Free-surface wave interaction with a thick flexible dock or very large floating platform

In this paper the recently developed semi-analytic method to solve the free-surface wave interaction with a thin elastic plate is extended to the case of a plate of finite thickness. The method used is based on the reformulation of the differential–integral equation for this problem. The thickness of the plate is chosen such that the elastic behavior of the plate can be described by means of thin-plate theory, while the water pressure at the plate is applied at finite depth. The water depth is finite.     

Towards a theory of granular plasticity

A theory of granular plasticity based on the time-averaged rigid-plastic flow equations is presented. Slow granular flows in hoppers are often modeled as rigid-plastic flows with frictional yield conditions. However, such constitutive relations lead to systems of partial differential equations that are ill-posed: they possess instabilities in the short-wavelength limit. In addition, features of these flows clearly depend on microstructure in a way not modeled by such continuum models. Here an attempt is made to address both short-comings by splitting variables into ‘fluctuating’ plus ‘average’ parts and time-averaging the rigid-plastic flow equations to produce effective equations which depend on the ‘average’ variables and variances of the ‘fluctuating’ variables. Microstructural physics can be introduced by appealing to the kinetic theory of inelastic hard-spheres to develop a constitutive relation for the new ‘fluctuating’ variables. The equations can then be closed by a suitable consitutive equation, requiring that this system of equations be stable in the short-wavelength limit. In this way a granular length-scale is introduced to the rigid-plastic flow equations.     

Towards a theory of granular plasticity

A theory of granular plasticity based on the time-averaged rigid-plastic flow equations is presented. Slow granular flows in hoppers are often modeled as rigid-plastic flows with frictional yield conditions. However, such constitutive relations lead to systems of partial differential equations that are ill-posed: they possess instabilities in the short-wavelength limit. In addition, features of these flows clearly depend on microstructure in a way not modeled by such continuum models. Here an attempt is made to address both short-comings by splitting variables into ‘fluctuating’ plus ‘average’ parts and time-averaging the rigid-plastic flow equations to produce effective equations which depend on the ‘average’ variables and variances of the ‘fluctuating’ variables. Microstructural physics can be introduced by appealing to the kinetic theory of inelastic hard-spheres to develop a constitutive relation for the new ‘fluctuating’ variables. The equations can then be closed by a suitable consitutive equation, requiring that this system of equations be stable in the short-wavelength limit. In this way a granular length-scale is introduced to the rigid-plastic flow equations.     

Density-driven sinking dynamics of a granular ring in sheared granular flows

This study reports experimental findings on the sinking dynamics of a heavy granular ring caused by the density-driven segregation effect in sheared granular flows. Specifically, this study systematically investigates the influences of the density ratio, shear rate, and solid fraction of the granular material on the sinking behavior of a heavy granular ring. The parameters of the dimensionless sinking depth and sinking rate, respectively, describe the change in the granular ring position and quantify the particle sinking speed. Experimental results show that both the dimensionless sinking depth and the sinking rate increase as the bottom wall velocity (shear rate) and solid fraction increase. The dimensionless sinking depth and the sinking rate also exhibit a linear relation. The dimensionless sinking depth does not increase monotonically as the density ratio increases. The sinking rate increases linearly with the final steady-state sinking depth for the same heavy granular ring structure, regardless of the wall velocity (shear rate), solid fraction, and density ratio.     

Depth-averaged relations for granular-liquid uniform flows over mobile bed in a wide range of slope values

