Trajectory analysis for non-Brownian inertial suspensions in simple shear flow

Subramanian, Ganesh; Brady, John F.

doi:10.1017/s0022112006000255

Cited by 27 publications

(17 citation statements)

References 53 publications

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“…The above indeterminacy associated with the pair probability on closed pair-particle pathlines prevents a straightforward determination of the stress tensor, at O(φ 2 ), for a range of linear flows that includes the rheologically important case of simple shear flow (Kao et al 1977). Any calculation of the O(φ 2 ) contribution in such flows must therefore appeal to physics outside of Stokesian hydrodynamics in the dilute regime, such as three-particle interactions, weak particle (Subramanian & Brady 2006) or fluid inertia (Morris, Yan & Koplik 2007) or weak Brownian motion (Morris & Brady 1997).…”

Section: Introductionmentioning

confidence: 99%

The effect of inertia on the orientation dynamics of anisotropic particles in simple shear flow

2016

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It is well known that, under inertialess conditions, the orientation vector of a torque-free neutrally buoyant spheroid in an ambient simple shear flow rotates along so-called Jeffery orbits, a one-parameter family of closed orbits on the unit sphere centred around the direction of the ambient vorticity (Jeffery, Proc. R. Soc. Lond. A, vol. 102, 1922, pp. 161-179). We characterize analytically the irreversible drift in the orientation of such torque-free spheroidal particles of an arbitrary aspect ratio, across Jeffery orbits, that arises due to weak inertial effects. The analysis is valid in the limit Re, St 1, where Re = (γ L 2 ρ f )/µ and St = (γ L 2 ρ p )/µ are the Reynolds and Stokes numbers, which, respectively, measure the importance of fluid inertial forces and particle inertia in relation to viscous forces at the particle scale. Here, L is the semimajor axis of the spheroid, ρ p and ρ f are the particle and fluid densities, γ is the ambient shear rate, and µ is the suspending fluid viscosity. A reciprocal theorem formulation is used to obtain the contributions to the drift due to particle and fluid inertia, the latter in terms of a volume integral over the entire fluid domain. The resulting drifts in orientation at O(Re) and O(St) are evaluated, as a function of the particle aspect ratio, for both prolate and oblate spheroids using a vector spheroidal harmonics formalism. It is found that particle inertia, at O(St), causes a prolate spheroid to drift towards an eventual tumbling motion in the flow-gradient plane. Oblate spheroids, on account of the O(St) drift, move in the opposite direction, approaching a steady spinning motion about the ambient vorticity axis. The period of rotation in the spinning mode must remain unaltered to all orders in St. For the tumbling mode, the period remains unaltered at O(St). At O(St 2 ), however, particle inertia speeds up the rotation of prolate spheroids. The O(Re) drift due to fluid inertia drives a prolate spheroid towards a tumbling motion in the flow-gradient plane for all initial orientations and for all aspect ratios. Interestingly, for oblate spheroids, there is a bifurcation in the orientation dynamics at a critical aspect ratio of approximately 0.14. Oblate spheroids with aspect ratios greater than this critical value drift in a direction opposite to that for prolate spheroids, and eventually approach a spinning motion about the ambient vorticity axis starting from any initial † Email address for correspondence: sganesh@jncasr.ac.in ‡ Present address: Aerospace Engineering and Mechanics, University of Minnesota, MN, USA. § These authors contributed equally to this work. 632 V. Dabade, N. K. Marath and G. Subramanian orientation. For smaller aspect ratios, a pair of non-trivial repelling orbits emerge from the flow-gradient plane, and divide the unit sphere into distinct basins of orientations that asymptote to the tumbling and spinning modes. With further decrease in the aspect ratio, these repellers move away from the flow-gradient plane, eventually coalesci...

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Section: Introductionmentioning

confidence: 99%

The effect of inertia on the orientation dynamics of anisotropic particles in simple shear flow

2016

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“…This recent work is numerical and we know of no experiments addressing the issue, although the influence of finiteinertial hydrodynamic interactions has been suggested to play a role in experimentally observed formation of chain-like structures in dilute pressure-driven suspension flow in a pipe by Matas et al (2004), a phenomenon replicated in square duct flow simulations (Chun & Ladd 2006). The numerical work noted includes the case † of St > 0 and † Subramanian & Brady (2006) used St * = 2St/9 and the same Re as presented in this work, but we prefer the definition of Stokes number given here as it indicates excess particle inertia if St exceeds Re, whereas the equivalent physically significant condition is found when St * > 2Re/9.…”

Section: Introductionmentioning

confidence: 99%

Pair-sphere trajectories in finite-Reynolds-number shear flow

Kulkarni

Morris

2008

J. Fluid Mech.

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The pair trajectories of neutrally buoyant rigid spheres immersed in finite-inertia simple-shear flow are described. The trajectories are obtained using the lattice-Boltzmann method to solve the fluid motion, with Newtonian dynamics describing the sphere motions. The inertia is characterized by the shear-flow Reynolds number ${\it Re} \,{=}\,\rho\dot{\gamma}a^2/\mu$, where μ and ρ are the viscosity and density of the fluid respectively, $\dot{\gamma}$ is the shear rate and a is the radius of the larger of the pair of spheres in the case of unequal sizes; the majority of results presented are for pairs of equal radii. Reynolds numbers of 0 ≤ Re ≤ 1 are considered with a focus on inertia at Re = O(0.1). At finite inertia, the topology of the pair trajectories is altered from that predicted at Re = 0, as closed trajectories found in Stokes flow vanish and two new forms of trajectories are observed. These include spiralling and reversing trajectories in addition to largely undisturbed open trajectories. For Re = O(0.1), the limits of the various regions in pair space yielding open, reversing and spiralling trajectories are roughly defined.

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“…where we have introduced a parameter ∈ (0, 1) (Subramanian & Brady 2006). In (C 1), the subscripts SS and dpT denote slender straight-swimmers and tumbling force-dipole swimmers; here, ζ SS (k, p 1 , p 2 ) and ζ dpT (k 1 , p 1 , p 2 ), respectively, represent the factors multiplying δ(k + k ) in (3.10) and (C 3) (see below).…”

Section: Resultsmentioning

confidence: 99%

“…We use the additive method of constructing a uniformly valid approximation of (C 8) at leading order in (Van Dyke 1964; Subramanian & Brady 2006). To do so we simplify (C 8) in the inner ( and ), outer ( and ) and matching region ( and ) to construct the leading-order -independent form of .…”

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confidence: 99%