A GENERAL APPROACH FOR ANALYZING THE ARBITRARY MOTION OF A CIRCULAR DISK IN A STOKES FLOW

A concise analytic method is developed to investigate the arbitrary motion of a circular disk through an unbounded fluid satisfying Stokes equation. Four elementary motions are considered within the same mathematical framework: broadside translation, edgewise translation, in-plane rotation and out-of-plane rotation of a disk. Stokes equations are reduced to a set of dual integral expressions relating the velocity and traction in the plane of the disk. The dual integral equations are solved exactly for each motion and lead, in turn, to closed-form analytical expressions for the velocity and pressure fields. Although many of these results have been previously reported, the approach described here unifies the analysis of the four different motions and presents a straightforward solution technique.     

Motions of a sphere in a time-dependent stokes flow: A generalization of Faxen’s law

A general solution of the unsteady Stokes equation in spherical coordinates is derived for flow in the exterior of a sphere, and then applied to study the arbitrary unsteady motion of a rigid sphere in an unbounded single fluid domain which is undergoing a time-dependent mean flow. Calculation of the hydrodynamic force and torque on the sphere leads to a generalization of the Faxen’s law to time-dependent flow fields which satisfy the unsteady Stokes equation. For illustrative purposes, we consider the relative motion of gas bubbles which undergo very rapid oscillations so that the generalized Faxen’s law derived for a solid sphere can be applied. We also demonstrate that our results reduce to those of Faxen for the steady flow limit.     

A NUMERICAL SOLUTION FOR THE ANOMALOUS SEDIMENTATION OF A SMALL BROWNIAN SPHERE IN A VERTICAL CYLINDER OF PERIODICALLY VARYING RADIUS

We consider the long-time asymptotic transport of a small Brownian sphere that is sedimenting through a viscous fluid within an infinite vertical cylinder whose radius changes periodically and abruptly between two values. Use of generalized Taylor dispersion theory gives rise to a set of steady-state differential equations that govern the long-time mean velocity. We prove that these equations possess a unique solution. Finite difference discretization of all governing equations and boundary conditions, except for the normalization condition, leads to a homogeneous system of equations that possesses an infinite number of solutions. Application of the normalization condition selects the particular solution of the set that satisfies all conditions. Despite the absence of classical wall effects, the long-time mean velocity of the particle through the pore is generally less than the Stokes sedimentation velocity that applies at every point within the fluid. This anomalous behavior could be used to enhance force-driven separation of Brownian particles that possess identical Stokes velocities but different diffusivities. Furthermore, this numerical approach may be applied to similar problems.     

A NUMERICAL SOLUTION FOR THE ANOMALOUS SEDIMENTATION OF A SMALL BROWNIAN SPHERE IN A VERTICAL CYLINDER OF PERIODICALLY VARYING RADIUS

We consider the long-time asymptotic transport of a small Brownian sphere that is sedimenting through a viscous fluid within an infinite vertical cylinder whose radius changes periodically and abruptly between two values. Use of generalized Taylor dispersion theory gives rise to a set of steady-state differential equations that govern the long-time mean velocity. We prove that these equations possess a unique solution. Finite difference discretization of all governing equations and boundary conditions, except for the normalization condition, leads to a homogeneous system of equations that possesses an infinite number of solutions. Application of the normalization condition selects the particular solution of the set that satisfies all conditions. Despite the absence of classical wall effects, the long-time mean velocity of the particle through the pore is generally less than the Stokes sedimentation velocity that applies at every point within the fluid. This anomalous behavior could be used to enhance force-driven separation of Brownian particles that possess identical Stokes velocities but different diffusivities. Furthermore, this numerical approach may be applied to similar problems.     