The behavior of liquid-granular flows, driven by gravity, is experimentally analyzed. Two types of free-surface uniform flow can take place, having different boundary conditions at the bottom. The first one runs over a fixed surface behaving as a solid (non-deformable) impermeable wall; the second one runs over a mobile-bed at rest, formed by the same loose grains and liquid of the flowing mixture. In the paper we will mark the differences between the two, but focus on the latter one. The experiments span over, and characterize, the possible flow regimes. In mobile-bed uniform flows it has been found that the Froude number reduces as the slope increases. Accordingly, there is an increment of the solid-concentration. These results are meaning that as slope increases a progressive dominance and thickening of frictional layers over collisional ones is taking place through the flow depth. Same behaviours have been observed by changing the type of grains in the flowing mixture. These findings contrast with the case of flows over a solid wall, where different trends are observed. Application of force balances by means of Coulomb law provides interesting confirmation of what observed and allows to take into account the surface-tension effects, which come into play when the particles on top are going to desaturate. Experimental data have also been employed to assess the applicability of kinetic theories to wet granular flows. Energy and momentum balances, under the hypothesis of no contribution in the liquid phase (except for the added mass concept) to shear stress and to the energy processes, are applied throughout the flow depth of the solid phase. Although depth-averaged quantities come out to have a trend similar to the experimental one, deficiencies in the theoretical approach, mainly due to its inability to represent frictional contacts, are clearly detected. Same conclusions may be drawn by applying the quite simple Bagnold theory. Altogether, a more appropriate theory able to deal with both collisional and frictional mechanisms, including the transition between, is demanded.     

Incompressible granular flow from wedge-shaped hoppers

The incompressible plastic flow equations for a Drucker-Prager yield law and a J₂ flow rule are shown not to allow a steady single radial velocity component, for flows from a wedge-shaped hopper. The corresponding equations for two components of velocity are considered, using a series expansion of Kaza and Jackson, which connects asymptotically to Jenike’s radial solution. This asymptotic solution gives a poor model of mass flows about the orifice, and an improvement is obtained by considering the pressure variation along the axis of the wedge, but using the angular variations determined by the power-series method. Numerical difficulties occurred for certain parameter values, when solving the two-point boundary-value problem resulting from the asymptotic series method. The region of this parametric sensitivity is associated with an internal maximum in the pressure field, whose appearance tends to offer a conservative estimate for the mass-funnel flow transition.     

Incompressible granular flow from wedge-shaped hoppers

The incompressible plastic flow equations for a Drucker-Prager yield law and aJ ₂ flow rule are shown not to allow a steady single radial velocity component, for flows from a wedge-shaped hopper. The corresponding equations for two components of velocity are considered, using a series expansion of Kaza and Jackson, which connects asymptotically to Jenike’s radial solution. This asymptotic solution gives a poor model of mass flows about the orifice, and an improvement is obtained by considering the pressure variation along the axis of the wedge, but using the angular variations determined by the power-series method. Numerical difficulties occurred for certain parameter values, when solving the two-point boundary-value problem resulting from the asymptotic series method. The region of this parametric sensitivity is associated with an internal maximum in the pressure field, whose appearance tends to offer a conservative estimate for the mass-funnel flow transition.     

Non-dilatant double-shearing theory applied to granular funnel-flow in hoppers

The storage and efficient withdrawal of material from silos and hoppers is basic to numerous industrial processes. Practising engineers classify two fundamental flows, namely mass-flow and funnel-flow. The former describes the situation when the bulk solid is in motion at every point in the silo or hopper, whenever material is drawn from the outlet. The latter describes the situation when a stable channel forms, called a rat-hole, and the flow is such that only material above the rat-hole is in motion. Funnel-flow occurs whenever the outlet walls are too rough and not sufficiently steeply sloped. Funnel-flow is generally erratic and can give rise either to segregation problems or may lead to complete blockage of the outlet. Here two relevant analytical solutions of the equations for the non-dilatant double-shearing model of granular flow are presented for both plane and axially symmetric funnel-flow. These solutions give rise to flow patterns which are similar to those observed in funnel-flow in the discharge of rectangular and circular cylindrical silos and hoppers.     

Cellular automata model of gravity-driven granular flows

A cellular automata model is used to simulate a variety of granular chute flows. The model is tested against several case studies: flow down a chute, flow past an obstacle, chute flow in which complex, counter-rotating vortices result in streamwise surface stripes and flow near a boundary. The model successfully reproduces experimental observations in all of these cases. These results lead us to propose that simple, rule-based, models such as this can improve our detailed understanding of dynamics and flow within an opaque granular bed.     

