Convection-Enhanced Transport into Open Cavities : Effect of Cavity Aspect Ratio.

Recirculating fluid regions occur in the human body both naturally and pathologically. Diffusion is commonly considered the predominant mechanism for mass transport into a recirculating flow region. While this may be true for steady flows, one must also consider the possibility of convective fluid exchange when the outer (free stream) flow is transient. In the case of an open cavity, convective exchange occurs via the formation of lobes at the downstream attachment point of the separating streamline. Previous studies revealed the effect of forcing amplitude and frequency on material transport rates into a square cavity (Horner in J Fluid Mech 452:199-229, 2002). This paper summarizes the effect of cavity aspect ratio on exchange rates. The transport process is characterized using both computational fluid dynamics modeling and dye-advection experiments. Lagrangian analysis of the computed flow field reveals the existence of turnstile lobe transport for this class of flows. Experiments show that material exchange rates do not vary linearly as a function of the cavity aspect ratio (A = W/H). Rather, optima are predicted for A ≈ 2 and A ≈ 2.73, with a minimum occurring at A ≈ 2.5. The minimum occurs at the point where the cavity flow structure bifurcates from a single recirculating flow cell into two corner eddies. These results have significant implications for mass transport environments where the geometry of the flow domain evolves with time, such as coronary stents and growing aneurysms. Indeed, device designers may be able to take advantage of the turnstile-lobe transport mechanism to tailor deposition rates near newly implanted medical devices.

Axial band scaling for bidisperse mixtures in granular tumblers.

Axial banding in rotating tumblers has been experimentally observed, but the dependence of band formation on the relative concentration of the bidisperse particles has not been thoroughly examined. We consider axial band formation and coarsening for dry and liquid granular systems of bidisperse mixtures of glass beads where the small particle volume fraction ranges from 10% to 90% in half-filled tumblers for several rotation rates. Single bands form for small particle volume fractions as low as 10% and as high as 90%, usually near the end walls. Band formation along the entire length of the tumbler is less likely at very low or very high volume fractions. After many rotations the segregation pattern coarsens, and for small particle volume fractions of 50% and greater, the coarsening is logarithmic. For very low or very high small particle volume fractions, the rate of coarsening is either not logarithmic or coarsening does not occur within the duration of the experiment (600 rotations). When bands form, the width of the band for either the small or large particles scales with the tumbler diameter.

Granular mixtures: analogy with chemical solution thermodynamics.

Particle dynamics simulations reveal parallels between granular mixtures and chemical solutions. Thermodynamic solution theory provides a connection between the interaction strength of the molecules, the concentration of each species in the solution, and bulk solution behavior. Particle dynamics simulations demonstrate a similar relationship between the interparticle forces, the composition of the granular mixture, and bulk flow behavior. The analogy holds true over different particle interaction types (friction or adhesion) and over different bulk properties (the angle of repose in a rotating drum and the viscosity of particles sheared between parallel plates). A solution theory for granular mixtures would provide a framework to study the properties of granular mixtures.

Encapsulated drop breakup in shear flow.

We investigate the deformation and breakup in shear flow of an encapsulated drop in which both the core and shell are Newtonian fluids. The equations of motion are solved numerically using a level set method to track interface motion. We consider the case of a drop stretched to a given length in constant shear and then allowed to relax. A range of morphologies is produced, and novel kinematics occur, due to the interaction of the core and outer interfaces. A phase diagram is presented to describe the morphologies produced over a range of capillary numbers and core interfacial tensions.

Dynamics of a drop at a fluid interface under shear.

We analyze the dynamics of a two-dimensional drop lying on a fluid interface, sometimes called a liquid lens, subjected to simple shear flow. The three fluids, the drop and the two external fluids, meet at a triple point (or a triple line in three dimensions). A requirement for steady drop shapes is that the triple points are stationary. This leads to a flow topology different than that of a freely suspended drop. Results are substantiated with numerical results using a level set method for interface evolution and treatment of triple points. Possible implications for new drop instabilities are also discussed.

Dynamics of axial segregation and coarsening of dry granular materials and slurries in circular and square tubes.

We study segregation and coarsening dynamics of dry granular materials and slurries in tubes with circular and square cross sections. Space-time plots show key differences between the four cases, including band splitting and wave formation, depending upon the rotational speed. However, the fraction of the surface occupied by bands of small-rich particles is nearly constant in all experiments, leading to quasi-1D behavior, and the rate of coarsening, when it occurs, is logarithmic in all cases. Coarsening rates are very similar except in the case with the longest development time.

Competition between chaos and order: mixing and segregation in a spherical tumbler.

