Steady Free Surface Flow of a Fluid-Solid Mixture Down an Inclined Plane

The present work is an extension of the investigations performed by Massoudi and Anand (2001) Massoudi, M. 2001. On the flow of granular materials with variable material properties. Inl. J. Non-linear Mech., 36: 25–37.  [Google Scholar]. The free surface flow problem is studied here. Numerical solutions for steady free surface flow of a solid-fluid mixture down an inclined plane are presented. The problem is formulated using the mixture theory framework. The resulting set of three coupled nonlinear differential equations is nondimensionalized. A parametric study is conducted to understand the influence of the dimensionless numbers on the velocity and volume fraction. The maximum fluid velocity is found to decrease with increase in the ratio of the drag force to the viscous forces within the fluid phase (D₁). The fluid phase velocity was found to decrease with increase in the ratio of the drag force to viscous force within the solid component (D₂), and the corresponding solid phase velocity was found to increase.     

A constitutive model of multiphase mixtures and its application in shearing flows of saturated solid-fluid mixtures

A continuum theory of a multiphase mixture is formulated. In the basic balance laws we introduce an additional balance of equilibrated forces to describe the microstructural response according to Goodman &; Cowin [11] and Passman et al. [23] for each constituent. Based on the Müler-Liu form of the second law of thermodynamics a set of constitutive equations for a viscous solid-fluid mixture with microstructure is derived. These relatively general equations are then reduced to a system of ordinary differential equations describing a steady flow of the solid-fluid mixture between two horizontal plates. The resulting boundary value problem is solved numerically and results are presented for various values of parameters and boundary conditions. It is shown that simple shearing generally does not occur. Typically, for the solid phase, in the vicinity of a boundary, if the solid-volume fraction is low, a layer of high shear rate occurs, whose thickness is nearly between 5 and 15 grain diameters, while if the solid-volume fraction is high, an interlock phenomenon occurs. The fluid velocity depends largely on the drag force between the constituents. If the drag coefficient is sufficiently large, the fluid flow is nearly the same as that of the solid, while for a small drag coefficient, the fluid shearing flow largely decouples from that of the solid in the entire flow region. Apart from this, there is a tendency for solid particles to accumulate in regions of low shear rate.     

Stokes flow past a swarm of porous approximately spheroidal particles with Kuwabara boundary condition

The solution of the problem of symmetrical creeping flow of an incompressible viscous fluid past a swarm of porous approximately spheroidal particles with Kuwabara boundary condition is investigated. The Brinkman equation for the flow inside the porous region and the Stokes equation for the outside region in their stream function formulations are used. As boundary conditions, continuity of velocity and surface stresses across the porous surface and Kuwabara boundary condition on the cell surface are employed. Explicit expressions are investigated for both inside and outside flow fields to the first order in a small parameter characterizing the deformation. As a particular case, the flow past a swarm of porous oblate spheroidal particles is considered and the drag force experienced by each porous oblate spheroid in a cell is evaluated. The dependence of the drag coefficient on permeability for a porous oblate spheroid in an unbounded medium and for a solid oblate spheroid in a cell on the solid volume fraction is discussed numerically an and graphically for various values of the deformation parameter. The earlier known results are then also deduced from the present analysis.     

Hydrodynamic permeability of membranes built up by spherical particles covered by porous shells: effect of stress jump condition

This paper concerns the flow of an incompressible, viscous fluid past a porous spherical particle enclosing a solid core, using particle-in-cell method. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid are used. At the fluid–porous interface, the stress jump boundary condition for the tangential stresses along with continuity of normal stress and velocity components are employed. No-slip and impenetrability boundary conditions on the solid spherical core have been used. The hydrodynamic drag force experienced by a porous spherical particle enclosing a solid core and permeability of membrane built up by solid particles with a porous shell are evaluated. It is found that the hydrodynamic drag force and dimensionless hydrodynamic permeability depends not only on the porous shell thickness, particle volume fraction γ and viscosities of porous and fluid medium, but also on the stress jump coefficient. Four known boundary conditions on the hypothetical surface are considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta–Morse’s condition). Some previous results for the hydrodynamic drag force and dimensionless hydrodynamic permeability have been verified.     