Micromechanic modeling and analysis of unsteady-state granular flow in a cylindrical hopper

This paper presents a numerical study of the micro- and macro-dynamic behavior of the unsteady-state granular flow in a cylindrical hopper with flat bottom by means of a modified discrete-element method (DEM) and an averaging method. The results show that the trends of the distributions of the microscopic properties such as the velocity and forces, and the macroscopic properties such as the velocity, mass density, stress and couple stress of the unsteady-state hopper flow are similar to those of steady-state hopper flow, and do not change much with the discharge of particles. However, the magnitudes of the macroscopic properties in different regions have different rates of variation. In particular, the magnitudes of the two normal stresses vary little with time in the orifice region, but decrease in other regions. The magnitude of the shear stress decreases with time when far from the bottom wall and central axis of the hopper. The results also indicate that DEM can capture the key features of the granular flow, and facilitated with a proper averaging method, can also generate information helpful to the test and development of an appropriate continuum model for granular flow.     

Micromechanic modeling and analysis of unsteady-state granular flow in a cylindrical hopper

This paper presents a numerical study of the micro- and macro-dynamic behavior of the unsteady-state granular flow in a cylindrical hopper with flat bottom by means of a modified discrete-element method (DEM) and an averaging method. The results show that the trends of the distributions of the microscopic properties such as the velocity and forces, and the macroscopic properties such as the velocity, mass density, stress and couple stress of the unsteady-state hopper flow are similar to those of steady-state hopper flow, and do not change much with the discharge of particles. However, the magnitudes of the macroscopic properties in different regions have different rates of variation. In particular, the magnitudes of the two normal stresses vary little with time in the orifice region, but decrease in other regions. The magnitude of the shear stress decreases with time when far from the bottom wall and central axis of the hopper. The results also indicate that DEM can capture the key features of the granular flow, and facilitated with a proper averaging method, can also generate information helpful to the test and development of an appropriate continuum model for granular flow.     

Exact results versus mean field solutions for binary granular gas mixtures

A novel method for obtaining the distribution functions from the Boltzmann equations for binary granular gas mixtures of smooth spheres has been developed. The method is capable of producing accurate results irrespective of the values of the parameters which define the system. Here we explain the method and present results obtained using it for the temperature ratio of the components, as well as for the flatness (and higher order measures of flatness) of the distribution functions, for the homogeneous cooling state. It turns out that the mean field approximation for the temperature ratio yields results which are within about 1% or better of the exact results for all checked values of the parameters (except when the mass ratio is very large (or small) and the system is very inelastic) even when the values of the flatness suggest that the distribution functions are not near-Maxwellian. The use of the method for obtaining constitutive relations is outlined but detailed results are deferred to another publication.     

A large deformation breakage model of granular materials including porosity and inelastic distortional deformation rate

A general constitutive model of crushable granular materials is developed within the context of large deformations. The time evolution equations for breakage, inelastic porous compaction and dilation, and distortional deformations are coupled by a yield surface and restrictions are imposed to ensure that these inelastic processes are dissipative. Some of the most salient mechanisms of such materials are described, including: (1) stiffness dependent on the breakage (a variable index of grading), porosity, and pressure; (2) critical comminution pressure and isotropic hardening, also dependent on the breakage and porosity; (3) jamming transition between solid and gaseous states; (4) a dilation law that embodies competition between porous compaction (due to the rate of breakage) and bulking (porous dilation at positive pressure due to the rate of inelastic distortional deformation); and finally, (5) the non-unique critical state relation between stress and porosity, in terms of the loading history and grading changes.     

Open channel flow past a bottom obstruction

A new nonlinear integral-equation model is derived in terms of hodograph variables for free-surface flow past an arbitrary bottom obstruction. A numerical method, carefully chosen to solve the resulting nonlinear algebraic equations and a simple, yet effective radiation condition have led to some very encouraging results. In this paper, results are presented for a semi-circular obstruction and are compared with those of Forbes and Schwartz [1]. It is shown that the wave resistance calculated from our nonlinear model exhibits a good agreement with that predicted by the linear model for a large range of Froude numbers for a small disturbance. The small-Froude-number non-uniformity associated with the linear model is also discussed.     

