Interactions between hard spheres sedimenting at low Reynolds number

In this chapter we shall consider the behaviour of two rigid or fluid particles moving through a viscous fluid under the action of external forces, typically the gravity force. When two rigid or fluid particles suspended in a viscous fluid approach one another, the motion of each one of them is influenced by the other, even in the absence of direct interparticle interactions, such as van-der-Waals and electrostatic forces. The velocity field caused by the motion of one of these particles is transmitted through the fluid medium and influences the motion and, therefore, the hydrodynamic force, torque and stresslet on the other one. This indirect particles interaction is known as a hydrodynamic interaction. The magnitude of the hydrodynamic interaction between rigid or fluid particles depends on the following variables: (a) their shapes and sizes; (b) the distance between them; (c) their orientation with respect to one another; (d) their individual orientation relative to the active external forces; (e) their velocities and spins relative to the fluid motion at infinity.

Two-dimensional direct numerical simulation and Floquet stability analysis have been performed at moderate Keulegan–Carpenter number ($KC$) and low Reynolds number ($Re$) for a square cross-section cylinder with its face normal to the oscillatory flow. Based on the numerical simulations a map of flow regimes is formed and compared to the map of flow around an oscillating circular cylinder by Tatsuno & Bearman (J. Fluid Mech., vol. 211, 1990, pp. 157–182). Two new flow regimes have been observed, namely A$^{\prime }$ and F$^{\prime }$. The regime A$^{\prime }$ found at low $KC$ is characterised by the transverse convection of fluid particles perpendicular to the motion; and the regime F$^{\prime }$ found at high $KC$ shows a quasi-periodic feature with a well-defined secondary period, which is larger than the oscillation period. The Floquet analysis demonstrates that when the two-dimensional flow breaks the reflection symmetry about the axis of oscillation, the quasi-periodic instability and the synchronous instability with the imposed oscillation occur alternately for the square cylinder along the curve of marginal stability. This alternate pattern in instabilities leads to four distinct flow regimes. When compared to the vortex shedding in otherwise unidirectional flow, the two quasi-periodic flow regimes are observed when the oscillation frequency is close to the Strouhal frequency (or to half of it). Both the flow regimes and marginal stability curve shift in the $(Re,KC)$-space compared to the oscillatory flow around a circular cylinder and this shift appears to be consistent with the change in vortex formation time associated with the lower Strouhal frequency of the square cylinder.

This paper discusses a fully decentralized algorithm able to create an adaptive swarm of autonomous mobile robots from the viewpoint of computational physics. To this end, we particularly focus on Stokesian Dynamics method. Simulation results indicate that the proposed algorithm can control the shape of the swarm appropriately according to the current situation without losing the coherence of the swarm.

We develop a model of contact interactions between two spherical particles immersed in a viscous fluid, under a very low Reynolds number. This model allows us to interpret results of our experiment, in which the settling motion of a ball in the vicinity of another fixed sphere is accurately measured with laser interferometry. Due to the symmetry of the experimental setup and the reversibility of the Stokes equations describing the fluid flow, the trajectory and velocity of the moving sphere center are expected to exhibit symmetry with respect to reflection in the horizontal plane containing the fixed sphere center. However, no such symmetry is observed if the particles ``touch'' each other. Our model accounts for symmetry breaking by the contact friction between surfaces---such a force appears only when the moving sphere center is above the horizontal plane containing the center of the fixed particle. The model predicts two intervals of motion, at contact: pure rolling and rolling with slip. The existence of both types of motion, with a sharp transition from one to another, has been verified experimentally.

The experimental investigation of the wall effect on the cluster particle settling in the Newtonian and non-Newtonian fluid medium is carried out in three different diameter flow channels. The cluster usually forms during the sedimentation of particles in a fluid medium, so it becomes necessary to study the behavior of the cluster particles. Different cluster particles are considered depending on the number of spheres (N) attached to the cluster and the cluster's shape. The present experiment covers the following range of conditions: 0.05 ≤ deq/D ≤ 0.24, 2 ≤ N ≤ 7, 0.64 ≤ n ≤ 1, and 0.14 ≤ K ≤ 1.81. The results reveal that the terminal velocity varies with the blockage ratio (deq/D), N, and cluster shape. For a particular deq/D ratio and the same N, the terminal velocity of polyhedron particles is high compared to the planar and chain shape particles. The blockage ratio and the Reynolds number affect the wall factor of the cluster particles. However, from the experiment, it is observed that the wall effect also depends on the orientation of the particles. The impact of the wall on the cluster particle is high in Newtonian fluid when compared to cluster particles in non-Newtonian fluid. The present work additionally investigates the influence of drag on cluster particles in the presence and absence of the wall effect. The numerical relationships are developed to predict experimental results in Newtonian and non-Newtonian fluid mediums.