We investigate the competition between granular mixing and segregation in a sphere rotating and rocking on two orthogonal axes. Operation corresponds to the continuous-flow regime and the flow within the sphere is three-dimensional and time-periodic. Experimental results are organized in a frequency/amplitude phase diagram showing modes of segregation (band formation/no axial bands); segregated bands are remarkably robust and survive rocking amplitudes as large as 60 degrees over a wide range of frequencies. Details differ, but the phenomenon occurs under both dry and slurry conditions, that is, when all air is replaced by a liquid. Experimental space-time plots of the stationary segregated patterns agree well with Poincaré maps obtained using a continuum model of the flow, suggesting that the final segregation patterns are relatively independent of materials tumbled.

Open problems in active chaotic flows: Competition between chaos and order in granular materials.

There are many systems where interaction among the elementary building blocks-no matter how well understood-does not even give a glimpse of the behavior of the global system itself. Characteristic for these systems is the ability to display structure without any external organizing principle being applied. They self-organize as a consequence of synthesis and collective phenomena and the behavior cannot be understood in terms of the systems' constitutive elements alone. A simple example is flowing granular materials, i.e., systems composed of particles or grains. How the grains interact with each other is reasonably well understood; as to how particles move, the governing law is Newton's second law. There are no surprises at this level. However, when the particles are many and the material is vibrated or tumbled, surprising behavior emerges. Systems self-organize in complex patterns that cannot be deduced from the behavior of the particles alone. Self-organization is often the result of competing effects; flowing granular matter displays both mixing and segregation. Small differences in either size or density lead to flow-induced segregation and order; similar to fluids, noncohesive granular materials can display chaotic mixing and disorder. Competition gives rise to a wealth of experimental outcomes. Equilibrium structures, obtained experimentally in quasi-two-dimensional systems, display organization in the presence of disorder, and are captured by a continuum flow model incorporating collisional diffusion and density-driven segregation. Several open issues remain to be addressed. These include analysis of segregating chaotic systems from a dynamical systems viewpoint, and understanding three-dimensional systems and wet granular systems (slurries). General aspects of the competition between chaos-enhanced mixing and properties-induced de-mixing go beyond granular materials and may offer a paradigm for other kinds of physical systems. (c) 2002 American Institute of Physics.

Modes of granular segregation in a noncircular rotating cylinder.

Axial segregation is a well-known example of segregation of granular materials. However, at present, there is no conclusive explanation as to why it occurs. Most studies of axial segregation to date are based on cylinders with circular cross sections, and models focus on the character of the surface flow without accounting explicitly for the influence of any subsurface detail. The present experiments demonstrate that the cross section of the mixer has a significant influence on axial segregation and that subsurface dynamics are, in fact, important. Unlike circular mixers, in square mixers the subsurface segregation patterns change with filling level, as does the time dependence of axial segregation. Furthermore, when radial segregation patterns in noncircular mixers most closely resemble that observed for circular cylinders, the time dependence for axial band formation deviates the most. These results challenge segregation theories of axial segregation that ignore subsurface effects.

Self-organization in granular slurries.

We report the existence of self-organization in wet granular media or slurries, mixtures of particles of different sizes dispersed in a lower density liquid. As in the case of dry granular mixtures, axial banding (alternating bands rich in small and large particles in a long rotating cylinder) and radial segregation (in quasi-2D containers) are observed in slurries. However, when compared with the dry counterpart axial segregation is significantly faster and the spectrum of outcomes is richer. Moreover, experiments with suitable fluids reveal, for the first time, the internal structure of axially segregated systems, something that up to now has been accessible only via magnetic resonance imaging experimentation.

Segregation-driven organization in chaotic granular flows.

An important industrial problem that provides fascinating puzzles in pattern formation is the tendency for granular mixtures to de-mix or segregate. Small differences in either size or density lead to flow-induced segregation. Similar to fluids, noncohesive granular materials can display chaotic advection; when this happens chaos and segregation compete with each other, giving rise to a wealth of experimental outcomes. Segregated structures, obtained experimentally, display organization in the presence of disorder and are captured by a continuum flow model incorporating collisional diffusion and density-driven segregation. Under certain conditions, structures never settle into a steady shape. This may be the simplest experimental example of a system displaying competition between chaos and order.

Chaotic mixing of granular materials in two-dimensional tumbling mixers.

