Natural convection cooling of a heat source embedded on the bottom of an enclosure filled with Cu-water nanofluid: effects of various thermal boundary conditions

Steady natural convection cooling of a localized heat source at the bottom wall of an enclosure filled with Cu-water nanofluid for a variety of thermal boundary conditions at the sidewalls has been studied in the present paper. Finite difference method was employed to solve the dimensionless governing equations of the problem. The effects of governing parameters, namely, solid volume fraction, the different values of the heat source length and the different locations of the heat source on the streamlines and isotherms contours as well as maximum temperature, Nusselt number and average Nusselt number along the heat source were considered. The present results are validated by favorable comparisons with previously published results. The results of the problem are presented in graphical and tabular forms and discussed.     

Double-diffusive natural convection in inclined finned triangular porous enclosures in the presence of heat generation/absorption effects

The problem of double-diffusive convection in inclined finned triangular porous enclosures for various thermal and concentration boundary conditions and in the presence of heat source or sink was studied. The finite difference method was employed to solve the dimensionless governing equations of the problem. The effects of the governing parameters, namely the dimensionless time parameter, the inclination angle, Darcy number, heat generation/absorption parameter, the buoyancy parameter and the Rayleigh number on the streamlines, temperature and concentration contours as well as selected velocity component in the x-direction, local and average Nusselt numbers and local and average Sherwood number at the heated and concentrated wall for various values of the aspect ratio and the position of the fin were considered. The present results are validated by favorable comparisons with previously published results. All the results of the problem were presented in graphical and tabular forms and discussed.     

On the spreading radius of surface tension driven oil on deep water

The spread of a thin oil film by surface tension gradients from an oil source of unlimited mass on deep water is considered. A similarity solution for the velocity fields of the oil and water, the oil thickness and the rate at which each grow is obtained both for axisymmetric and the previously explored planar spreading. The dimensionless size of the spread, which is oil type independent, is shown to be 1.0754 and 1.4150 for axisymmetric and planar spreading respectively. It is further shown that the oil film equation of state, which relates surface tension to oil thickness, is unique to each oil or oil-surfactant mixture.     

粘弹性地基上粘弹性输流管道的稳定性分析

从Winkler假设和单轴线性粘弹性本构方程出发,推导了Kelvin-Voigt粘弹性地基上三参量固体模型输流管道的运动微分方程,采用改进的有限差分法,分析了管道和地基的粘弹性参数对输流管道无量纲复频率和无量纲流速之间的变化关系的影响。     

An Analytic Solution for the Frontal Flow Period in 1D Counter-Current Spontaneous Imbibition into Fractured Porous Media Including Gravity and Wettability Effects

Including gravity and wettability effects, a full analytical solution for the frontal flow period for 1D counter-current spontaneous imbibition of a wetting phase into a porous medium saturated initially with non-wetting phase at initial wetting phase saturation is presented. The analytical solution applicable for liquid–liquid and liquid–gas systems is essentially valid for the cases when the gravity forces are relatively large and before the wetting phase front hits the no-flow boundary in the capillary-dominated regime. The new analytical solution free of any arbitrary parameters can also be utilized for predicting non-wetting phase recovery by spontaneous imbibition. In addition, a new dimensionless time equation for predicting dimensionless distances travelled by the wetting phase front versus dimensionless time is presented. Dimensionless distance travelled by the waterfront versus time was calculated varying the non-wetting phase viscosity between 1 and 100 mPas. The new dimensionless time expression was able to perfectly scale all these calculated dimensionless distance versus time responses into one single curve confirming the ability for the new scaling equation to properly account for variations in non-wetting phase viscosities. The dimensionless stabilization time, defined as the time at which the capillary forces are balanced by the gravity forces, was calculated to be approximately 0.6. The full analytical solution was finally used to derive a new transfer function with application to dual-porosity simulation.     

Transient Heat and Moisture Flow Around Heat Source Buried in an Unsaturated Half Space

This article presents solutions for the transient heat and moisture transport due to both disk heat source and cylindrical heat source buried in an unsaturated half space. The solutions are presented in Hankel–Laplace transform domain and in dimensionless style. Coupled effect of thermally driven moisture transport is especially investigated because of its importance to alter the flow field in low-permeability medium. Parametric study has been performed to assess the effects of five independent dimensionless parameters on flow field. The stability and accuracy of the present solutions are demonstrated from the comparison between the results obtained from these solutions and those by using a well-established finite element code CODE_BRIGHT. Despite the simplified assumptions required in order to obtain analytical solutions in Hankel–Laplace transform domain, the results incorporate the main mechanisms involved in the coupled thermo-hydraulic (T-H) problem, and they may be eventually used for validation purposes.     