Wave propagation in micropolar mixture of porous media

This paper is concerned with the possible propagation of waves in an infinite porous continuum consisting of a micropolar elastic solid and a micropolar viscous fluid. Micropolar mixture theory of porous media developed by Eringen [A.C. Eringen, Micropolar mixture theory of porous media, J. Appl. Phys. 94 (2003) 4184–4190] is employed. It is found that there exist four coupled longitudinal waves (two coupled longitudinal displacement waves and two coupled longitudinal microrotational waves) and six coupled transverse waves in a continuum of this micropolar mixture. All the waves are found to attenuate and dispersive in nature. A problem of reflection of coupled longitudinal waves from a free boundary surface of a half-space consisting the mixture of a micropolar elastic solid and Newtonian liquid, is investigated. The expressions of various amplitude ratios and surface responses are derived. Numerical computations are performed to find out the phase velocity and attenuation of the waves. The variation of amplitude ratios, energy ratios and surface responses are also computed for a specific model. All the numerical results are depicted graphically. Some limiting cases have also been discussed.     

Stokes flow of micropolar fluid past a viscous fluid spheroid with non-zero boundary condition for microrotation

The Stokes axisymmetric flow of an incompressible micropolar fluid past a viscous fluid spheroid whose shape deviates slightly from that of a sphere is studied analytically. The boundary conditions used are the vanishing of the normal velocities, the continuity of the tangential velocities, continuity of shear stresses and spin–vorticity relation at the surface of the spheroid. The hydrodynamic drag force acting on the spheroid is calculated. An exact solution of the problem is obtained to the first order in the small parameter characterizing the deformation. It is observed that due to increased spin parameter value, the drag coefficient decreases. Well-known results are deduced and comparisons are made with classical viscous fluid and micropolar fluids.     

雷诺数对沟槽减阻特性影响的数值分析

采用雷诺平均N—S方程和RNG κ-ε湍流模型计算V型沟槽面的湍流边界层流动和黏性阻力，通过改变来流速度大小和沟槽面布置位置，研究了雷诺数对沟槽减阻特性的影响规律。计算结果表明，来流速度对沟槽减阻率的影响很大，对于一种尺度的V型沟槽，存在着一个具有较好减阻效果的来流速度范围，最大减阻率可迭8．6％；沟槽面在沿来流方向上的布置位置对其减阻效果的影响则非常小。     

Flow of a viscous fluid at small Reynolds number past a porous sphere with a solid core

Summary Uniform flow of an incompressible viscous fluid at small Reynolds number past a porous sphere of radius a with a solid concentric spherical core of radius b has been discussed. The region of the porous shell is called zone I which is fully saturated with the viscous fluid, and the flow in this zone is governed by the Brinkman equation. The space outside the shell where clear fluid flows is divided into two zones (II and III). In these zones the flow is discussed following Proudman and Pearson's method of expanding Stokes' stream function in powers of Reynolds number and then matching Stokes' solution with Oseen's solution. The stream function of zone II is matched with that of zone I at the surface of the shell by the condition suggested by Ochoa – Tapia and Whitaker. It is found that the drag on the spherical shell increases with the increase of the λ (=b/a) and also with the increase of the Darcy number. The graph of dimensionless drag against λ for various values of Reynolds number shows that the drag increases with the increase of the Reynolds number for all values of λ.     