Inspired by the tubercles on the pectoral flipper leading-edge of the humpback whale, the most acrobatic of baleen whales, sinusoidal leading-edge have been used as a passive flow control method to enhance the stall characteristics of lifting surfaces in different applications. They are supposed to reduce the lift gradually during the stall with an increase in the post-stall lift, however, their enhancement ability in pre- and post-stall is highly dependent on the geometrical parameters and the flow regime. Two different design schemes: nonlinear shear transformation design scheme (NLST) and channelled-like scheme, of the sinusoidal leading edge of the high lift airfoil S1223 are tested and compared with the unmodified airfoil at angles of attack of −6 to 30 degrees experimentally inside a wind tunnel at low Reynolds number Re = 100,000. Experimental data show that Tubercles implemented on S1223 at such a flow regime with either design scheme enhance the post-stall characteristics as much as a 42% increase in lift at some angles for NLST with pre-stall lift penalties. NLST, compared with baseline and channelled models, exhibit a significant increase in aerodynamic efficiency which make it more preferable for wind turbines applications.

We show that discrete element simulations of colloidal gelation must account for hydrodynamic interactions between suspended particles through investigation of gelation in a dispersion of colloids interacting pair-wise via short-ranged attraction and long-ranged repulsion (SALR). These dynamic simulations juxtapose self-assembly with and without hydrodynamic interactions between the particles. The long-ranged repulsion impacts the relative rates of coagulation and compaction of colloidal aggregates pre-gel, and introduces a surprising sensitivity to the nature of hydrodynamic interactions between the suspended colloids. For such SALR dispersions, we observe a significant disparity between the percolation boundaries predicted by simulations including and neglecting long-ranged hydrodynamic interactions. Additionally, we find that the percolation boundaries predicted by simulations including hydrodynamic interactions agree well with those measured experimentally. Long-ranged repulsion promotes gelation via growth of anisotropic clusters regardless of the hydrodynamic model employed. However, differences between the models, which persist far from the percolation boundary, are apparent via measurements of the fractal dimension, local bond order parameters, and the collective relaxation dynamics. Notably, the growth of elongated clusters is augmented in simulations that incorporate long-ranged hydrodynamic interactions due to the anisotropic diffusion of elongated bodies at low Reynolds numbers, which favors percolation over a transition of anisotropic clusters to their more isotropic ground states. It is only in relatively dense suspensions that a combination of hydrodynamic screening and significantly faster aggregation combine to bring the two simulation methods into agreement. These results demonstrate the necessity of long-ranged hydrodynamic forces in discrete element simulations of heterogeneous gelation at the colloidal scale.

Speed and efficiency of two swimming planar sheets in hydrodynamic interaction at low Reynolds number

The dynamics of flexible filament in a viscous fluid is a complex fluid–structure interaction problem that has wide scientific and engineering applications in emerging fields such as biomimetics and biotechnology. Coupling the structural equations with fluid flow poses a number of challenges for numerical simulation. In this regard, techniques like immersed boundary method (IBM) have been quite successful. In the present study, a two-dimensional numerical simulation of flexible filament in a rectangular channel with an oscillating fluid flow at low Reynolds number is carried out using IBM. The discretization of governing continuity and Navier–Stokes equation is done by finite volume method on a staggered Cartesian grid. SIMPLE algorithm is used to solve fluid velocity and pressure terms. The filament mechanical properties like stiffness and bending rigidity are incorporated into the governing equation via Eulerian forcing term. An oscillating pressure gradient drives the fluid while the flexible filament is fixed to the bottom channel wall. The simulation results are validated with filament dynamic studies of previous researchers. The interaction of the filament with nearby oscillating fluid motion is well captured by the developed numerical model.

We study both experimentally and theoretically the rheological behavior of isotropic bidisperse suspensions of noncolloidal particles in yield stress fluids. We focus on materials in which noncolloidal particles interact with the suspending fluid only through hydrodynamical interactions. We observe that both the elastic modulus and yield stress of bidisperse suspensions are lower than those of monodisperse suspensions of same solid volume fraction. Moreover, we show that the dimensionless yield stress of such suspensions is linked to their dimensionless elastic modulus and to their solid volume fraction through the simple equation of Chateau et al.[J. rheol. 52, 489-506 (2008)]. We also show that the effect of the particle size heterogeneity can be described by means of a packing model developed to estimate random loose packing of assemblies of dry particles. All these observations finally allow us to propose simple closed form estimates for both the elastic modulus and the yield stress of bidisperse suspensions: while the elastic modulus is a function of the reduced volume fraction $\phi/\phi_m$ only, where $\phi_m$ is the estimated random loose packing, the yield stress is a function of both the volume fraction $\phi$ and the reduced volume fraction.