Heat transfer and pressure distributions of an impinging jet on a flat surface

Experiments were conducted to determine the heat transfer and surface pressure characteristics of a round jet impinging normal on isothermal flat plate. Three nozzles of exit diameters 3, 5 and 7?mm have been used. The local heat transfer rates have been estimated from the outputs of three-wire differential thermocouple heat flux sensors. The results cover a Reynolds number range of 3400 to 41?000 and dimensionless separation distances varies from 6 to 58. The static pressure distributions along the impingement surface are found to be similar and closer to the heat transfer variations at the same configurations. A simple correlation is derived for the average heat transfer coefficients within ±10% deviation from the output data covering the complete range of experimental limits. The predicted values of Nusselt number have also been compared with the results obtained from the literature. The agreement was generally good.     

Asymptotic behavior of the Taylor-expansion method of moments for solving a coagulation equation for Brownian particles

The evolution equations of moments for the Brownian coagulation of nanoparticles in both continuum and free molecule regimes are analytically studied. These equations are derived using a Taylor-expansion technique. The self-preserving size distribution is investigated using a newly defined dimensionless parameter, and the asymptotic values for this parameter are theoretically determined. The dimensionless time required for an initial size distribution to achieve self-preservation is also derived in both regimes. Once the size distribution becomes self-preserving, the time evolution of the zeroth and second moments can be theoretically obtained, and it is found that the second moment varies linearly with time in the continuum regime. Equivalent equations, rather than the original ones from which they are derived, can be employed to improve the accuracy of the results and reduce the computational cost for Brownian coagulation in the continuum regime as well as the free molecule regime.     

Analytical solution of a double moving boundary problem for nonlinear flows in one-dimensional semi-infinite long porous media with low permeability简

Based on Huang＇s accurate tri-sectional nonlin- ear kinematic equation （1997）, a dimensionless simplified mathematical model for nonlinear flow in one-dimensional semi-infinite long porous media with low permeability is presented for the case of a constant flow rate on the inner boundary. This model contains double moving boundaries, including an internal moving boundary and an external mov- ing boundary, which are different from the classical Stefan problem in heat conduction： The velocity of the external moving boundary is proportional to the second derivative of the unknown pressure function with respect to the distance parameter on this boundary. Through a similarity transfor- mation, the nonlinear partial differential equation （PDE） sys- tem is transformed into a linear PDE system. Then an ana- lytical solution is obtained for the dimensionless simplified mathematical model. This solution can be used for strictly checking the validity of numerical methods in solving such nonlinear mathematical models for flows in low-permeable porous media for petroleum engineering applications. Finally, through plotted comparison curves from the exact an- alytical solution, the sensitive effects of three characteristic parameters are discussed. It is concluded that with a decrease in the dimensionless critical pressure gradient, the sensi- tive effects of the dimensionless variable on the dimension- less pressure distribution and dimensionless pressure gradi- ent distribution become more serious; with an increase in the dimensionless pseudo threshold pressure gradient, the sensi- tive effects of the dimensionless variable become more serious; the dimensionless threshold pressure gradient （TPG） has a great effect on the external moving boundary but has little effect on the internal moving boundary.     

Impact of thermal radiation on MHD slip flow over a flat plate with variable fluid properties

The influence of partial slip, thermal radiation and temperature dependent fluid properties on the hydro-magnetic fluid flow and heat transfer over a flat plate with convective surface heat flux at the boundary and non-uniform heat source/sink is studied. The transverse magnetic field is assumed as a function of the distance from the origin. Also it is assumed that the fluid viscosity and the thermal conductivity vary as an inverse function and linear function of temperature respectively. Using the similarity transformation, the governing system of non-linear partial differential equations are transformed into similarity non-linear ordinary differential equations and are solved numerically using symbolic software MATHEMATICA 7.0. The numerical values obtained within the boundary layer for the dimensionless velocity, temperature, skin friction coefficient and the Nusselt number are presented through graphs and tables for several sets of values of the parameters. The effects of various physical parameters on the flow and heat transfer characteristics are discussed from the physical point of view.     

