It has been recognized that particle inertia throws dense particles out of regions of high vorticity and leads to an accumulation of particles in the straining-flow regions of a turbulent flow field. However, recent direct numerical simulations (DNS) indicate that the tendency to cluster is evident even at particle separations smaller than the size of the smallest eddy. Indeed, the particle radial distribution function (RDF), an important measure of clustering, increases as an inverse power of the interparticle separation for separations much smaller than the Kolmogorov length scale. Motivated by this observation, we have developed an analytical theory to predict the RDF in a turbulent flow for particles with a small, but non-zero Stokes number. Here, the Stokes number ($\hbox{\it St}$) is the ratio of the particle's viscous relaxation time to the Kolmogorov time. The theory approximates the turbulent flow in a reference frame following an aerosol particle as a local linear flow field with a velocity gradient tensor and acceleration that vary stochastically in time. In monodisperse suspensions, the power-law dependence of the pair probability is seen to arise from a balance of an inward drift caused by the particles' inertia that scales linearly with the particle separation distance and a pairwise diffusion owing to the random nature of the flow with a diffusivity that scales quadratically with the particle separation distance. The combined effect leads to a power law behaviour for the RDF with an exponent, $c_1$, that is proportional to $\hbox{\it St}^2$. Predictions of the analytical theory are compared with two types of numerical simulation: (i) particle pairs interacting in a local linear flow whose velocity varies according to a stochastic velocity gradient model; (ii) particles interacting in a flow field obtained from DNS of isotropic turbulence. The agreement with both types of simulation is very good. The theory also predicts the RDF for unlike particle pairs (particle pairs with different Stokes numbers). In this case, a second diffusion process occurs owing to the difference in the response of the pair to local fluid accelerations. The acceleration diffusivity is independent of the pair separation distance; thus, the RDF of particles with even slightly different viscous relaxation times undergoes a transition from the power law behaviour at large separations to a constant value at sufficiently small separations. The radial separation corresponding to the transition between these two behaviours is predicted to be proportional to the difference between the Stokes numbers of the two particles. Once again, the agreement between the theory and simulations is found to be very good. Clustering of particles enhances their rate of coagulation or coalescence. The theory and linear flow simulations are used to obtain predictions for the rate of coagulation of particles in the absence of hydrodynamic and colloidal particle interactions.

Experimental observations indicate that, at sufficiently high cell densities, swimming bacteria exhibit coordinated motions on length scales (10 to 100 μm) that are large compared with the size of an individual cell but too small to yield significant gravitational or inertial effects. We discuss simulations of hydrodynamically interacting self-propelled particles as well as stability analyses and numerical solutions of averaged equations of motion for low Reynolds number swimmers. It has been found that spontaneous motions can arise in such systems from the coupling between the stresses the bacteria induce in the fluid as they swim and the rotation of the bacteria due to the resulting fluid velocity disturbances.

Lattice-Boltzmann simulations are used to examine the effects of fluid inertia, at moderate Reynolds numbers, on flows in simple cubic, face-centred cubic and random arrays of spheres. The drag force on the spheres, and hence the permeability of the arrays, is calculated as a function of the Reynolds number at solid volume fractions up to the close-packed limits of the arrays. At Reynolds numbers up to O(102), the non-dimensional drag force has a more complex dependence on the Reynolds number and the solid volume fraction than suggested by the well-known Ergun correlation, particularly at solid volume fractions smaller than those that can be achieved in physical experiments. However, good agreement is found between the simulations and Ergun's correlation at solid volume fractions approaching the close-packed limit. For ordered arrays, the drag force is further complicated by its dependence on the direction of the flow relative to the axes of the arrays, even though in the absence of fluid inertia the permeability is isotropic. Visualizations of the flows are used to help interpret the numerical results. For random arrays, the transition to unsteady flow and the effect of moderate Reynolds numbers on hydrodynamic dispersion are discussed.