ON THE MODELLING OF PARTICLE-BODY INTERACTIONS IN STOKES FLOWS INVOLVING A SPHERE AND CIRCULAR DISC OR A TORUS AND CIRCULAR CYLINDER USING POINT SINGULARITIES

Few exact solutions of the Stokes equations are known, even for steady or quasi-steady flows, involving finite sized bodies, and numerical techniques generally have to be resorted to for finding solutions. However, quite effective modelling of flows involving complicated boundary geometries is possible using the three-dimensional Stokeslet and rotlet point singularities. Two problems are studied in detail. In the first example, exact solutions for the three-dimensional Stokeslet and rotlet placed axisymmetrically along the axis of a circular disc are found and combined with Brenner's first order interaction formulae to determine the effect of the presence of the disc on the force and torque acting on a particle whose dimensions are small compared with its distance from the disc. The results are compared with those of a full numerical integration of the Stokes equations for a sphere translating towards a disc. In the second example, Brenner's first order wall correction theory is applied to the motion of a particle in a circular cylinder using the exact solutions for a torus translating or rotating in isolation. The theoretical predictions for the drag on a torus settling symmetrically in a circular cylinder are compared with those determined experimentally.     

FORCE AND TORQUE ON A BODY IN AN UNBOUNDED STOKES FLOW EXPRESSED EXPLICITLY IN TERMS OF RESPECTIVE QUADRATURES OF THE PRESSURE AND VORTICITY FIELDS†

The hydrodynamic force and torque on a body in an external Stokes flow are explicitly expressed in terms of respective quadratures of the pressure and vorticity fields analytically continued into the body interior. These expressions are used to demonstrate that a body cannot experience a force in a uniform pressure field nor a torque in a uniform vorticity field.     

THE TRANSIENT MOTION OF A SPHERICAL FLUID DROPLET

A numerical method is developed for investigation of the unsteady motion of a spherical fluid droplet under the influence of gravity. This study extends previous work valid for creeping flow to moderate Reynolds number. The unsteady flow fields inside and outside of the fluid sphere are described by the two-dimensional, axisymmetric Navier-Stokes equations in the form of vorticity and stream function, along with the equation of motion of the droplet. The governing equations are approximated by a central difference and a second-order upwind difference, and are solved iteratively using the Gauss-Siedel and secant methods. Numerical results of the time-dependent vorticity, stream function and drop velocity are presented for a water droplet moving through air and for an air bubble rising in water. The steady state drop velocity and the drag coefficient at various Reynolds numbers are examined, and they are shown to agree very well with previous results.     

水性聚氨酯羟基组分的制备及其稳定性研究

采用聚丙二醇(N210)、2,4-甲苯二异氰酸酯(TDI)、二羟甲基丙酸(DMPA)、三羟甲基丙烷(TMP)为主要原料合成具有支化结构的水性聚氨酯羟基组分。探讨了工艺条件对羟基组分中—NCO含量的影响,采用红外光谱(IR)对产物进行了表征;用亚微观可视化反应装置测试了乳液粒径;用Stokes自由沉降速率方程求出了乳液的沉降速率。结果表明,适宜的工艺条件是反应温度为75℃,反应时间为4h;乳液的稳定性与DMPA和TMP的含量密切相关,随DMPA含量的增加乳液稳定性增加;当TMP的质量分数为0～4.7%时,其沉降速率为(0.041×10-13)～(2.124×10-13)m/s,乳液较稳定。     

MODELING VACUUM-CONTACT DRYING OF WOOD: THE WATER POTENTIAL APPROACH

A two-dimensional mathematical model for vacuum-contact drying of wood is presented. The moisture and heat equations are based on the water potential concept whereas the pressure equation is formulated considering unsteady state conservation equation of dry air. Most of the model parameters were determined during independent experiments. The set of equations is then solved in a coupled form using the finite element method. The validation of the model is performed using experimental results obtained during vacuum-contact drying of sugar maple sapwood. The experimental and calculated data are in good agreement. Nevertheless, some discrepancies are observed which can be attributed to the boundary conditions used and to the fact that heat transfer by convection was neglected.     

