotspots must be reduced by a coolant flowing in a microchannel. One of the most meaningful microscale effects is the emergence of slip flow. The present analysis is concerned with the 2-D velocity distribution of steady-state, hydrodynamically developed, laminar slip flow, for Newtonian fluids in rectangular ducts. The solutions describing velocity profiles, friction factors, shear stresses, momentum flux, and kinetic energy correction factors are derived resorting to the integral transform technique. The results are shown as functions of the aspect ratio and the Knudsen number, in the limit of Kn 0 they reproduce perfectly the well-known results of macroscale fluid dynamics.     

THERMAL SHOCK OF ORTHOTROPIC RECTANGULAR PLATES

The dynamic response of homogeneous, orthotropic plates having two parallel, simply supported edges and exposed to rapid surface heating is examined. Uncoupled, thin-plate theory is used to determine the inducedflexural vibrations. The solution is obtained as the superposition of two displacement fields, representing the quasi-static and the dynamic behaviors. An exact Levy-type solution is derived for the quasi-static problem. For the dynamic case, the governing partial differential equation is first reduced to a system of ordinary differential equations in time by means of the Galerkin method. For special situations in which the latter equations are uncoupled (e.g., plates having all edges simply supported), the solution is obtained using the Laplace transform. For cases involving more general support conditions, the equations are integrated numerically using the Runge-Kutta-Gill technique. Numerical results are presented for both isotropic and orthotropic plates having various combinations of edge conditions     

FLOW DEVELOPMENT OF CO-FLOWING STREAMS IN RECTANGULAR MICRO-CHANNELS

The diffusion and flow development characteristics of two co-flowing, laminar streams in a high aspect ratio rectangular micro-channel have been examined. A long, thin splitter plate initially separates the two streams such that fully developed flow in each of the two channels is established prior to merging. The co-flowing micro-channel has an aspect ratio of 16 with a width of 1006 μm and a height of 63 μm. Micro-Particle Image Velocimetry (μPIV) was utilized to observe the interaction between the streams for a range of flow rate ratios ranging from one to nine, for Reynolds numbers of one and ten. For flow rate ratios greater than one, a cross-stream pressure gradient exists immediately downstream of the splitter plate, which results in a strong lateral flow of the faster moving fluid into the slower moving fluid. Despite this rapid expansion, the fluids in the two streams do not mix. The two streams eventually recover a fully developed velocity profile across the entire channel. A model is presented to predict this development length based on the pressure imbalance between the two streams. The model is expressed in terms of the flow rate ratio between the streams, which is shown to be a function of channel aspect ratio. An asymptotic condition for the development length is found for high flow rate ratios and high aspect ratio channels. It is shown that existing entrance length relationships greatly underpredict this development length.     

HYDROMAGNETIC NATURAL CONVECTION IN A TILTED RECTANGULAR POROUS ENCLOSURE

A study is made of natural convection within an inclined porous layer saturated by an electrically conducting fluid in the presence of a magnetic field. The long side walls of the cavity are maintained at a uniform heat flux condition, while the short side walls are thermally insulated. On the basis of a parallel flow model, the problem is solved analytically to obtain a set of closed-form solutions. Scale analysis is applied to the case of a boundary layer flow regime in a vertical enclosure. Comparison between the fully numerical and analytical solutions is presented for 0 ≤, Ra ≤, 10³ ≤, Ha ≤, 10, and -180 ° ≤, Φ ≤, 180°, where Ra, Ha, and Φ denote the Rayleigh number, Hartmann number, and inclination of the enclosure, respectively. It is found that the analytical solutions can faithfully predict the influence of a magnetic field on the flow structure and heat transfer for a wide range of the governing parameters. For a boundary layer flow regime in a vertical cavity the results of the scale analysis agree well with approximations of the analytical solution. For this situation it is found that the Nusselt number is Nu = O.5Ra^2/5 / ( 1 + Ha²) ^2/5. For a horizontal cavity heated from below the critical Rayleigh number for the onset of motion, determined from a stability analysis, corresponds to that for the existence of unicellular convection using the parallel flow approximation. In general, it is demonstrated that, with the application of an external magnetic field, the temperature and velocity fields are significantly modified and the Nusselt number is decreased with increasing Ha.     