A macroscopic equation of mass conservation is obtained by ensemble-averaging the basic conservation laws in a porous medium. In the long-time limit this 'macrotransport' equation takes the form of a macroscopic Fick's law with a constant effective diffusivity tensor. An asymptotic analysis in low volume fraction of the effective diffusivity in a bed of fixed spheres is carried out for all values of the PBclet number P = Ua/D,, where Uis the average velocity through the bed, a is the particle radius and D, is the molecular diffusivity of the solute in the fluid. Several physical mechanisms causing dispersion are revealed by this analysis. The stochastic velocity fluctuations induced in the fluid by the randomly positioned bed particles give rise to a convectively driven contribution to dispersion. A t high PBclet numbers, this convective dispersion mechanism is purely mechanical, and the resulting effective diffusivities are independent of molecular diffusion and grow linearly with P. The region of zero velocity in and near the bed particles gives rise to non-mechanical dispersion mechanisms that dominate the longitudinal diffusivity at very high PBclet numbers. One such mechanism involves the retention of the diffusing species in permeable particles, from which it can escape only by molecular diffusion, leading to a diffusion coefficient that grows aa P2. Even if the bed particles are impermeable, non-mechanical contributions that grow as P l n P and P2 at high P arise from a diffusive boundary layer near the solid surfaces and from regions of closed streamlines respectively. The results for the longitudinal and transverse effective diffusivities as functions of the PBclet number are summarized in tabular form in $6. Because the same physical mechanisms promote dispersion in dilute and dense fixed beds, the predicted PBclet-number dependences of the effective diffusivities are applicable to all porous media. The theoretical predictions are compared with experiments in densely packed beds of impermeable particles, and the agreement is shown to be remarkably good.

We examine the problem of determining the particle-phase velocity variance and rhe-ology of sheared gas-solid suspensions at small Reynolds numbers and finite Stokes numbers. Our numerical simulations take into account the Stokes flow interactions among particles except for pairs of particles with a minimum gap width comparable to or smaller than the mean free path of the gas molecules for which the usual lubrication approximation breaks down and particle collisions occur in a finite time. The simulation results are compared to the predictions of two theories. The first is an asymptotic theory for large Stokes number St and nearly elastic collisions, i.e. St [Gt ] 1 and 0 ≤ 1 - e [Lt ] 1, e being the coefficient of restitution. In this limit, the particle velocity distribution is close to an isotropic Maxwellian and the velocity variance is determined by equating the energy input in shearing the suspension to the energy dissipation by inelastic collisions and viscous effects. The latter are estimated by solving the Stokes equations of motion in suspensions with the hard-sphere equilibrium spatial and velocity distribution while the shear energy input and energy dissipation by inelastic effects are estimated using the standard granular flow theory (i.e. St = ∞). The second is an approximate theory based on Grad's moments method for which St and 1 – e are O(1). The two theories agree well with each other at higher values of volume fraction ϕ of particles over a surprisingly large range of values of St. For smaller ϕ however, the two theories deviate significantly except at sufficiently large St. A detailed comparison shows that the predictions of the approximate theory based on Grad's method are in excellent agreement with the results of numerical simulations.

We examine the stability of a suspension of swimming bacteria in a Newtonian medium. The bacteria execute a run-and-tumble motion, runs being periods when a bacterium on average swims in a given direction; runs are interrupted by tumbles, leading to an abrupt, albeit correlated, change in the swimming direction. An instability is predicted to occur in a suspension of ‘pushers’ (e.g.E. Coli,Bacillus subtilis, etc.), and owes its origin to the intrinsic force dipoles of such bacteria. Unlike the dipole induced in an inextensible fibre subject to an axial straining flow, the forces constituting the dipole of a pusher are directed outward along its axis. As a result, the anisotropy in the orientation distribution of bacteria due to an imposed velocity perturbation drives a disturbance velocity field that acts to reinforce the perturbation. For long wavelengths, the resulting destabilizing bacterial stress is Newtonian but with a negative viscosity. The suspension becomes unstable when the total viscosity becomes negative. In the dilute limit (nL3≪ 1), a linear stability analysis gives the threshold concentration for instability as (nL3)crit= ((30/Cℱ(r))(DrL/U)(1 + 1/(6τDr)))/(1−(15𝒢(r)/Cℱ(r))(DrL/U)(1 + 1/(6τDr))) for perfectly random tumbles; here,LandUare the length and swimming velocity of a bacterium,nis the bacterial number density,Drcharacterizes the rotary diffusion during a run and τ−1is the average tumbling frequency. The function ℱ(r) characterizes the rotation of a bacterium of aspect ratiorin an imposed linear flow; ℱ(r) = (r2−1)/(r2+ 1) for a spheroid, and ℱ(r) ≈ 1 for a slender bacterium (r≫ 1). The function 𝒢(r) characterizes the stabilizing viscous response arising from the resistance of a bacterium to a deforming ambient flow; 𝒢(r) = 5π/6 for a rigid spherical bacterium, and 𝒢(r)≈ π/45(lnr) for a slender bacterium. Finally, the constantCdenotes the dimensionless strength of the bacterial force dipole in units of μU L2; forE. Coli,C≈ 0.57. The threshold concentration diverges in the limit ((15𝒢(r)/Cℱ(r)) (DrL/U)(1 + 1/(6τDr))) → 1. This limit defines a critical swimming speed,Ucrit= (DrL)(15𝒢(r)/Cℱ(r))(1 + 1/(6τDr)). For speeds smaller than this critical value, the destabilizing bacterial stress remains subdominant and a dilute suspension of these swimmers therefore responds to long-wavelength perturbations in a manner similar to a suspension of passive rigid particles, that is, with a net enhancement in viscosity proportional to the bacterial concentration.On the other hand, the stability analysis predicts that the above threshold concentration reduces to zero in the limitDr→ 0, τ → ∞, and a suspension of non-interacting straight swimmers is therefore always unstable. It is then argued that the dominant effect of hydrodynamic interactions in a dilute suspension of such swimmers is via an interaction-driven orientation decorrelation mechanism. The latter arises from uncorrelated pair interactions in the limitnL3≪ 1, and for slender bacteria in particular, it takes the form of a hydrodynamic rotary diffusivity (Dhr); forE. Coli, we findDhr= 9.4 × 10−5(nUL2). From the above expression for the threshold concentration, it may be shown that even a weakly interacting suspension of slender smooth-swimming bacteria (r≫ 1, ℱ(r) ≈ 1, τ → ∞) will be stable providedDhr> (C/30)(nUL2) in the limitnL3≪ 1. The hydrodynamic rotary diffusivity ofE. Coliis, however, too small to stabilize a dilute suspension of these swimmers, and a weakly interacting suspension ofE. Coliremains unstable.