Proton-exchanged Z-cut LiNbO3 waveguides for surfaceacoustic waves

Proton-exchanged (PE) waveguides in Z-cut LiNbO₃ have been fabricated using benzoic acid. Secondary-ion mass spectrometry (SIMS) measurements show that the distribution of hydrogen in the PE Z-cut LiNbO₃ samples exhibits a step-like profile with the diffusion constant D₀ and the activation energy Q of about 2.82×10⁸ μm²/h and 87.76 kJ/mol, respectively. On the other hand, the important parameters for the design of surface acoustic wave (SAW) devices are measured and discussed. The results show that the phase velocity and electromechanical coupling coefficient decrease with the increase of kd, where k is the wavenumber and d is the waveguide depth. The variation of insertion loss becomes saturated at about kd=0.068 with a maximum increase of about 4~5 dB. The temperature coefficient of delay calculated from the frequency change of the output of SAW delay line shows an evident increase in the PE layer. Moreover, the effects of postannealing can result in a restoration of the decreased velocity and an improvement of the insertion loss     

Stokes flow with slip caused by the axisymmetric motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid

The first part of this paper investigates the motion of a solid spherical particle in an incompressible axisymmetric micropolar Stokes flow. A linear slip, Basset-type, boundary condition has been used. Expressions for the drag force and terminal velocity has been obtained in terms of the parameter characterizing the slip friction. In the second part, we consider the flow of an incompressible axisymmetrical steady semi-infinite micropolar fluid arising from the motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid. Two cases are considered for the motion of the sphere: perpendicular translation to the free surface and rotation about a diameter which is also perpendicular to the free surface. The speed of the translational motion and the angular speed for the rotational motion of the sphere are assumed to be small so that the nonlinear terms in the equations of motion can be neglected under the usual Stokesian approximation. Also a linear slip, Basset-type, has been used. The analytical expressions for velocity and microrotation components are determined in terms of modified Bessel functions of second kind and Legendre polynomials. The drag for the translation case and the couple for the rotational motion on the submerged half sphere are calculated and expressed in terms of nondimensional coefficients whose variation is studied numerically. The variations of the drag and couple coefficients with respect to the micropolarity parameter and slip parameter are tabulated and displayed graphically.     

Experimental investigation of bubble and particle motion behaviors in a gas-solid fluidized bed with side wall liquid spray

Bubble and particle motion behaviors are investigated experimentally in a gas solid fluidized bed with liquid spray on the side wall. The particles used in the experiment are classified as Geldart B particles. The results reveal that when the fluid drag force is less than the liquid bridge force between particles, liquid distribute all over the bed. Bubble size increases as the increase of inter-particle force, then decreases owing to the increase of particle weight with increasing liquid flow rate. When the fluid drag force is greater than the liquid bridge force, liquid mainly distribute in the upper part of the bed. And it is difficult for the wet particles to form agglomerates. Bubble size decreases with increasing liquid flow rate due to the increasing of minimum fluidization velocity. Besides, the acoustic emission (AE) measurements illustrate that the liquid adhesion and evaporation on particles could enhance the particles motion intensity. Consequently, the bubble and particle behaviors change due to the variation in fluidized gas velocity and liquid flow rate should be seriously considered when attempting to successfully design and operate the side wall liquid spray gas solid fluidized bed.     

Energy Dissipation and Apparent Viscosity of Semi-solid Metal during Rheological Processes Part Ⅰ： Energy Dissipation

The energy dissipation caused by the viscous force has great effects on the flow property of semi-solid metal during rheological processes such as slurry preparing, delivering and cavity filling. Experimental results in this paper indicate that the viscous friction between semi-solid metal and pipe wall, the collisions among the solid particles, and the liquid flow around particles are the three main types of energy dissipation. On the basis of the hydromechanics, the energy dissipation calculation model is built. It is demonstrated that the micro-structural parameters such as effective solid fraction, particle size and shape, and flow parameters such as the mean velocity, the fluctuant velocity of particles and the relative velocity between the fluid and solid phase, affect the energy dissipation of semi-solid metal.     