Numerical study of the axially symmetric motion of an incompressible viscous fluid in an annulus between two concentric rotating spheres

The research reported herein involved the study of the transient motion of a system consisting of an incompressible Newtonian fluid in an annulus between two concentric, rotating, rigid spheres. The primary purpose of the research was to study the use of a numerical method for analysing the transient motion that results from the interaction between the fluid in the annulus and the spheres which are started suddenly by the action of prescribed torques. The problems considered in this research included cases where: (a) one or both spheres rotate with prescribed constant angular velocities and (b) one sphere rotates due to the action of an applied constant or impulsive t?orque. In this research the coupled solid and fluid equations were solved numerically by employing the finite difference technique. With the approach adopted in this research, only the derivatives with respect to spatial variables were approximated with the use of the finite difference formulae. The steady state problem was also solved as a separate problem (for verification purposes), and the results were compared with those obtained from the solution of the transient problem. Newton's algorithm was employed to solve the algebraic equations which resulted from the steady state problem, and the Adams fourth-order predictor–corrector method was employed to solve the ordinary differential equations for the transient problem. Results were obtained for the streamfunction, circumferential function, angular velocity of the spheres and viscous torques acting on the spheres as a function of time for various values of the system dimensionless parameters.     

传热传质对有抽吸的收缩薄片上的非线性磁流体动力学边界层流动的影响

This work is concerned with Magnetohydrodynamic viscous flow due to a shrinking sheet in the presence of suction. The cases of two dimensional and axisymmetric shrinking are discussed. The governing boundary layer equations are written into a dimensionless form by similarity transformations. The transformed coupled nonlinear ordinary differential equations are numerically solved by using an advanced numeric technique. Favorability comparisons with previously published work are presented. Numerical results for the dimensionless velocity, temperature and concentration profiles as well as for the skin friction, heat and mass transfer and deposition rate are obtained and displayed graphically for pertinent parameters to show interesting aspects of the solution.     

Computation of V-notch shape factors in four-point bend specimen for fracture tests on brittle materials

Four-point bend (FPB) specimen is an important test sample in mixed mode fracture study of notched components made from brittle materials like rocks, brittle polymers, ceramics, etc. On the other hand, the notch stress intensity factors (NSIFs) are vital parameters in brittle fracture assessment of V-notched structures. Therefore, computation of NSIFs in FPB specimens is of practical interest to engineers and researchers. Since the available methods for calculating the NSIFs are often cumbersome and need complicated calculations, it is preferred to show them as a set of dimensionless parameters for the FPB specimen. In this research, the finite element method coupled with a recently developed algorithm called FEOD is employed to calculate the NSIFs of a FPB specimen for several V-shape notches and for different combinations of mode I and mode II. The obtained NSIFs are then converted to dimensionless parameters called notch shape factors and are illustrated in a number of figures. It is shown that depending on the notch depth and the location of loading points, full mode mixity from pure mode I to pure mode II can be provided in the FPB specimen. The numerical results obtained in this research are verified by using very limited results reported earlier in literature.     

Influence of the variable thermophysical properties on the turbulent buoyancy-driven airflow inside open square cavities

The effects of the air variable properties (density, viscosity and thermal conductivity) on the buoyancy-driven flows established in open square cavities are investigated, as well as the influence of the stated boundary conditions at open edges and the employed differencing scheme. Two-dimensional, laminar, transitional and turbulent simulations are obtained, considering both uniform wall temperature and uniform heat flux heating conditions. In transitional and turbulent cases, the low-Reynolds k − ω turbulence model is employed. The average Nusselt number and the dimensionless mass-flow rate have been obtained for a wide and not yet covered range of the Rayleigh number varying from 10³ to 10¹⁶. The results obtained taking into account variable properties effects are compared with those calculated assuming constant properties and the Boussinesq approximation. For uniform heat flux heating, a correlation for the critical heating parameter above which the burnout phenomenon can be obtained is presented, not reported in previous works. The effects of variable properties on the flow patterns are analyzed.     

An asymptotic solution for fluid production from an elliptical hydraulic fracture at early-times

The early-time transient flow during the start-up of fluid production from a porous medium by a well intersected by a vertical elliptical hydraulic fracture is studied using an asymptotic analysis. The analysis is focused on the situation of practical interest where the fracture conductivity is high so that production from the fracture dominates. The first three terms in a short-time asymptotic expansion for the production rate during constant-pressure production, and for the well-pressure during constant-rate production, are obtained. It is shown that the fracture tip starts to influence the production rate only when the dimensionless time is increased to the square of the reciprocal of the dimensionless fracture conductivity. The asymptotic results also show that geometric factors of an elliptical fracture introduce non-negligible corrections to the so-called bilinear flow in the early times, which were previously erroneously associated with the effect of the fracture tip.     

