Session EE: Suspensions II

Inertial migration of neutrally buoyant particles in Poiseuille flow: An investigation of multiple equilibrium positions.

In Poiseuille flow, a neutrally-buoyant particle migrates to a position that is determined by the balance of forces generated by the gradient of the shear rate and interactions of a particle with the container wall. In a cylindrical geometry, uniformly distributed particles migrate to form a stable ring located at approximately 0.6R, where R is radius of the cylinder. However, a recent experiment shows two interesting observations. First, the suspended particles tend to align near the walls to make linear chains of more or less equally-spaced particles. Second, at high Reynolds numbers (Reynolds number is about 1000), an additional inner ring of particles was observed, when the ratio of particle diameter to cylinder diameter was of the order of 1:10. We have therefore investigated inertial migration of the neutrally buoyant particles by the lattice Boltzmann method in the range of Reynolds numbers from 100 to 1000. Our numerical results show linear trains of particles along the axis of the flow, near the equilibrium positions of single particles. At Reynolds number greater than 700, particles were also seen near the center of the duct. We will present a new mechanism to explain these results by comparing the migration of single particles and rigid dumbbells, Reynolds numbers in the range of 100 to 1000.   

Lattice-Boltzmann method for Stokes flow of Deformable Particles

Predicting the rheology of deformable particles suspended in viscous fluid is important in industrial and biological transport applications. We combine a Stokes lattice-Boltzmann Model (SLBM) and Lattice Spring Model (LSM) to capture the dynamics of particles with deformable membrane with enclosed fluid. We simulate blood flow in a small vessel which has deformable arterial wall and red blood cell (RBC) membrane. We find that the flow is affected by the physical properties of wall, membrane and enclosed fluid and these are very important in the hemodynamic applications. The combination of SLBM-LSM can calculate the mesoscale blood flow close to the endothelial surface layer (ESL) or its subset, the glycocalyx layer. And it is also a strong tool to capture the multiphase flow with large number of deformable particles.   

A ‘Universal’ Scale for Cluster Size Distribution of solid spheres and red blood cells Suspended in Pressure-Driven Flow

The focus of the present work is on clustering of neutrally buoyant solid particles suspended in viscous liquid under purely hydrodynamic interactions in the absence of Brownian motion. The approach is based on direct numerical simulation of the particle motion and interaction, considering the full hydrodynamic forces on the particle and particle-particle interaction. Rigid particles in the shape of sphere and red blood cell are considered. A transition in flow regime occurs at a critical concentration, which is characterized by a fundamental change in cluster size distribution. The post-critical state is characterized by a large cluster dominating the flow. We have found a `universal' scaling relation for the cluster size distribution in the subcritical regime\footnote{E.-J. Ding and C. K. Cyrus, ``Cluster size distribution and scaling for spherical particles and red blood cells in pressure-driven flows at small Reynolds number'' Phys. Rev. Lett. 96 (2006) 204502.} in all the cases considered here. The underlying physics will be presented.   

Spreading fronts in sedimentation of dilute suspension of spheres

When very dilute suspensions settle, a wide diffuse front can form between the clear fluid above and the uniform suspension below. The width of this front grows through two mechanisms - hydrodynamic interactions and polydispersity in the size of the spheres. We explore experimentally and by numerical simulations how these two effects combine.   

Relaxation of Large-Scale Fluctuations in a Well-Stirred Suspension

Long-wavelength density fluctuations in a well-stirred suspension under the influence of gravity induce velocity fluctuations with a magnitude proportional to the linear dimensions of the container. The resulting flow suppresses the density fluctuations and results in a relaxation of the well-stirred suspension to a weakly stratified state with relatively small velocity fluctuations. This process is modelled as Stokes flow of an incompressible, highly viscous fluid with a variable density; the initial density is a realization of a random Gaussian field. The relaxation process is examined through theoretical analysis and numerical simulation. Results concerning the degree of stratification and the decay of the various Fourier components of the flow are presented.   

Stratification and wavenumber selection in the instability of sedimenting spheroids

A dilute suspension of spheroids settling under gravity is unstable to density fluctuations as a result of hydrodynamic interactions. Whereas the original stability analysis of Koch \& Shaqfeh (\textit{JFM} 1989) predicts a maximum growth rate at zero wavenumber, recent experiments (Metzger et al.~\textit{PRL} 2005) and simulations (Saintillan et al.~\textit{JFM} 2006) both suggest that the instability in fact selects a finite wavelength corresponding to a typical cluster size. To elucidate the mechanism for this wavenumber selection, we revisit the linear stability analysis of Koch \& Shaqfeh by including the effects of a stable stratification and of hydrodynamic center-of-mass diffusion. The analysis shows that the growth rates for the concentration fluctuations are damped at low wavenumbers by stratification and at high wavenumbers by diffusion, resulting in a maximum growth rate at an intermediate wavenumber. A scaling for the dependence of the cluster size on the vertical density gradient and on the local volume fraction is inferred. To validate the theory, numerical simulations are performed in stratified suspensions, where good agreement is found with the proposed scaling. In initially homogeneous suspensions, we argue that the process leading to the wavenumber selection may be controlled by the formation of vertical density gradients over the course of sedimentation.   

Motion of flexible cylindrical fibers in shearing and sedimenting flows with inertia

A method for direct simulations of interactions of multi flexible fibers in flows with inertia is developed. The simulation method is based on a lattice Boltzmann equation and a flexible fiber model. A slender solid body is discretized into a chain of cylindrical segments contacting each other at the fiber ends through ball and socket joints. A constraint force is imposed at each joint. To linearize the rotational matrix, quaternion parameters could be expanded in a power series of the length of time step up to a second order. A half leap frog algorithm is modified to ensure the ball and socket joint conditions to be satisfied at each time step. A single and multi fibers in shearing and sedimenting flows are simulated. In a shearing flow, the period increases as Reynolds number increases and flexibility reduces. The effective viscosity increase as fiber volume fraction increases. In a sedimenting flow, a lower Reynolds number fiber will settle down with original orientation, while at a large Reynolds number the fiber will turn its long body perpendicular to the gravity. There is a transition Reynolds number between the two settling states, the transition Reynolds number increases as the fiber aspect ratio increases. A wall effect is observed. Interactions between multi fibers are analyzed using angular distribution functions.   

Short-time diffusivities of suspensions of rigid fibers with hydrodynamic interactions: The influence on the scaling of long-time rotational diffusivities

Brownian dynamics simulations indicate that the scaling behavior of the long-time rotational diffusion of concentrated, isotropic suspensions of rigid fibers depends upon the ratio of short-time rotational and translational velocities.\footnote{Cobb \& Butler, J. Chem. Phys. 2005} However, the previous work used infinite dilution values for the short-time diffusivities, ignoring the effects of concentration. We have calculated short-time diffusivities for concentrated suspensions of rigid fibers which include multibody hydrodynamic interactions between fibers. These improved values of the short-time diffusivities are utilized to calculate the long-time rotational diffusion. We discuss the effect of hydrodynamic interactions on the short-time diffusivities and the consequent scaling behavior of the long-time rotational diffusvity.