HIGH-ORDER BOUNDARY CONDITIONS FOR GASEOUS FLOWS IN RECTANGULAR MICRODUCTS

Heat conduction of finite-length single-walled carbon nanotubes (SWNTs) was simulated by the molecular dynamics method with Tersoff-Brenner bond order potential. Temperature at each end of an SWNT was controlled by the phantom technique, and the thermal conductivity was calculated with Fourier’s law from the measured temperature gradient and the energy budgets in phantom molecules. The measured thermal conductivity did not converge to a finite value with increase in tube length up to 404 nm, but an interesting power law relation was observed. The phonon density of states and photon dispersion relations were directly extracted from simulation results for further analysis of heat conduction mechanism based on the phonon concept.     

MIXED CONVECTION IN A RECTANGULAR ENCLOSURE WITH DISCRETE HEAT SOURCES

A numerical study is carried out on mixed convective heat transfer in an enclosure. The discrete heat sources are embedded on a vertical board, which is situated on the bottom wall of an enclosure. An external airflow enters the enclosure through an opening in one vertical wall and exits from another opening in the opposite wall. This study simulates a practical system, such as air-cooled electronic devices with heated elements. Emphasis is placed on the influence of the governing parameters, such as Reynolds number, Re, buoyancy parameter, Gr/Re², location of the heat sources, and the conductivity ratio, rk, on the thermal phenomenon in the enclosure. The coupled equations of the simulated model are solved numerically using the cubic spline collocation method. The computational results indicate that both the thermal field and the average Nusselt number (Nu) depend strongly on the governing parameters, position of the heat sources, as well as the property of the heat-source-embedded board.     

HEAT TRANSFER CHARACTERISTICS OF FLOW PAST A RECTANGULAR PROTRUDING BODY

Flow over a protruding body is an attractive research field in thermal engineering. In the present study, laminar flow over a protruding body is considered. Unsteady two-dimensional Navier-Strokes and energy equations are solved numerically using a control volume approach. The heat transfer characteristics due to vortex shedding are examined in detail. The flow and heat transfer characteristics are compared with their counterparts obtained from the steady flow case. The entropy analysis is carried out and irreversibility generated because of fluid friction is computed. The relative heat transfer and irreversibility ratios are introduced to compare the heat transfer performance characteristics of unsteady and steady flow cases. It is found that the relative heat transfer ratio attains higher value, whereas relative irreversibility ratio becomes less for unsteady flow as compared with that obtained for the steady flow case.     

CONTROL OF VORTEX SHEDDING BEHIND A RECTANGULAR CYLINDER NEAR THE GROUND

ABSTRACT

The present study focuses on the effects of the aspect ratio of a rectangular cylinder and the gap height from the ground to the unsteady wake fields around the rectangular cylinder near the ground. The results indicate that the vortex begins to shed by the interaction of the separated shear layer on the upper surface of the cylinder with the upwash flow in the gap region. Based on this investigation, simple passive control methods consisting of attached vertical or horizontal plates on the lower surface of the rectangular cylinder are introduced to reduce the aerodynamic drag as well as to suppress the vortex-induced oscillation.     

MIXED CONVECTION HEAT AND MASS TRANSFER IN RADIALLY ROTATING RECTANGULAR DUCTS

The present work investigates mixed convection heat and mass transfer in the entrance region of radially rotating rectangular ducts with water film evaporation along the porous duct watts. Mechanisms of secondary vortex development in the ducts under various conditions are examined by a vorticity-velocity numerical method. Emphasis is placed on the rotation effects, including both Coriolis and centrifugal buoyancy forces, and the mass diffusion effect on the flow structure and heat transfer characteristics. Results are presented in particular for an air-water vapor system under various conditions. Predicted results show that the effects of liquid film evaporation along the porous duct walls on the mixed convection neat transfer are rather substantial. The magnitude of the evaporative latent heal transfer may be 10 times greater than that of sensible heat transfer. The predictions also demonstrate that the distributions of Nu, Sh?z?, and fRe are closely related to the emergence, disappearance, growth, and decay of the rotating-induced secondary vortices. Additionally, a higher Nu?z? is found for a rectangular duct with a larger aspect ratio ( γ = 2) due to the relatively stronger secondary flows.     