A note on the velocity of granular flow down a bumpy inclined plane

The velocity distribution of granular flow down a bumpy inclined plane is theoretically studied. The characteristic length scale of local transient cluster plays an important role in determining the flow rheology. After discussing the factors influencing the cluster size, we reproduce all observed velocity distributions successfully.This research was supported by the National Key Basic Research and Development Foundation of the Ministry of Science and Technology of China No. G2000048702.     

APPROACHES TO PARAMAGNETIC SEPARATIONS IN BIOLOGY AND MEDICINE

ABSTRACT

Exposure to paramagnetic cations, such as Er³⁺, endows many types of particles with positive magnetic susceptibilities. This discovery has served as the basis for our investigations into methods of separating particulate biological materials magnetically. As part of this enterprise, the “isomagnetic” method has been developed to permit the accurate determination of the magnetic susceptibilities of individual particles. By manipulating the experimental conditions, the particles' magnetic behavior can be controlled to permit the isolation of specific populations of particles from heterogeneous mixtures. Using the Ferro-graph Analyzer, we have applied these principles to the magnetic retrieval and separation of wear particles from human synovial fluid and of various types of eukaryotic and prokaryotic cells.     

Refined resultant thermomechanics of shells

The resultant two-dimensional (2D) balance laws of mass, linear and angular momentum, and energy as well as the entropy inequality for shells are derived by direct through-the-thickness integration of corresponding 3D laws of continuum thermomechanics. It is indicated that the resultant shell stress power cannot be expressed exactly through the 2D shell stress and strain measures alone. Hence, an additional stress power called an interstitial working is added to the resultant 2D balance of energy. The new, refined, resultant balance of energy and entropy inequality derived here are regarded to be exact implications of corresponding global 3D laws of rational thermodynamics. The kinematic structure of our shell theory is that of the Cosserat surface, while our refined resultant laws of thermomechanics contain three additional surface fields somewhat similar to those present in 3D extended thermodynamics. We briefly analyse the restrictions imposed by our refined resultant entropy inequality on the forms of 2D constitutive equations of viscous shells with heat conduction and of thermoelastic shells. It is shown, in particular, that in such shells the refined resultant entropy inequality allows one to account for some longer-range spatial interactions. We also present several novel forms of 2D kinetic constitutive equations compatible with the resultant shell equations.     

Double-shearing theory applied to instability and strain localization in granular materials

An account is given of the non-dilatant double-shearing theory of plane flow of granular materials, and it is shown that the theory may be formulated as a special form of hypoplasticity theory. It is shown that according to this theory, simple shearing flows may be supported by a time-independent stress field, but that this solution is unstable. An alternative solution in which the stress in time-dependent is also derived, and shear flow takes place under decreasing shear stress. The strain localization theory of Rudnicki and Rice is applied in conjunction with the double-shearing theory, and it is shown that the theory admits bifurcations in which shear bands form on planes that coincide with the shear plane. Similarly, in pure shear, there exists an unstable solution with time-independent stress, and a solution with time-dependent stress in which the compressive load falls as the deformation increases, and shear bands may form at surfaces on which, according to the Coulomb criterion, the critical shear stress is mobilized. The double-shearing theory for axially symmetric flow is summarized, and applied to compression of a circular cylinder. Again there is an unstable constant stress solution, a time-dependent stress solution in which the axial pressure decreases as the compression of the cylinder increases, and conical shear bands may form on conical surfaces on which the critical shear stress is mobilized.     

Effect of energy nonequipartition on the transport properties in a granular mixture

The Boltzmann kinetic theory is used to analyze the effect of energy nonequipartition on the pressure and the shear viscosity of a granular binary mixture under simple shear flow. Theory and Monte Carlo simulations show that both quantities exhibit a non–monotonic behaviour with the mass ratio in contrast to the predictions made from previous theories based on the equipartition assumption. Our results agree qualitatively well with recent molecular dynamics simulations performed by Alam and Luding [Granular Matter 4, 139 (2002)]. The authors acknowledge partial support from the MCYT (Spain) through Grants No. BFM2001-0718 and ESP2003-02859.