Since its development, Stokesian dynamics has been a leading approach for the dynamic simulation of suspensions of particles at arbitrary concentrations with full hydrodynamic interactions. Although developed originally for the simulation of passive particle suspensions, the Stokesian dynamics framework is equally well suited to the analysis and dynamic simulation of suspensions of active particles, as we elucidate here. We show how the reciprocal theorem can be used to formulate the exact dynamics for a suspension of arbitrary active particles, and then show how the Stokesian dynamics method provides a rigorous way to approximate and compute the dynamics of dense active suspensions where many-body hydrodynamic interactions are important.

The characteristics and differences in the two approximations. i.e., the additivity of forces and the additivity of velocities, on which Stokesian dynamics methods are based, have been investigated. To do so, Stokesian dynamics simulations of a ferromagnetic colloidal dispersion have been carried out for a simple shear flow, and the aggregate structures and the averaged viscosities have been evaluated, From the results of aggregate structures in equilibrium, the results obtained by the additivity of forces are quantaively different from those by the approximation of ignoring hydrodynamic interactions, although a qualitative agreement is recognized. The results obtained by the additivity of velocities give medium characteristics between the two above approximations. This difference in characeristics between the two additivity approximations and that of ignoring hydrodynamics becomes clearer as the influences of the shear flow dominate those of magnetic forces more significantly. From the results of transient characteristics from an initial state, the results of the additivity of velocities agree well with those of the additivity of forces. In contrast, the results without hydrodynamic interactions deviate from those of the two additivity approximations at a time step when a nearly particle particle touching situation starts to appear.

The advent of micro aerial vehicles (MAV) and advanced flow control methods, using microelectromechanical systems (MEMS) and micro-blowing or suction, has significantly increased the need for a better understanding of very low Reynolds number flows. Aerodynamic testing in this flow regime using conventional facilities, such as wind or water tunnels, is seriously hampered by the necessary small model sizes and low testing speeds, rendering it very difficult to diagnose the flow with adequate spatial resolution and to measure aerodynamic loads with sufficient accuracy. A proof-of-concept pilot of a novel facility specifically intended for low Reynolds number testing has been built and is briefly described. It is based on the use of a high kinematic viscosity test medium, which allows testing superscale models at the correct Reynolds number. The facility is akin to a towing tank, but does not have a free surface with its associated shortcomings. Implementation of the particle image velocimetry (PIV) technique in this type of facility is complicated by the fact that the optics, rather than being stationary as is the case in conventional aerodynamic testing facilities, move in synchronism with the model. Mistracking directly results in velocity-field errors which must be corrected. Methods to do so are described.

This paper concerns the hydrodynamic interactions on a cylindrical particle in non-dilute regime at low Reynolds numbers. The particle moves between two parallel walls with its axis parallel to the boundaries. A numerical finite-volume procedure is implemented and a generalized resistance matrix is built by means of the superposition principle. Three problems are solved: the settling of the particle, the transport of a neutrally and of a non-neutrally buoyant particle in a Poiseuille flow. Concerning sedimentation, the settling velocity is maximal off the symmetry plane and decreases when the confinement increases. The particle rotates in the direction opposite to that of contact rolling. The particle induces a high pressure zone in the front and a low pressure zone in the back, the difference of which is maximal in the symmetry plane. For a neutrally-buoyant particle, the hydrodynamic interactions lead to a velocity lag between the particle and the undisturbed flow. The magnitude of the velocity lag increases with confinement and eccentricity. The angular velocity and pressure difference are opposite to the previous case. For a non-neutrally buoyant particle, three situations are found depending on a dimensionless parameter similar to an inverse Shields number. For its extreme low and high values, the particle is respectively either carried by the flow or settles against it whatever its position. For intermediate values, the particle either settles close to the walls or is dragged by the flow close to the symmetry plane. Similar results are obtained for the angular velocity and the pressure difference. All these results questionthe assumption usually met in particulate transport in which the kinematics of the particle is often supposed to be that of the flow.

We investigate the hydrodynamic interactions between micro-organisms swimming at low Reynolds number. By considering simple model swimmers, and combining analytic and numerical approaches, we investigate the time-averaged flow field around a swimmer. At short distances the swimmer behaves like a pump. At large distances the velocity field depends on whether the swimming stroke is invariant under a combined time-reversal and parity transformation. We then consider two swimmers and find that the interaction between them consists of two parts: a passive term, independent of the motion of the second swimmer, and an active term resulting from the simultaneous swimming action of both swimmers. The swimmer-swimmer interaction is a complicated function of their relative displacement, orientation, and phase, leading to motion that can be attractive, repulsive, or oscillatory.