Unsteady Natural Convection,Heat and Mass Transfer in Inclined Triangular Porous Enclosures in the Presence of Heat Source or Sink: Effect of Sinusoidal Variation of Boundary Conditions

The problem of double-diffusive convection in inclined triangular porous enclosures with sinusoidal variation of boundary conditions in the presence of heat source or sink was discussed numerically. The dimensionless governing equations of the problem were solved numerically by using finite difference method. The effects of governing parameters, namely, the dimensionless time parameter, various values of the inclination angle, Darcy number, the heat generation/absorption parameter, the buoyancy parameter and the amplitude wave length ratio on the streamlines, temperature and concentration contours as well as the velocity component in the x-direction at the triangle mid-section, the average Nusselt and Sherwood numbers at the bottom wall of the triangle for various values of aspect ratio were considered. The present results are validated by favorable comparisons with previously published results. All the results of the problem were presented in graphical and tabular forms and discussed.     

Propagation of a penny-shaped fluid-driven fracture in an impermeable rock: asymptotic solutions

This paper presents an analysis of the propagation of a penny-shaped hydraulic fracture in an impermeable elastic rock. The fracture is driven by an incompressible Newtonian fluid injected from a source at the center of the fracture. The fluid flow is modeled according to lubrication theory, while the elastic response is governed by a singular integral equation relating the crack opening and the fluid pressure. It is shown that the scaled equations contain only one parameter, a dimensionless toughness, which controls the regimes of fracture propagation. Asymptotic solutions for zero and large dimensionless toughness are constructed. Finally, the regimes of fracture propagation are analyzed by matching the asymptotic solutions with results of a numerical algorithm applicable to arbitrary toughness.     

承压热冲击下反应堆压力容器的塑性失效分析

承压热冲击（PTS）使反应堆压力容器（RPV）的完整性面临极大的挑战,尤其是在喷嘴周围的环带区。考虑到瞬态温度处于零韧性参考温度之上,引入非线性材料性能来模拟一个真实RPV的混合温度场和应力场。应用扩展有限元法对喷嘴区中的裂纹扩展过程进行模拟,并得到了承压热冲击下的临界裂纹尺寸。结果显示,PTS初期的热应力效应显著,后期热-机械耦合作用产生的峰值应力很可能会引起结构失效。应用直接和间接耦合法得到的数值结果吻合,且后者的计算效率较高。在塑性极限承载状况下,裂纹尖端在靠近内壁的位置容易产生扩展。离内壁较远的裂纹尖端由于受热冲击影响较小,发生裂纹扩展的可能性相对较低。随着基体墙厚度的减小,容许的裂纹尺寸急剧缩小,接近临界承载状态时稳态裂纹扩展的程度明显降低。     

Local Nonsimilarity Solution for the Impact of a Chemical Reaction in an MHD Mixed Convection Heat and Mass Transfer Flow over a Porous Wedge in the Presence Of Suction/Injection

Combined heat and mass transfer in free, forced, and mixed convection flows along a porous wedge with a magnetic effect in the presence of a chemical reaction is investigated. The flow field characteristics are analyzed with the Runge—Kutta—Gill method in conjunction with the shooting method, and local nonsimilarity method. The governing boundary-layer equations are written in a dimensionless form with the use of the Falkner—Skan transformations. Owing to the effect of the buoyancy force, the power law of temperature and concentration, and suction/injection on the wall of the wedge, the flow field is locally nonsimilar. Numerical calculations up to the third-order level of truncation are carried out for different values of dimensionless parameters as a special case. Effects of the magnetic field strength in the presence of a chemical reaction with a variable wall temperature and concentration on the dimensionless velocity, temperature, and concentration profiles are shown graphically. Comparisons with previously published works are performed, and excellent agreement between the results is obtained.     

Analytical solution to stagnation-point flow and heat transfer over a stretching sheet based on homotopy analysis

This paper is concerned with two-dimensional stagnation-point steady flow of an incompressible viscous fluid towards a stretching sheet whose velocity is proportional to the distance from the slit. The governing system of partial differential equations is first transformed into a system of dimensionless ordinary differential equations. Analytical solutions of the velocity distribution and dimensionless temperature profiles are obtained for different ratios of free stream velocity and stretching velocity, Prandtl number, Eckert number and dimensionality index in series forms using homotopy analysis method（HAM）. It is shown that a boundary layer is formed when the free stream velocity exceeds the stretching velocity, and an inverted boundary layer is formed when the free stream velocity is less than the stretching velocity. Graphs are presented to show the effects of different parameters.