We consider the mixing of similar, cohesionless granular materials in quasi-two-dimensional rotating containers by means of theory and experiment. A mathematical model is presented for the flow in containers of arbitrary shape but which are symmetric with respect to rotation by 180 degrees and half-filled with solids. The flow comprises a thin cascading layer at the flat free surface, and a fixed bed which rotates as a solid body. The layer thickness and length change slowly with mixer rotation, but the layer geometry remains similar at all orientations. Flow visualization experiments using glass beads in an elliptical mixer show good agreement with model predictions. Studies of mixing are presented for circular, elliptical, and square containers. The flow in circular containers is steady, and computations involving advection alone (no particle diffusion generated by interparticle collisions) show poor mixing. In contrast, the flow in elliptical and square mixers is time periodic and results in chaotic advection and rapid mixing. Computational evidence for chaos in noncircular mixers is presented in terms of Poincare sections and blob deformation. Poincare sections show regions of regular and chaotic motion, and blobs deform into homoclinic tendrils with an exponential growth of the perimeter length with time. In contrast, in circular mixers, the motion is regular everywhere and the perimeter length increases linearly with time. Including particle diffusion obliterates the typical chaotic structures formed on mixing; predictions of the mixing model including diffusion are in good qualitative and quantitative (in terms of the intensity of segregation variation with time) agreement with experimental results for mixing of an initially circular blob in elliptical and square mixers. Scaling analysis and computations show that mixing in noncircular mixers is faster than that in circular mixers, and the difference in mixing times increases with mixer size. (c) 1999 American Institute of Physics.

Mixing and segregation of granular materials in chute flows.

Mixing of granular solids is invariably accompanied by segregation, however, the fundamentals of the process are not well understood. We analyze density and size segregation in a chute flow of cohesionless spherical particles by means of computations and theory based on the transport equations for a mixture of nearly elastic particles. Computations for elastic particles (Monte Carlo simulations), nearly elastic particles, and inelastic, frictional particles (particle dynamics simulations) are carried out. General expressions for the segregation fluxes due to pressure gradients and temperature gradients are derived. Simplified equations are obtained for the limiting cases of low volume fractions (ideal gas limit) and equal sized particles. Theoretical predictions of equilibrium number density profiles are in good agreement with computations for mixtures of equal sized particles with different density for all solids volume fractions, and for mixtures of different sized particles at low volume fractions (nu<0.2), when the particles are elastic or nearly elastic. In the case of inelastic, frictional particles the theory gives reasonable predictions if an appropriate effective granular temperature is assumed. The relative importance of pressure diffusion and temperature diffusion for the cases considered is discussed. (c) 1999 American Institute of Physics.

Transport in a grooved perfusion flat-bed bioreactor for cell therapy applications.

This study considers the transport of oxygen (a growth-associated solute) and lactate (a metabolic byproduct) in a flat-bed perfusion chamber modified to retain cells through the addition of grooves, perpendicular to the direction of flow, at the chamber bottom. The chamber has been successfully applied to hematopoietic cell culture and may be useful for other basic and applied biomedical applications. The objective of this study is to characterize the culture environment in terms of solute transport under various operational conditions. This will allow one to improve the design and operating strategy of the perfusion system for maximizing cell numbers. The system is numerically simulated using the finite element package FIDAP. The reaction kinetics describing oxygen uptake by cells are simplified to zero order to give an upper bound for the oxygen consumption. A flat-bed chamber without grooves is considered here as a benchmark. We show that the growth environment is not oxygen limited (local oxygen concentration above 10 microM) for a variety of flow rates and culture conditions (qO2 = 0.1 micromol/(10(6) cells h)). With a medium flow rate of 2.5 mL/min through the reactor, the model predicts that the 29-cm2 reactor can support at least 33.4 x 10(6) total cells when the inlet medium is in equilibrium with high (20%) oxygen concentration. The culture becomes oxygen limited however for the same flow rate for low (5%) oxygen concentration and can only support 7.2 x 10(6) total cells. Comparison of grooved vs nongrooved chambers reveals that the presence of grooves only affects solute transport on a local scale. This result is attributed to the small size (200 microgram) of the cavities relative to the chamber dimensions. The comparison also yields an empirical relation that allows for rapid estimation of oxygen and lactate concentrations in the grooves using only the numerical simulation of the simpler nongrooved chamber. Finally, our investigation shows that, while decreasing the spacing between cavities decreases the total number of cells the reactor can support, the efficiency of the reactor is increased by 25% (on an area basis) without growth restriction.

Chaos, symmetry, and self-similarity: exploiting order and disorder in mixing processes.

Fluid mixing is a successful application of chaos. Theory anticipates the coexistence of order and disorder-symmetry and chaos-as well as self-similarity and multifractality arising from repeated stretching and folding. Experiments and computations, in turn, provide a point of confluence and a visual analog for chaotic behavior, multiplicative processes, and scaling behavior. All these concepts have conceptual engineering counterparts: examples arise in the context of flow classification, design of mixing devices, enhancement of transport processes, and controlled structure formation in two-phase systems.