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Hydrodynamic force on a sphere normal to an obstacle due to a non-uniform flow

Published online by Cambridge University Press:  04 April 2017

Bhargav Rallabandi Open the ORCID record for Bhargav Rallabandi [Opens in a new window] ,
Sascha Hilgenfeldt  and
Howard A. Stone
Bhargav Rallabandi*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Sascha Hilgenfeldt
Affiliation:
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206 West Green Street, Urbana, IL 61801, USA
Howard A. Stone*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email addresses for correspondence: vbr@princeton.edu, hastone@princeton.edu
†Email addresses for correspondence: vbr@princeton.edu, hastone@princeton.edu
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Abstract

For a small sphere suspended in a background fluid flow near an obstacle, we calculate the hydrodynamic force on the sphere in the direction normal to the boundary of the obstacle. Using the Lorentz reciprocal theorem, we obtain analytical expressions for the normal force in the Stokes flow limit, valid for arbitrary separations of the particle from the obstacle, both for solid obstacles and those with free surfaces. The main effect of the boundary is to produce a normal force proportional to extensional flow gradients in the vicinity of the particle. The strength of this force is greatest when the separation between the surfaces of the particle and the obstacle is small relative to the particle size. While the magnitude of the force weakens for large separations between the sphere and the obstacle (decaying quadratically with separation distance), it can significantly modify Faxén’s law even at modestly large separation distances. In addition, we find a second force contribution due to the curvature of the background flow normal to the obstacle, which is also important when the sphere is close to the obstacle. The results of the theory are of importance to the dynamics of particles in confined geometries, whether bounded by a solid obstacle, the wall of a channel or a gas bubble.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Multiphase and Particle-laden Flows: Particle/fluid flow Complex Fluids: Suspensions
Type
Papers
Information
Journal of Fluid Mechanics , Volume 818 , 10 May 2017 , pp. 407 - 434
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Copyright
© 2017 Cambridge University Press 

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