H. Brenner

Shear flow over a self-similar expanding pulmonary alveolus during rhythmical breathing

Alternating shear flow over a self-similar, rhythmically expanding hemispherical depression is investigated. It provides a fluid-mechanical model for an alveolated respiratory unit, by means of which the eect of lung rhythmical expansion on gas mixing as well as aerosol dispersion and deposition can be studied. The flow is assumed to be very slow and governed by the quasi-steady linear Stokes equations. Consequently, superposition of the following two cases provides an easy route toward characterizing the aforementioned flow eld. The rst case treats the flow eld that is generated by a rhythmically expanding spherical cap (the alveolus). The cap is attached at its rim to a circular opening in an expanding unbounded plane bounding a semi-innite fluid region. The rate of expansion of the cap and the plane are chosen such as to maintain the system’s congurational self-similarity. The second case addresses the flow disturbance that is generated by an alternating shear flow encountering a rigid hemispherical cavity in a plane bounding a semi-innite fluid domain. For the rst case, a stream-function representation employing toroidal coordinates furnishes an analytical solution, whereas the second case was solved numerically by Pozrikidis (1994). Linear superposition of the two flow cases results in particularly rich streamline maps. In the symmetry plane (bisecting the cap and parallel to the mean shear flow), for a certain range of shear to expansion-rate ratios, the streamline maps are self-similar and display closed orbits and two internal stagnation points. One of the stagnation points is a ‘centre’ surrounded by closed streamlines whereas the other constitutes a ‘saddle point’. For other planes, no stagnation points exist in the eld, but the streamlines associated with the saddle point display complex looping patterns. These unique flow structures, when subjected to a small perturbation (e.g. a small asynchrony between ductal and alveolar entering flows) cause highly complex stochastic particle trajectories even in the quasi-static Stokes alveolar flow. The observed irreversible flow phenomena in a rhythmically expanding alveolus may be partially responsible for the ‘stretch-and-fold’ flow mixing patterns found in our recent flow visualization studies performed in excised animal lung acini.

Shear-induced particle migration in suspensions of rods

Shear‐induced migration of particles occurs in suspensions of neutrally buoyant spheres in Newtonian fluids undergoing shear in the annular space between two rotating, coaxial cylinders (a wide‐gap Couette), even when the suspension is in creeping flow. Previous studies have shown that the rate of migration of spherical particles from the high‐shear‐rate region near the inner (rotating) cylinder to the low‐shear‐rate region near the outer (stationary) cylinder increases rapidly with increasing sphere size. To determine the effect of particle shape, the migration of rods suspended in Newtonian fluids was recently measured. The behavior of several suspensions was studied. Each suspension contained well‐characterized, uniform rods with aspect ratios ranging from 2 to 18 at either 0.30 or 0.40 volume fraction. At the same volume fraction of solids, the steady‐state, radial concentration profiles for rods were independent of aspect ratio and were indistinguishable from those obtained from suspended spheres. On...