FAST CONVERGENT SERIES SOLUTIONS FOR THERMAL BENDING OF THIN RECTANGULAR PLATES

Thermally induced bending of thin rectangular plates with one clamped and three simply supported edges is studied in detail for the case of a spacewise constant thermal moment. Using this sample problem, it is demonstrated that classical series representations for thermally induced bending moments and shear forces may exhibit numerical instabilities, slow convergence, and divergence. Fast convergent solutions are developed by replacing hyperbolic functions in the classical series representations by means of exponential functions with a negative argument and by utilizing Kummer's transformation. Divergence is overcome using Cesaro's generalized C₁-summation method. The presented series solutions are checked numerically via finite element computations. Symbolic computation is used to derive and to evaluate the series solutions and to derive limiting values at the plate corners. For practical use, tables and graphical representations of results are presented in the form ofCzerny's Tables for force-loaded rectangular plates.     

矩形腔内相变材料接触熔化的分析

对矩形腔内相变材料紧密接触熔化过程进行了理论分析。应用努谢尔特液体边界层理论，求得了便于工程计算用的接触熔化传热过程的理论解。分析结果与实验数据进行了比较，吻合程度良好。     

THERMOELASTIC VIBRATIONS OF A LAMINATED RECTANGULAR PLATE SUBJECTED TO A THERMAL SHOCK

In this work the response of a rectangular, simply supported, symmetrically laminated, cross-ply composite plate subjected to a thermal shock is developed. The analysis includes the interaction between the strain and temperature fields and investigates the effect of accounting for the orthotropic material properties in the governing elastic and thermal equations. The resulting solution for the vibration of the plate is compared to a previous analysis of a homogeneous, isotropic, rectangular plate. Comparison indicates that while the solutions have similar forms, there are two key quantitative differences between them.     

NATURAL CONVECTION OVER A ROTATING CYLINDRICAL HEAT SOURCE IN A RECTANGULAR ENCLOSURE

A numerical study of laminar two-dimensional natural convection heat transfer from a uniformly heated horizontal cylinder rotating about its center, and placed in an isothermal rectangular enclosure, is performed using a spectral element method. The physical aspects of the flow and its thermal behavior are studied for a wide range of pure natural convection to mixed convection at low and high rotational speeds of the cylinder. The computer program has been validated against experimental correlations available on pure natural convection of heated bodies in enclosures. The rotation of the cylinder has been found to enhance the heat transfer. At low ratios of Rayleigh number to the square of the rotational Reynolds number, Ra / Re_ω ², the maximum temperature on the cylinder surface is decreased by as much as 25–35% from similar cases with fixed cylinders. At moderate values of Ra/ Re_ω ², the thermal plume rising above the cylinder is shifted in the rotation direction and the angular shift decreases as Ra / Re_ω increases. The rotation produces more uniform temperature and shear stress distributions around the cylinder surface. At high Rayleigh numbers the increase in rotation reduces the cylinder mean Nusselt number by 2–10% as compared with the fixed cylinder.     

THERMAL BUCKLING ANALYSIS OF INHOMOGENEOUS RECTANGULAR PLATE DUE TO UNIFORM HEAT SUPPLY

This paper is concerned with the thermal buckling analysis of an isotropic inhomogeneous rectangular plate subjected to the arbitrary thermal loads. The fundamental equations system is derived by introducing the technique of the newly defined position of the reference plane, which allows us to analyze the problem using an elementary plate theory. It is assumed that the material properties such as the coefficient of linear thermal expansion α, the thermal conductivity λ, and Young's modulus of elasticity, E, are changed in the thickness direction with the power law of the coordinate variable, whereas Poisson's ratio ν is assumed to be constant. As an illustrative example, we consider the thermal buckling problem of a simply supported inhomogeneous rectangular plate due to uniform heat supply. Numerical calculations are carried out for several cases taking into account the variations of the inhomogeneous material properties, aspect ratio, and width-to-thickness ratio.     

“NEALE PLATE“ EQUATIONS AND APPLICATIONS TO BUCKLING OF RECTANGULAR PLATES

The field equations associated with Neale's variational theorem are developed and applied to the problem of buckling and postbuckling of heated, constrained flat plates. The resulting equations are a generalization of Karman's equations in rate form, having a constitutive equation in which the strain/curvature rates are a linear combination of the stress resultant rates. In the immediate neighborhood of a critical point, the theory predicts a substantial reduction of the buckling temperature due to plasticity effects. Further, for long strips, the force stress resultant in the short direction appears to decrease in absolute value after buckling, while the force stress resultant in the long direction appears to increase in absolute value.     

