Resistance functions for spherical particles, droplets and bubbles in cylindrical tubes

Abstract

Numerical computations are performed to evaluate the resistance functions for low Reynolds number flow past spherical particles, droplets and bubbles in cylindrical domains. Spheres of arbitrary radius a and radial position b move with arbitrary velocity U within a cylinder of radius R. The undisturbed fluid may be at rest, or subject to a pressure-driven flow with maximum velocity U0. The spectral boundary element method is employed to compute the resistance force for torque-free bodies in three cases: rigid solids, fluid droplets with viscosity ratio λ = 1, and bubbles with viscosity ratio λ = 0. A lubrication theory is developed to predict the limiting resistance of bodies near contact with the cylinder walls. Compact algebraic expressions are developed which accurately represent the numerical data over the entire range of particle positions 0 < b/(R − a) < 1 for all particle sizes in the range 0 < a/R < 0.9. The resistance functions are consistent with known analytical results and are presented in a form suitable for further studies of particle migration in cylindrical vessels.

Membrane rejection of nonspherical particles: Modeling and experiment

“…As an initial approximation for G(a, b, w), expressions developed by Higdon and Muldowney 16 for spherical particles (i.e., for G(a, b)) in a cylindrical domain were incorporated into Eq. 2.…”

“…Ennis et al 15 used a Pad e approximation to combine the small particle expression developed by Brenner and Gaydos 14 with a large particle expression developed by Bungay and Brenner. 13 Numerical simulations were carried out by Higdon and Muldowney 16 who also fit their results to provide an analytical expression for G(X). Dechadilok and Deen 17 used this expression to determine radially averaged hindrance factors as a function of relative particle size which were in excellent agreement with the expression developed by Ennis et al 15 Anderson 11 theoretically examined the transport of capsule shaped particles in a cylindrical pore by considering only configurational effects (i.e., G(X, w) 5 1 was assumed for all particle positions and orientations).…”

“…Specifically, we have determined G(X, w) using two different approaches. As a first approximation, the results of Higdon and Muldowny 16 describing G(X) for spherical particles were incorporated into the Anderson model 11 that considers the steric limitations for a capsule in a pore. Because only one size parameter characterizes the sphere geometry, we considered different size parameters when using the sphere model to quantify the hydrodynamic interactions for a capsule: the diameter of a sphere with volume equivalent to the capsule volume, the diameter of a sphere equal to the capsule length and the diameter of a sphere equal to the capsule diameter.…”

Computation of Stokes Flow due to the Motion or Presence of a Particle in a Tube

“…Asymptotic results for small particles and numerical results for arbitrarily sized particles positioned along the tube axis in axisymmetric flow were presented by Happel and Byrne [1], Haberman and Sayre [2], and Wang and Skalak [3]. Asymptotic results for particles translating off the tube axis in non-axisymmetric flow were presented by Greenstein and Happel [4] and Tözeren [5,6], and numerical results based on finite-element and boundary-element methods were presented by Tözeren [7], Graham et al [8], Ingber [9] Mondy et al [10], and Higdon and Muldowney [11]. The complementary problem of flow due to particle rotation was also addressed by several authors for axisymmetric and non-axisymmetric flow, as reviewed by Zheng et al [12].…”

“…Higdon and Muldowney [11] computed resistance functions for spherical particles in cylindrical tubes using a highly accurate spectral boundary-element method, and deduced p as part of the solution by imposing a constraint on the axial flow rate. The authors argued that using the traction boundary condition involving p instead of the velocity boundary condition at the tube caps minimizes the effect of domain truncation.…”