Antisymmetric stresses in suspensions: vortex viscosity and energy dissipation

When the individual particles in an otherwise quiescent suspension of freely suspended spherical particles are acted upon by external couples, the resulting suspension-scale fluid motion is characterized by a non-symmetric state of stress. Viewed at the interstitial scale (i.e. microscopic scale), this coupling between translational and rotational particle motions is a manifestation of particle–particle hydrodynamic interactions and vanishes with the volume fraction φ of suspended spheres. The antisymmetric portion of the stress is quantified by the suspension-scale vortex viscosity µv, different from the suspension’s shear viscosity µ. Numerical boundary element method (BEM) simulations of such force-free suspensions of spheres uniformly dispersed in incompressible Newtonian liquids of viscosity µ0 are performed for circumstances in which external couples (of any specified suspension-scale position-dependence) are applied individually to each of the suspended particles in order to cause them to rotate in otherwise quiescent fluids. In the absence of external forces acting on either the spheres or boundaries, such rotations indirectly, through interparticle coupling, cause translational motions of the individual spheres which, owing to the no-slip boundary condition, drag neighbouring fluid along with them. In turn, this combined particle–interstitial fluid movement is manifested as a suspension-scale velocity field, generated exclusively by the action of external couples. Use of this scheme to create suspension-scale particle-phase spin fields Ω and concomitant velocity fields v enables both the vortex and shear viscosities of suspensions to be determined as functions of φ in disordered systems. This scheme is shown, inter alia, to confirm the constitutive equation, T a =2 µve · [(1/2)∇ × v − Ω], proposed in the continuum mechanics literature for the linear relation between the antisymmetric stress T a and the disparity existing between the particle-phase spin rate Ω and half the suspension’s vorticity, ∇ × v (with the third-rank pseudotensor e the permutation triadic). Our dynamically based BEM simulations confirm the previous computations of the Prosperetti et al. group for the dependence of the vortex viscosity upon the solids volume fraction in concentrated disordered suspensions, obtained by a rather different simulation scheme. Moreover, our dynamically based rheological calculations are confirmed by our semi-independent, energetically based, calculations that equate the rates of working (equivalently, the energy dissipation rates) at the respective interstitial and suspension scales. As an incidental by-product, the same BEM simulation results also verify the suspension-scale Newtonian constitutive equation, T s = µ[∇v +( ∇v) † ],

Particle tracking in three‐dimensional Stokes flow

The three‐dimensional motions of particles settling in Stokes flow are tracked numerically and experimentally. The numerical simulations are performed using a boundary element representation of the Stokes flow equations coupled with an Euler parameter representation of the particle kinematics. The experiments provide quantitative, three‐dimensional results for particle trajectories in Stokes flow. The experimental and numerical results are compared and found to agree to within the experimental uncertainty. The numerical results are also benchmarked against previously reported results for two test problems.

Hydrodynamic particle migration in small-amplitude, oscillatory, circular couette flow: The limits of reversibility

Initially well-mixed suspensions of large spherical particles in viscous Newtonian fluids subjected to continuous nonhomogeneous shear flows demix and establish large concentration gradients. A number of experimental studies have determined that the particles migrate to the low shear-rate region in one-dimensional flow fields. Other investigators have proposed mechanisms to model the migration caused by the presence of nonuniform shear gradients. Here, we report on the shear-induced migration of particles in the annular space between two coaxial cylinders (a wide-gap Couette), with the inner cylinder oscillating and the outer cylinder fixed. In these experiments, the conditions are such that colloidal and inertial forces do not exert an appreciable effect on the suspensions of neutrally bouyant spherical particles in Newtonian liquids. Nuclear magnetic resonance (NMR) imaging is used to measure the concentration profile during the demixing of the initially well-mixed suspensions.

The pressure drop created by a ball settling in a quiescent suspension of comparably sized spheres

Measurements are reported of the pressure differences Δ P existing at large distances above and below a ball settling along the axis of a circular cylinder filled with an otherwise quiescent viscous Newtonian liquid in which identical particles, comparable in size to the settling ball, are suspended. The suspensions ranged in solids volume fraction ϕ from 0.30 to 0.50 and consisted of 0.635 cm diameter spheres density-matched to the suspending oil. The settling balls varied in diameter from 0.318 to 1.27 cm, resulting in particle Reynolds numbers always less than about 0.4 based upon ball diameter and the effective viscosity of the suspension. For the moderately concentrated suspension (ϕ=0.30), the product of Δ P with the cross-sectional area A of the containing cylinder was observed to be equal to twice the drag force D on the settling sphere, in accord with theory. In the more concentrated suspension (ϕ=0.50) this product was found to be slightly, but significantly, less than twice the drag on the settling sphere. It is speculated that this lower pressure drop may result from the presence of one or more of the following phenomena: (i) migration of the falling ball off the cylinder axis ; (ii) apparent slip of the suspension at the cylinder wall; (iii) blunting of the otherwise Poiseuillian parabolic velocity profile, the latter phenomenon being known to occur during the creeping flow of concentrated suspensions through circular tubes. Incidental to the suspension experiments, for a homogeneous fluid we verify the classical theoretical formula for the off-axis pressure drop when the sphere settles at a non-concentric position in the cylinder.