Numerical characterisation of folding flow microchannel mixers

Micromixers have been considered in numerous recent studies with the aim of mixing different liquid streams for the common circumstance of non-inertial flow, i.e., in the Stokes flow regime. Under such conditions, the diffusion of momentum is dominant but the diffusion of species remains weak because the Schmidt number of liquids is large. Most mixers that have potential for application in the Stokes regime make use of a folding flow pattern that approximates the baker's transformation. In the work presented here, the general scaling of mixers of this type is developed from the exact equation for species transport and computations are made for a specimen mixer geometry to test the effectiveness of the resulting scaling. The scaling relation developed is found to give an excellent representation of the actual mixing characteristics of the specimen mixer over the entire range of Péclet number of practical interest. Finite volume computations are employed to solve the governing equations up to around Pe=10³. At higher Péclet numbers, where finite volume numerical solution becomes inaccurate with affordable mesh sizes, the species equation is solved using a Monte Carlo method instead. Finally, the scaling relation is used to develop the design relations needed to determine the number of mixing elements, the pressure drop incurred and the Péclet number of operation to achieve a given mixture uniformity within a specified mixing time.     

MODELING VACUUM-CONTACT DRYING OF WOOD: THE WATER POTENTIAL APPROACH

ABSTRACT

A two-dimensional mathematical model for vacuum-contact drying of wood is presented. The moisture and heat equations are based on the water potential concept whereas the pressure equation is formulated considering unsteady state conservation equation of dry air. Most of the model parameters were determined during independent experiments. The set of equations is then solved in a coupled form using the finite element method. The validation of the model is performed using experimental results obtained during vacuum-contact drying of sugar maple sapwood. The experimental and calculated data are in good agreement. Nevertheless, some discrepancies are observed which can be attributed to the boundary conditions used and to the fact that heat transfer by convection was neglected.     

Unsteady conjugate mass transfer of a 2D deformable droplet in a modest extensional flow in across-slot

This work aims to investigate the unsteady conjugate interphase mass transfer between a stationary deformed drop and the modest extensional flow in a cross-intersected 2D channel. It is very difficult to accurately quantify the transient mass transfer rate of solute in such a geometry. Therefore, we established a mathematical model on the basic of the Stokes equation and solved it by the boundary element method, which could deal precisely with a two-phase flow system with a deformable interface; meanwhile, the convection-diffusion equation was solved by the finite difference method to calculate the unsteady conjugate interphase mass transfer. The simulation results showed that the mass transfer rate, analyzed and characterized in terms of mean concentration variation and Sherwood number Sh, was affected by capillary number Ca, Peclet number Pe, viscosity ratio λ , interior-to-exterior diffusivity ratio K, distribution coefficient m, and wall effect factor W.     

INTEGRAL EQUATIONS OF THE SECOND KIND FOR STOKES FLOW: DIRECT SOLUTION FOR PHYSICAL VARIABLES AND REMOVAL OF INHERENT ACCURACY LIMITATIONS

Boundary integral methods offer the most attractive combination of generality and computational efficiency for a wide class of particulate Stokes flow problems. Integral equations of the first kind have been numerically applied for more than a decade, whereas those of the second kind are numerically better behaved but involve abstract nonphysical density distributions and have not gained much popularity in applications. We show how the latter may be used for the direct solution of mobility problems, and how the surface tractions corresponding to rigid body motion of a particle may be easily found, thus removing the major disadvantages of the second kind formulations. For the numerical examples we also show how Fourier analysis may be applied to non-axisymmetric problems with axisymmetric boundaries to yield one-dimensional Fredholm integral equations of the second kind. As an application we solve the resistance problem with a numerically efficient quadrature collocation method that avoids the complications of element methods and difficulties with the integrations near the kernel singularities.     