Energy Dissipation and Apparent Viscosity of Semi-solid Metal during Rheological Processes Part Ⅰ: Energy Dissipation

The energy dissipation caused by the viscous force has great effects on the flow property of semi-solid metal during rheological processes such as slurry preparing, delivering and cavity filling. Experimental results in this paper indicate that the viscous friction between semi-solid metal and pipe wall, the collisions among the solid particles, and the liquid flow around particles are the three main types of energy dissipation. On the basis of the hydromechanics, the energy dissipation calculation model is built. It is demonstrated that the micro-structural parameters such as effective solid fraction, particle size and shape, and flow parameters such as the mean velocity, the fluctuant velocity of particles and the relative velocity between the fluid and solid phase, affect the energy dissipation of semi-solid metal.     

Immersed-boundary/soft-sphere method for particle–particle-fluid interaction in a viscous flow: An OpenFOAM solver

We developed a stable OpenFOAM solver for Immersed Boundary Method based on direct forcing and regularized delta function. The soft-sphere model and a lubrication model were implemented to consider particle–particle collision in a viscous flow. We proposed a fluid–structure interaction (FSI) coupling method to accurately calculate the fluid forcing term and particle velocity. Our solver was validated for fixed and moving bodies, including rotation. The accuracy of various FSI schemes was evaluated in predicting the solid and fluid flow behavior in a viscous flow. It was demonstrated that neglecting or simplifying the fluid momentum change affects the accuracy of the solid velocity and fluid flow dynamic; for higher solid-to-fluid density ratios, a larger deviation was predicted. Furthermore, the FSI schemes highly influenced the behavior of the formed vortices.The solver was validated to predict the effective restitution coefficient of particles in a viscous flow as a function of the Stokes number. We also thoroughly analyzed the dynamic flow behavior of colliding particles through the pressure and velocity field and fluid force. This analysis helped us accurately determine the rebound velocity of particles in case of high Stokes numbers when the effect of viscous force is significant.     

MHD flow over a moving plate in a rotating fluid with magnetic field, Hall currents and free stream velocity

The non-similar boundary layer flow of a viscous incompressible electrically conducting fluid over a moving surface in a rotating fluid, in the presence of a magnetic field, Hall currents and the free stream velocity has been studied. The parabolic partial differential equations governing the flow are solved numerically using an implicit finite-difference scheme. The Coriolis force induces overshoot in the velocity profile of the primary flow and the magnetic field reduces/removes the velocity overshoot. The local skin friction coefficient for the primary flow increases with the magnetic field, but the skin friction coefficient for the secondary flow reduces it. Also the local skin friction coefficients for the primary and secondary flows are reduced due to the Hall currents. The effects of the magnetic field, Hall currents and the wall velocity, on the skin friction coefficients for the primary and secondary flows increase with the Coriolis force. The wall velocity strongly affects the flow field. When the wall velocity is equal to the free stream velocity, the skin friction coefficients for the primary and secondary flows vanish, but this does not imply separation.     

Interfacial capillary–gravity waves due to a fundamental singularity in a system of two semi-infinite fluids

The interfacial capillary–gravity waves due to a transient fundamental singularity immersed in a system of two semi-infinite immiscible fluids of different densities are investigated analytically for two- and three- dimensional cases. The two-fluid system, which consists of an inviscid fluid overlying a viscous fluid, is assumed to be incompressible and initially quiescent. The two fluids are each homogeneous, and separated by a sharp and stable interface. The Laplace equation is taken as the governing equation for the inviscid flow, while the linearized unsteady Navier–Stokes equations are used for the viscous flow. With surface tension taken into consideration, the kinematic and dynamic conditions on the interface are linearized for small-amplitude waves. The singularity is modeled as a simple mass source when immersed in the inviscid fluid above the interface, or as a vertical point force when immersed in the viscous fluid beneath the interface. Based on the integral solutions for the interfacial waves, the asymptotic wave profiles are derived for large times with a fixed distance-to-time ratio by means of the generalized method of stationary phase. It is found that there exists a minimum group velocity, and the wave system observed will depend on the moving speed of the observer. Two schemes of expansion of the phase function are proposed for the two cases when the moving speed of an observer is larger than, or close to the minimum group velocity. Explicit analytical solutions are presented for the long gravity-dominant and the short capillary-dominant wave systems, incorporating the effects of density ratio, surface tension, viscosity and immersion depth of the singularity.     