Theory and lattice-Boltzmann simulations are used to examine the effects of fluid inertia, at small Reynolds numbers, on flows in simple cubic, face-centred cubic and random arrays of spheres. The drag force on the spheres, and hence the permeability of the arrays, is determined at small but finite Reynolds numbers, at solid volume fractions up to the close-packed limits of the arrays. For small solid volume fraction, the simulations are compared to theory, showing that the first inertial contribution to the drag force, when scaled with the Stokes drag force on a single sphere in an unbounded fluid, is proportional to the square of the Reynolds number. The simulations show that this scaling persists at solid volume fractions up to the close-packed limits of the arrays, and that the first inertial contribution to the drag force relative to the Stokes-flow drag force decreases with increasing solid volume fraction. The temporal evolution of the spatially averaged velocity and the drag force is examined when the fluid is accelerated from rest by a constant average pressure gradient toward a steady Stokes flow. Theory for the short- and long-time behaviour is in good agreement with simulations, showing that the unsteady force is dominated by quasi-steady drag and added-mass forces. The short- and long-time added-mass coefficients are obtained from potential-flow and quasi-steady viscous-flow approximations, respectively.

A linear stability analysis is performed for the homogeneous state of a monodisperse gas-fluidized bed of spherical particles undergoing hydrodynamic interactions and solid-body collisions at small particle Reynolds number and finite Stokes number. A prerequisite for the stability analysis is the determination of the particle velocity variance which controls the particle-phase pressure. In the absence of an imposed shear, this velocity variance arises solely due to the hydrodynamic interactions among the particles. Since the uniform state of these suspensions is unstable over a wide range of values of particle volume fraction φ and Stokes number St, full dynamic simulations cannot be used in general to characterize the properties of the homogeneous state. Instead, we use an asymptotic analysis for large Stokes numbers together with numerical simulations of the hydrodynamic interactions among particles with specified velocities to determine the hydrodynamic sources and sinks of particle-phase energy. In this limit, the velocity distribution to leading order is Maxwellian and therefore standard kinetic theories for granular/hard-sphere molecular systems can be used to predict the particle-phase pressure and rheology of the bed once the velocity variance of the particles is determined. The analysis is then extended to moderately large Stokes numbers for which the anisotropy of the velocity distribution is considerable by using a kinetic theory which combines the theoretical analysis of Koch (1990) for dilute suspensions (φ 1) with numerical simulation results for non-dilute suspensions at large Stokes numbers. A linear stability analysis of the resulting equations of motion provides the first a priori predictions of the marginal stability limits for the homogeneous state of a gas-fluidized bed. Dynamical simulations following the detailed motions of the particles in small periodic unit cells confirm the theoretical predictions for the particle velocity variance. Simulations using larger unit cells exhibit an inhomogeneous structure consistent with the predicted instability of the homogeneous gas-solid suspension.