Analysis for concentric-double inclusions dispersed in continuous media

The behaviors of concentric-double inclusions dispersed in continuous media are investigated theoretically to find some possibilities of improving toughness of composite materials by dispersing double-inc’usions instead of single-inclusions. The general solutions of the Stokes equation, expressed in terms of the spherical harmonics, are used for analyzing the problems that are related to the concentric-double inclusions. From the analysis, it is found that the pressure and stress fields inside and outside the inclusion can be modified by changing the modulus ratios and the thickness of shell layer. Especially, the positions of the minimum pressure points and the maximum stress points turn out to be controllable with some degree of freedom.     

Numerical study of a permeable capsule under Stokes flows by the immersed interface method

A two-dimensional model to simulate the mass transfer of a permeable, deformable, and adhesive capsule flowing in a binary solution of a vessel is proposed using the immersed interface method (IIM). The fluid flow is governed by the full Navier–Stokes equations and the solute distribution is governed by the advection–diffusion equation. Mass transport across the capsule membrane is computed using the Kedem–Katchalsky equations while the adhesion between the capsule and the walls is introduced via a potential function. The model is first validated for the simple shear flow away from the substrate walls and then for capsule adhesion and deformation next to a substrate wall. It is next used to study solute transfer between the capsule and the vessel walls with and without a flow field. In the absence of a flow field, the results show that the transient of the solute transfer between the capsule and the vessel walls depends on the membrane diffusive permeability. In the presence of a Stokes flow field, behavior of the solute transfer seems to be fairly similar to that found for the stationary capsule for the same physical parameters. Moreover, the results suggest that the total solute transfer between the capsule and the vessel walls is enhanced when the capsule moves near to one wall. The increased adhesion strength between the capsule and walls would further increase the total solute transfer to the vessel walls although quite marginal.     

多组分射流床中的射流深度

采用多路毕托管、电磁阀、微压传感器检测技术，考察了气固射流流化床中GeldartB、D类的单一物料及混合物的射流深度．指出物料的平均密度或粒径的增加都会造成射流深度的降低；得到的包括气团惯性准数在内的经验关联式能成功地预测多组分床层的射流深度．     

Mixing performance of a planar micromixer with circular obstructions in a curved microchannel

A numerical investigation of the mixing and fluid flow in a new design of passive micromixer employing several cylindrical obstructions within a curved microchannel is presented in this work. Mixing in the channels is analyzed using Navier–Stokes equations and the diffusion equation between two working fluids (water and ethanol) for Reynolds numbers from 0.1 to 60. The proposed micromixer shows far better mixing performance than a T-micromixer with circular obstructions and a simple curved micromixer. The effects of cross-sectional shape, height, and placement of the obstructions on mixing performance and the pressure drop of the proposed micromixer are evaluated.     

内校正2N牛顿-拉甫逊法与新松弛法结合用于多级多组分分离过程的计算

提出了将“内校正”观点引入2N牛顿-拉甫逊法的新算法,其主要特点是:①在雅可毕(Jacobian)矩阵的计算中同时考虑了温度、汽相流率和液相组成对目标函数的影响,在校正温度、汽相流率的同时,也隐含着对液相组成的校正。②ME方程系由非稳态的物料衡算式导出。③允许有多种设计变量指定方案。计算结果表明:该法兼具松弛法稳定性好和牛顿-拉甫逊法收敛速度快的优点,能同时适用于烃类物系和非理想物系的分离计算。     

超声辐照前后煤浆粒度分布变化研究

选用红会、大同及兖州煤制备水煤浆,考察了超声辐照前后煤浆粒度分布变化,并研究了超声辐照前后粒度分布变化对煤浆表观粘度、流变物质及静态稳定性影响,在超声辐照强度为８０Ｗ／ｃｍ＾２,频率为２０ｋＨｚ的条件下,超声辐照导致煤浆中１００μｍ以下的细级煤粒的比例明显增加,实验结果表明,超声辐照对煤浆具有很强的分散作用,在一定的粒度范围,细级煤粒的增加有利于煤浆流变性能的改善,同时细级煤粒的增加是煤浆静态 稳