THE DISSOLUTION OR GROWTH OF A SPHERE

The problem of the dissolution or growth of an isolated, stationary, sphere in a large fluid body is analyzed. The motion of the boundary as well as the resulting motion in the liquid are properly taken into account. The governing equations are solved using a recently developed technique (Subramanian and Weinberg, 1981) which employs an asymptotic expansion in time. Results for the radius of the sphere as a function of time are calculated. The range of utility of the present solution is established by comparison with a numerical solution of the governing equations obtained by the method of finite differences.     

A NUMERICAL SOLUTION OF THE FLOW AROUND A SPHERE IN A CIRCULAR CYLINDER

A numerical solution of the Navier-Stokes equations is presented for Poiseuille flow around an axially placed, fixed sphere in a circular cylinder. Streamlines and isovorticity lines are calculated from the governing equations for the strearnfunction and the vorticity. Isobars are calculated from a Poisson equation, derived from the Navier-Stokes equations. The pressure and vorticity distribution on the surface of the sphere, the additional pressure drop and the drag coefficients are presented. Solutions are obtained for Reynolds numbers up to 150 (based on cylinder diameter and mean velocity). The wall effects are examined by comparison with results of previous investigations for an unbounded flow around a sphere.     

A NUMERICAL SOLUTION OF THE FLOW AROUND A SPHERE IN A CIRCULAR CYLINDER

A numerical solution of the Navier-Stokes equations is presented for Poiseuille flow around an axially placed, fixed sphere in a circular cylinder. Streamlines and isovorticity lines are calculated from the governing equations for the strearnfunction and the vorticity. Isobars are calculated from a Poisson equation, derived from the Navier-Stokes equations. The pressure and vorticity distribution on the surface of the sphere, the additional pressure drop and the drag coefficients are presented. Solutions are obtained for Reynolds numbers up to 150 (based on cylinder diameter and mean velocity). The wall effects are examined by comparison with results of previous investigations for an unbounded flow around a sphere.     

UNSTEADY MHD HEAT AND MASS TRANSFER BY MIXED CONVECTION FLOW IN THE FORWARD STAGNATION REGION OF A ROTATING SPHERE AT DIFFERENT WALL CONDITIONS

This study is focused on the problem of MHD heat and mass transfer by mixed convection flow in the forward stagnation region of a rotating sphere in the presence of heat generation and chemical reaction effects. The surface of the sphere is maintained at constant fluid temperature and species concentration. The governing equations of the problem are converted into ordinary differential equations by using suitable similarity transformations. Two cases are considered, namely, constant wall temperature and mass (CWTM) and constant heat and mass fluxes (CHMF). The obtained self-similar equations for both cases are solved numerically using an efficient iterative implicit finite-difference method. The numerical results are compared with previously published results on special cases of the problem and found to be in excellent agreement. The obtained results are displayed graphically to illustrate the influence of the different physical parameters on the velocity components in x- and y-directions, temperature, and concentration profiles as well as the local surface shear stresses and local heat and mass transfer coefficients.     

数值模拟中低Reynolds数下简单剪切流流过单液滴的稳态流场(英文)

This work presents a numerical investigation on steady internal, external and surface flows of a liquid sphere im-mersed in a simple shear flow at low and intermediate Reynolds numbers. The control volume formulation is adopted to solve the governing equations of two-phase flow in a 3-D spherical coordinate system. Numerical re-sults show that the streamlines for Re=0 are closed Jeffery orbits on the surface of the liquid sphere, and also closed curves outside and inside the liquid sphere. However, the streamlines have intricate and non-closed struc-tures for Re≠0. The flow structure is dependent on the values of Reynolds number and interior-to-exterior vis-cosity ratio.     

Numerical simulation of steady flow past a liquid sphere immersed in simple shear flow at low and moderate Re

This work presents a numerical investigation on steady internal, external and surface flows of a liquid sphere im-mersed in a simple shear flow at low and intermediate Reynolds numbers. The control vol...     

A numerical study of the accelerating motion of a dense rigid sphere in non-newtonian power law fluids

The equations of motion of an accelerating sphere falling through non-Newtonian fluids with power law index n in the range 0.2 ≤ n ≤ 1.8 were integrated numerically using the assumption that the drag on the sphere was a function of both power law index and terminal Reynolds number, Re_t For 10^?2 ≤ Re_t ≤ 10³ both dimensionless time and distance travelled by the sphere under transient conditions showed a much stronger dependence on the flow behaviour index, n, for shear-thinning than for shear-thickening fluids. The form of this dependence is investigated here. Furthermore, results in four typical shear-thinning fluids suggested a strong correlation between the distance and time travelled by the sphere under transient conditions and the value of the fluid consistency index. The analysis reported herein is, however, restricted to dense spheres falling in less dense fluids, when additional effects arising from the Basset forces can be neelected.     