PRESSURE EFFECTS ON NATURAL CONVECTION FOR NON-BOUSSINESQ FLUID IN A RECTANGULAR ENCLOSURE

Numerical predictions of pressure effects on natural convection for a non-Boussinesq fluid in the rectangular enclosures are presented. A solution method based on a compressible flow model is employed to simultaneously determine the absolute pressure, density, temperature, and velocity distributions in the enclosures. Discretization equations are derived from the integral mass, momentum, and energy equations on a staggered grid. The fluid pressure in the enclosure is varied from 20 to 300 kPa such that the flow behavior in a vacuum or pressurized system can be observed. Physical situations investigated also include cases in a wide range of wall temperature difference associated with respective length scales, corresponding to an equivalent modified Rayleigh number ranging from 10 4 to 10 6 . The validity of the incompressible flow model coupled with the Boussinesq approximation for the fluid density, which is commonly used in the existing studies of the buoyant flows, is discussed.     

EVOLUTION TO CHAOTIC NATURAL CONVECTION IN A RECTANGULAR ENCLOSURE WITH MIXED BOUNDARY CONDITIONS

The purpose of this numerical study is to analyze the characteristics of transition from laminar to chaotic natural convection in a fluid-filled two-dimensional, unity aspect ratio rectangular cavity with mixed thermal boundary conditions. For a medium Prandtl number fluid ( Pr ) the numerical solution of the two-dimensional Navier-Stokes momentum and energy equation with Bousinessq approximation, it is found that there are finite Rayleigh numbers Ral Ra2, and Ra3 for the onset of single-, double-, and multiple-frequency oscillatory motion at different spatial locations in the enclosure. As Ra increases, the flow exhibits a change from steady convection to periodic to quasi-periodic flow, while no period doubting is observed. The onset of strong chaos appears when Ra = 57,000 Rac. This system does not revert to steady state convection for Rac as high as 285,000. As Ra increases, various measures of chaos, such as power spectrum, Poincare sections, phase portrait, and time series of various dynamical variable signals, all show an increasing degree of characteristics of chaos,     

A RECTANGULAR SOLAR POND MODEL INCORPORATING EMPIRICAL FUNCTIONS FOR AIR AND SOIL TEMPERATURES

A novel theoretical model, capable of giving the temporal temperature variation at any point inside or outside a non-insulated rectangular solar pond at any time, is presented. Incorporating the finite difference approach, the model makes use of one- and two-dimensional heat balances written on discrete regions in the brine and in the soil adjacent to the pond. These simultaneous equations are solved for the local temperatures, using a computer program. Values of hourly averaged air temperature and daily averaged soil temperature for the site were used as input parameters, and empirical functions for the time-dependence of these variables were incorporated into the theoretical model. It was found necessary to use this level of detail of the meteorological data for reliable predictions on the solar ponds. The model results are compared with measured results on an actual solar pond built in Cukurova, Turkey. The modelled and experimental temperature profiles are found to be in a very good agreement. The results indicate that the thickness of the salt gradient region of a solar pond should not be less than 1.3 m. Heat losses form the pond side-walls was found not to effect the performance of solar ponds when the surface area is greater than 100 m².     

TRANSIENT THERMAL STRESSES IN A RECTANGULAR PLATE DUE TO NONUNIFORM HEAT TRANSFER COEFFICIENTS

Abstract

The linear problems of transient temperature and thermal stresses in a thin, finite, rectangular plate subjected to heat losses due to nonuniform heat transfer coefficients on the upper and lower plate surfaces are solved by a direct power series approach through the application of the Lanczos-Chebyshev and the discrete least-squares methods. A numerical example demonstrates the accuracies that can be achieved by using only a small number of terms.     

EXPERIMENTAL STUDIES ON FLOW IN CONVERGENT AND DIVERGENT DUCTS OF RECTANGULAR CROSS SECTION

This paper describes the experimental examination of the pressure drop and heat transfer of the flow in convergent and divergent ducts of rectangular cross section. The aspect ratio based on the dimensions of the large end of the duct was 0⋅1. It has been found that at a given convergent or divergent angle pressure drop decreases while heat transfer increases with increasing Reynolds number. Along a given duct of small convergent angle, pressure drop increases while heat transfer decreases along the duct. However, heat transfer may increase near the downstream end of ducts of high convergent angle. At a given Reynolds number, both pressure drop and heat transfer increase with increasing convergent angle. As for flow in divergent ducts, the effects of Reynolds number on pressure drop and heat transfer are somewhat similar to those of flow in a convergent duct. © 1997 by John Wiley & Sons, Ltd.