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Trajectory and distribution of suspended non-Brownian particles moving past a fixed spherical or cylindrical obstacle
- Sumedh R. Risbud, German Drazer
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- Journal:
- Journal of Fluid Mechanics / Volume 714 / 10 January 2013
- Published online by Cambridge University Press:
- 02 January 2013, pp. 213-237
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We investigate the motion of a suspended non-Brownian sphere past a fixed cylindrical or spherical obstacle in the limit of zero Reynolds number for arbitrary particle–obstacle aspect ratios. We consider both a suspended sphere moving in a quiescent fluid under the action of a uniform force as well as a uniform ambient velocity field driving a freely suspended particle. We determine the distribution of particles around a single obstacle and solve for the individual particle trajectories to comment on the transport of dilute suspensions past an array of fixed obstacles. First, we obtain an expression for the probability density function governing the distribution of a dilute suspension of particles around an isolated obstacle, and we show that it is isotropic. We then present an analytical expression – derived using both Eulerian and Lagrangian approaches – for the minimum particle–obstacle separation attained during the motion, as a function of the incoming impact parameter, i.e. the initial offset between the line of motion far from the obstacle and a parallel line that goes through its centre. Further, we derive the asymptotic behaviour for small initial offsets and show that the minimum separation decays exponentially. Finally we use this analytical expression to define an effective hydrodynamic surface roughness based on the net lateral displacement experienced by a suspended sphere moving past an obstacle.
Directional locking and deterministic separation in periodic arrays
- JOELLE FRECHETTE, GERMAN DRAZER
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- Journal:
- Journal of Fluid Mechanics / Volume 627 / 25 May 2009
- Published online by Cambridge University Press:
- 25 May 2009, pp. 379-401
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We investigate the dynamics of a non-Brownian sphere suspended in a quiescent fluid and moving through a periodic array of solid obstacles under the action of a constant external force by means of Stokesian dynamics simulations. We show that in the presence of non-hydrodynamic, short-range interactions between the solid obstacles and the suspended sphere, the moving particle becomes locked into periodic trajectories with an average orientation that coincides with one of the lattice directions and is, in general, different from the direction of the driving force. The locking angle depends on the details of the non-hydrodynamic interactions and could lead to vector separation of different species for certain orientations of the external force. We explicitly show the presence of separation for a mixture of suspended particles with different roughness, moving through a square lattice of spherical obstacles. We also present a dilute model based on the two-particle mobility and resistance functions for the collision between spheres of different sizes. This simple model predicts the separation of particles of different size and also suggests that microdevices that maximize the differences in interaction area between the different particles and the solid obstacles would be more sensitive for size separation based on non-hydrodynamic interactions.
Microstructure and velocity fluctuations in sheared suspensions
- GERMAN DRAZER, JOEL KOPLIK, BORIS KHUSID, ANDREAS ACRIVOS
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- Journal:
- Journal of Fluid Mechanics / Volume 511 / 25 July 2004
- Published online by Cambridge University Press:
- 12 July 2004, pp. 237-263
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The velocity fluctuations present in macroscopically homogeneous suspensions of neutrally buoyant non-Brownian spheres undergoing simple shear flow, and their dependence on the microstructure developed by the suspensions, are investigated in the limit of vanishingly small Reynolds numbers using Stokesian dynamics simulations. We show that, in the dilute limit, the standard deviation of the velocity fluctuations (the so-called suspension temperature) is proportional to the volume fraction, in both the transverse and the flow directions, and that a theoretical prediction, which considers only the hydrodynamic interactions between isolated pairs of spheres, is in good agreement with the numerical results at low concentrations. We also simulate the velocity fluctuations that would result from a random hard-sphere distribution of spheres in simple shear flow, and thereby investigate the effects of the microstructure on the velocity fluctuations. Analogous results are discussed for the fluctuations in the angular velocity of the suspended spheres. In addition, we present the probability density functions for all the linear and angular velocity components, and for three different concentrations, showing a transition from a Gaussian to an exponential and finally to a stretched exponential functional form as the volume fraction is decreased.
The simulations include a non-hydrodynamic repulsive force between the spheres which, although extremely short range, leads to the development of fore–aft asymmetric distributions for large enough volume fractions, if the range of that force is kept unchanged. On the other hand, we show that, although the pair distribution function recovers its fore–aft symmetry in dilute suspensions, it remains anisotropic and that this anisotropy can be accurately predicted theoretically from the two-sphere solution by assuming the complete absence of any permanent doublets of spheres.
We also present a simple correction to the analysis of laser-Doppler velocimetry measurements, which substantially improves the interpretation of these measurements at low volume fractions even though it involves only the angular velocity of a single sphere in the vorticity direction.
Finally, in an Appendix, we show that, in the dilute limit, the whole velocity autocorrelation function can be predicted using again only two-particle encounters.
Deterministic and stochastic behaviour of non-Brownian spheres in sheared suspensions
- GERMAN DRAZER, JOEL KOPLIK, BORIS KHUSID, ANDREAS ACRIVOS
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- Journal:
- Journal of Fluid Mechanics / Volume 460 / 10 June 2002
- Published online by Cambridge University Press:
- 25 June 2002, pp. 307-335
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The dynamics of macroscopically homogeneous sheared suspensions of neutrally buoyant, non-Brownian spheres is investigated in the limit of vanishingly small Reynolds numbers using Stokesian dynamics. We show that the complex dynamics of sheared suspensions can be characterized as a chaotic motion in phase space and determine the dependence of the largest Lyapunov exponent on the volume fraction ϕ. We also offer evidence that the chaotic motion is responsible for the loss of memory in the evolution of the system and demonstrate this loss of correlation in phase space. The loss of memory at the microscopic level of individual particles is also shown in terms of the autocorrelation functions for the two transverse velocity components. Moreover, a negative correlation in the transverse particle velocities is seen to exist at the lower concentrations, an effect which we explain on the basis of the dynamics of two isolated spheres undergoing simple shear. In addition, we calculate the probability distribution function of the transverse velocity fluctuations and observe, with increasing ϕ, a transition from exponential to Gaussian distributions.
The simulations include a non-hydrodynamic repulsive interaction between the spheres which qualitatively models the effects of surface roughness and other irreversible effects, such as residual Brownian displacements, that become particularly important whenever pairs of spheres are nearly touching. We investigate, for very dilute suspensions, the effects of such a non-hydrodynamic interparticle force on the scaling of the particle tracer diffusion coefficients Dy and Dz, respectively, along and normal to the plane of shear, and show that, when this force is very short-ranged, both are proportional to ϕ2 as ϕ → 0. In contrast, when the range of the non-hydrodynamic interaction is increased, we observe a crossover in the dependence of Dy on ϕ, from ϕ2 to ϕ as ϕ → 0. We also estimate that a similar crossover exists for Dz but at a value of ϕ one order of magnitude lower than that which we were able to reach in our simulations.