SYMBOLIC OPERATOR SOLUTIONS OF LAPLACE'S AND STOKES' EQUATIONS PART II STOKES FLOW PAST A RIGID SPHERE

Symbolic operator methods are employed to derive a general solution of the Stokes' and continuity equations of low Reynolds number hydrodynamics for arbitrary flow past an immobile sphere. The novelty of the final working formula lies in the fact that the velocity and pressure fields are explicitly and directly expressed in terms of the prescribed velocity field and its derivatives at infinity. Hence, apart from these trivial differentiations, no further operations of any kind are required to effect a solution

The computational scheme is illustrated for the case of a rigid sphere embedded in a general undisturbed quadratic velocity field, of which Poiseuille flow in a circular tube is the pre-eminent example. Results obtained by this method accord with those obtained by other, more traditional, schemes requiring preliminary expansion of (certain functions of) the prescribed data into surface spherical harmonics.     

SYMBOLIC OPERATOR SOLUTIONS OF LAPLACE'S AND STOKES’ EQUATIONS PART II STOKES FLOW PAST A RIGID SPHERE

Symbolic operator methods are employed to derive a general solution of the Stokes' and continuity equations of low Reynolds number hydrodynamics for arbitrary flow past an immobile sphere. The novelty of the final working formula lies in the fact that the velocity and pressure fields are explicitly and directly expressed in terms of the prescribed velocity field and its derivatives at infinity. Hence, apart from these trivial differentiations, no further operations of any kind are required to effect a solution

The computational scheme is illustrated for the case of a rigid sphere embedded in a general undisturbed quadratic velocity field, of which Poiseuille flow in a circular tube is the pre-eminent example. Results obtained by this method accord with those obtained by other, more traditional, schemes requiring preliminary expansion of (certain functions of) the prescribed data into surface spherical harmonics.     

THE MEASUREMENT OF THE FLOW AROUND A SPHERE SETTLING IN A RECTANGULAR BOX USING 3-DIMENSIONAL PARTICLE IMAGE VELOCIMETRY

A novel three-dimensional particle image velocimetry technique is used to measure the planar three-dimensional flow field about the centreline of a sphere sedimenting in a rectangular shaped box. Measurements are made in the center of the container and also one diameter from a plane wall. Results are presented for a sphere falling in both a constant viscosity elastic (Boger) fluid and a shear-thinning elastic liquid. In the center of the box, the flow field is essentially two-dimensional as expected. Near the wall, there is substantial out-of-plane motion in the shear-thinning solution due to the presence of the wall. Surprisingly, there is little out-of-plane motion for a sphere sedimenting near the wall in the Boger fluid. There are significant qualitative differences in the flow field for the sphere sedimenting in the shear-thinning and constant viscosity elastic liquids. The results are compared with previously published work for a sphere settling in a non-Newtonian fluid and also with results obtained in an identical geometry for a Newtonian fluid. Reasons for the differences in the velocity maps are discussed. The drag coefficient for each geometry and fluid is calculated.     

THE MEASUREMENT OF THE FLOW AROUND A SPHERE SETTLING IN A RECTANGULAR BOX USING 3-DIMENSIONAL PARTICLE IMAGE VELOCIMETRY

A novel three-dimensional particle image velocimetry technique is used to measure the planar three-dimensional flow field about the centreline of a sphere sedimenting in a rectangular shaped box. Measurements are made in the center of the container and also one diameter from a plane wall. Results are presented for a sphere falling in both a constant viscosity elastic (Boger) fluid and a shear-thinning elastic liquid. In the center of the box, the flow field is essentially two-dimensional as expected. Near the wall, there is substantial out-of-plane motion in the shear-thinning solution due to the presence of the wall. Surprisingly, there is little out-of-plane motion for a sphere sedimenting near the wall in the Boger fluid. There are significant qualitative differences in the flow field for the sphere sedimenting in the shear-thinning and constant viscosity elastic liquids. The results are compared with previously published work for a sphere settling in a non-Newtonian fluid and also with results obtained in an identical geometry for a Newtonian fluid. Reasons for the differences in the velocity maps are discussed. The drag coefficient for each geometry and fluid is calculated.     

Slip in Flow through Assemblages of Spherical Particles at Low to Moderate Reynolds Numbers

Effects of slip velocity and volume fraction of slip spheres on the momentum transfer characteristics of assemblages of slip spheres are numerically investigated. The fluid slip along the surface of the sphere is considered by Navier's linear slip model. The dimensionless governing continuity and momentum equations are solved using a semi‐implicit marker and cell method implemented on a staggered grid arrangement in spherical coordinates. The convection and viscous terms of momentum equations are discretized by means of the QUICK scheme and a second‐order central differencing scheme, respectively. The present numerical solver is benchmarked via grid independence and comparisons with the existing literature values. Results were obtained over a wide range of pertinent dimensionless numbers such as the Reynolds number, volume fraction of the dispersed phase, and dimensionless slip parameter.     

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.     

Étude de l'évaporation par convection forcée au‐dessus de corps tournants à symétrie de révolution dans un fluide au repos

This paper presents a theoretical and experimental study of the vaporization over rotating bodies of revolution, which are subjected to a constant wall heat flux. The laminar boundary‐layer approach and the first stage of drying conditions are considered when writing the governing equations. Numerical results are presented for the disc and sphere geometries. The experimental procedure is based upon infrared thermography and pyrometry techniques which help to obtain some quantitative comparisons with the theoretical results for the case of the rotating disc.     