RE-ENTRAINMENT OF DUSTS FROM THE DUST LAYER ON THE THE DEPOSITED DUSTS IN THE CYCLONE DUST COLLECTORS

There were many papers concerning the experimental results of the collection efficiency, but up to this time there are a few papers concerning the experimental results of the re-entrainment or dispersion of the dust particles from the dust layer by the turbulent rotational air flow in the dust bunker for the cyclone dust collector. Then in this paper, the author described the experimental results of the re-entrainment of the test dust ( talc X_R50 = 8.O µm ) for the four kinds of the throat diameter D₃ = 50, 80, 100 and 150 mm. Especially it is very importance to take into consideration of flow rate Qb into the dust bunker which is a function of D₃ and cyclone diameter D₁ and the maximum tangential velocity Vet in the dust bunker which depends on D₁,D₃ and Qb.     

The flow of liquid past a plate with boiling and injection of gas

A classical model boundary layer problem is considered for the flow of liquid past a plate in view of injection of a vapor-gas mixture from its surface. The obtained self-similar solutions enable one to estimate the typical values of thickness of the vapor-gas layer, the value of heat-transfer coefficient as a function of temperature of liquid, intensity of injection and composition of mixture being injected, and the velocity of flow past the plate. In addition, the problem is considered of reducing the hydrodynamic drag owing to vapor and vapor-gas “lubrication” because of boiling of liquid and injection of vapor-gas mixture from the plate surface. The possibility is analyzed of the emergence of vapor film due to viscous friction forces in the case where the liquid is in the vicinity of the boiling point.     

Experimental and numerical analysis on the hydrodynamic behaviors of permeable microbial granules

The microbial granules are found to be porous and permeable, which leads to a different drag force coefficient from the rigid sphere granules. Discrete element method (DEM) was employed to establish geometric models of porous microbial granules for the first time in this study. And computational fluid dynamics (CFD) was applied to simulate the effects of porosity and Reynolds number on the fluid flow, shear stress, pressure and drag force based on the established geometric models. The results showed that both the Reynolds number and the porosity of microbial granules significantly affect the fluid velocity distribution inside the granules. The porosity shows less effect on maximum shear stress than Reynolds number. It s well known that drag force consists form drag and skin drag. The ratio of form drag to drag force increased, while the skin drag force ratio decreased with the increasing Reynolds number. The porosity will enhance the skin drag and weaken the form drag at the same Reynolds number. A drag force coefficient equation was established based on the simulated results in order for engineering application. The correctness of the equation was confirmed by comparing with experimental results. The results from this study may provide valuable information for operation and designing of a granule-based bioreactor in wastewater treatment.     

Numerical solution of gas–solid flow in fluidised bed at sub-atmospheric pressures

Fluidised beds are characterised by excellent thermal and chemical uniformity and have a wide application range including heat and surface treatment, ore roasting and catalyst production. However, compared to other gas-based systems, to fluidise a particulate mass, a significant quantity of gas is required. To conserve gas there is potential to operate the fluid bed under low-pressure conditions. It is also observed that heat transfer remains constant with reduction in pressure. The present work has numerically studied the nature of hydrodynamics in fluidised bed at sub-atmospheric conditions and a new drag law is proposed to account for the increased mean free path of the fluid. A wide range of sub-atmospheric pressures were considered such that slip flow regime, which is characterised with Kn ～ 1, is applicable. An open source code (MFIX) is used to numerically solve the multiphase problem of a jet in the fluidised bed column with an immersed surface at vacuum pressure conditions. Bubbling fluidisation in shallow and deep beds are also solved. The new drag model takes into consideration the effect of slip flow to model drag force on the particles and the results of velocity distributions in the column and around the submerged surface is presented. The results of velocity distributions from the slip flow model are compared with the existing Gidaspow’s model. Significant differences were observed in the simulation results of velocity distributions and flow structure in the fluidised bed under vacuum conditions.