Hydrodynamic interaction of two unequal-sized spheres in a slightly rarefied gas: resistance and mobility functions

Ying, Ruoxian; Peters, Michael

doi:10.1017/s0022112089002612

Cited by 18 publications

(8 citation statements)

References 15 publications

(20 reference statements)

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“…However, results for the mobility were not available previously over the entire range of particle separations. Ying & Peters (1989) have provided results for / l o~a e , with E -O(1). These results together with ours for e + l with af/Ao -O( 1 ) enable the calculation of mobility for all separations.…”

Section: Discussionmentioning

confidence: 99%

Non-continuum lubrication flows between particles colliding in a gas

Sundararajakumar

Koch

1996

J. Fluid Mech.

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Solid-body collisions between smooth particles in a gas would not occur if the lubrication force for a continuum incompressible fluid were to hold at all particle separations. When the gap between the particles is of the order of the mean free path λ0 of the gas, the discrete molecular nature of the gas becomes important. For particles of radii a smaller than about 50 μm colliding in air at a relative velocity comparable to their terminal velocity, the effects of compressibility of the gas in the gap are not important.The nature of the flow in the gap depends on the relative magnitudes of the minimum gap thickness h0 ≡ aε, the mean-free path λ0, and the distance aε1/2 over which the effects of curvature become important. The slip-flow regime, a[Gt ]λ0, was analysed by Hocking (1973) using the Maxwell slip boundary condition at the particle surface. To find the lubrication force in the transition regime (aε ∼ O(λ0)), we use the results of Cercignani & Daneri (1963) for the flux as a function of the pressure gradient in a Poiseuille channel flow. When aε[Lt ]λ0[Lt ]aε1/2, one might expect the local flow in the gap to be governed by Knudsen diffusion. However, an attempt to calculate the Knudsen diffusivity between parallel plates leads to a logarithmic divergence, which is cut off by intermolecular collisions, and the flux is therefore proportional to h0c log(λ0/h0), where c is the mean molecular speed. The non-continuum lubrication force is shown to have a weak, log - log divergence as the particle separation goes to zero. As a result, the energy dissipated in the collision is finite. In the limit of large particle inertia, the energy dissipated is 6πμU0a2(log h0/λ0 – 1.28), where 2U0 is the relative velocity of the particles.When λ0[Gt ]aε1/2, we have a free molecular flow in the gap. In this case, owing to the curvature of the particles, the flux versus pressure gradient relation is non-local. We analyse the free molecular flow between two cylinders and obtain scalings for the lubrication force.

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Section: Discussionmentioning

confidence: 99%

Non-continuum lubrication flows between particles colliding in a gas

Sundararajakumar

Koch

1996

J. Fluid Mech.

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“…(29) and K in Eq. (38) with the Oseen-Burgers tensor in real space J 1 in Eq. (78), whose explicit from is given by…”

Section: Appendix A: Lamb's General Solutionmentioning

confidence: 99%

Targeting Transport Properties in Nanofluidics: Hydrodynamic Interaction Among Slip Surface Nanoparticles in Solution

Ichiki¹,

Kobryn²,

Kovalenko³

2008

Jnl of Comp & Theo Nano

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Hydrodynamic interaction among rigid nanoparticles in laminar flow with Navier's slip boundary condition on the nanoparticles' surfaces is formulated. The single-particle problem under the general linear flow is solved in terms of the Lamb general solution, and the velocity field exerted by the slip particle is expressed in terms of the multipole expansion in the force moments. Thereby, the mobility matrix for a many-body system is constructed with Faxén's laws for the force, torque and the stresslet, and is extended to periodic systems by the Ewald summation technique. Using this formulation, the Stokesian dynamics method is generalized to slip particles with arbitrary slip length. The method is applied to a system in an unbounded fluid and to a system with periodic boundary conditions. The mobility problem with constant force for the former and sedimentation velocity (drag coefficient) and spin and shear viscosities for the latter are solved. A comparison is made with the existing results for no-slip particles. According to the surface slip, the reductions of friction (drag force), spin and shear viscosities are observed for the problems with the applied force, torque, and shear, respectively. In particular, we show that just changing the slip properties of the nanoparticle surface, one can control the drag force within an order of magnitude. The slip-length dependences of the drag coefficient and other rheological properties are useful for rational design of nanofluidic devices, including controllable manipulation and separation of large biomolecules in nanofluidic channels.

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“…Points on the plane are parameterized by r p = Xe x + Ye y and points on the sphere by r s = De z + Rn s (θ, φ), where n p = e z and n s (θ, φ) = e x sin θ cos φ + e y sin θ sin φ + e z cos θ are the unit normals on the plane and sphere, respectively. & Pozrikidis 2008) or following the singular perturbation scheme of Sone & Onishi (1978) in Ying & Peters (1989, 1991. In this case a much weaker logarithmic divergence of the force on the width of the gap is found on close approach, but these results are limited to non-overlapping Knudsen layers, H, in the gap.…”

Section: Introductionmentioning

confidence: 99%