Thermophoresis of Liquid Airborne Particles in Gravity at Continuum Regime

An analytical study is presented for the thermophoretic motion of an aerosol droplet in a prescribed temperature gradient under the influence of gravity. The Peclet and Reynolds numbers are assumed small, so that the temperature distributions are governed by the Laplace equation and the flow fields are governed by the Stokes equation. The temperature discontinuity, thermal creep, and hydrodynamic slip features that occurred at the droplet surface are considered. The slow motion of a liquid aerosol sphere subject to the combination of thermophoresis and sedimentation is obtained by superposition of the individual solutions for pure thermophoresis and pure body-force-driven motion, since both the governing equations and boundary conditions in this problem are linear. The stream functions of the internal and external flows, which are displayed in both the laboratory frame and a reference frame moving with the droplet, as well as the migrating velocity of a droplet, are formulated generally. The flow structures manifest more remarkable topologies than do those of common intuition developed from sedimentation. Our results can be simplified to the corresponding motion for solid aerosol particles.     

废钢在气体搅拌容器中运动融化的水模型实验

由于经济原因,在炼钢过程如转炉和精炼钢包中,废钢的用量有所增加。废钢在上述反应器中的熔化机理仍然不明确。水模型研究可以直观地模拟废钢的运动和熔化过程。通过水模型实验,以冰球和饱和KCl溶液制成的盐球分别模拟轻、重废钢,研究其在类转炉及钢包中的运动过程和融化规律。结果表明,在本实验中,冷冻时长超过18 h后,冰样的融化时间受影响较小。冰球和盐球融化过程呈球形或椭球形,且其直径随时间呈线性减小。随着熔池温度的提高,冰样融化加快。不同的液面高度及吹气流量会形成不同特征的流场,使冰球与盐球呈现不同的运动规律。冰球的融化时间随液面高度增加而降低,盐球呈先大幅度降低再小幅度增加的趋势。在本研究的吹气流量下,液面高度和直径比为0.94时,盐球的融化时间同比最低。在液面高度较低时(高度与直径比为0.42～0.73),在羽流区上方加入盐球会显著降低融化时间;液面高度较高时(高度与直径比为0.83～1.04),加入位置对融化时间无显著影响。     

Creeping motions of a composite sphere in a concentric spherical cavity

An analytical study is presented for the quasisteady translation and steady rotation of a spherically symmetric composite particle composed of a solid core and a surrounding porous shell located at the center of a spherical cavity filled with an incompressible Newtonian fluid. In the fluid-permeable porous shell, idealized hydrodynamic frictional segments are assumed to distribute uniformly. In the limit of small Reynolds number, the Stokes and Brinkman equations are solved for the flow field of the system, and the hydrodynamic drag force and torque exerted by the fluid on the particle which is proportional to the translational and angular velocities, respectively, are obtained in closed forms. For a given geometry, the normalized wall-corrected translational and rotational mobilities of the particle decrease monotonically with a decrease in the permeability of its porous shell. The boundary effects of the cavity wall on the creeping motions of a composite sphere can be quite significant in appropriate situations. In the limiting cases, the analytical solutions describing the drag force and torque or mobilities for a composite sphere in the cavity reduce to those for a solid sphere and for a porous sphere.     

Numerical simulation of motion of rigid spherical particles in a rotating tumbler with an inner wavelike surface

The present work is a numerical simulation of motion of rigid spherical particles within a 2-D tumbler with an inner wavelike surface. The rotation of the tumbler is simulated as a traveling sine wave around a circle. The discrete element method (DEM, a hard sphere approach) is used. The particle-wall interactions are taken into account in a changed numerical approach of hard sphere model. The effects of two basic factors of the rotating velocity (phase velocity) and the wave numbers are separately investigated. A simple but useful method for cluster identification is provided and used. The energy-based analysis of particle clusters and the motion pattern study indicate the existence of a pulsed variation in the kinetic energy of the clusters at low wave numbers and a cyclic bulk motion of the clusters at high wave numbers. The necessary conditions for the pulsed variation of motion of particle clusters at low wave number are analyzed and a mode for industrial application, e.g. coal grinding process in power plant, is demonstrated.     

Eulerian-Lagrangian simulation of aeolian sand transport

A numerical investigation of aeolian sand transport is performed with an Eulerian-Lagrangian model. In this model, the gas phase is described by the volume-averaged Navier-Stokes equations of two-phase flow. The particle motion is obtained by solving Newton's second law of motion taking into account the inter-particle collisions, where a soft sphere model is used to describe inter-particle collisions. The dynamic process of aeolian sand transport is simulated. The simulation results show that the variation of mean horizontal velocity of the particles with height can be expressed by a logarithmical function or a power function at h > 0.02 m, and the power function can be described below 0.02 m. The sand mass flux decreases exponentially with height for h > 0.02 m, but there is a deviation from the exponential decay due to the creep grains in the near-bed region. It is also shown that the inter-particle collisions play an important role in sand saltation. Therefore the present numerical model is capable of being applied to the study of windblown sand movement.