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Reciprocal theorem for the prediction of the normal force induced on a particle translating parallel to an elastic membrane

Abdallah Daddi-Moussa-Ider1,2,*, Bhargav Rallabandi3,4, Stephan Gekle2, and Howard A. Stone3,†

  • 1Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, Düsseldorf 40225, Germany
  • 2Biofluid Simulation and Modeling, Theoretische Physik, Universität Bayreuth, Universitätsstraße 30, Bayreuth 95440, Germany
  • 3Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, New Jersey 08544, USA
  • 4Department of Mechanical Engineering, University of California, Riverside, California 92521, USA

  • *ider@thphy.uni-duesseldorf.de
  • †hastone@princeton.edu
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Phys. Rev. Fluids 3, 084101 – Published 17 August, 2018

DOI: https://doi.org/10.1103/PhysRevFluids.3.084101

Abstract

When an elastic object is dragged through a viscous fluid tangent to a rigid boundary, it experiences a lift force perpendicular to its direction of motion. An analogous lift occurs when a rigid symmetric object translates parallel to an elastic interface or a soft substrate. The induced lift force is attributed to an elastohydrodynamic coupling that arises from the breaking of the flow reversal symmetry produced by the elastic deformation of the translating object or the interface. Here we derive explicit analytical expressions for the quasi-steady-state lift force exerted on a rigid spherical particle translating parallel to a finite-sized membrane exhibiting a resistance toward both shear and bending. Our analytical approach applies the Lorentz reciprocal theorem so as to obtain the solution of the flow problem using a perturbation technique for small deformations of the membrane. We find that the shear-related contribution to the normal force leads to an attractive interaction between the particle and the membrane. This emerging attractive force decreases quadratically with the system size to eventually vanish in the limit of an infinitely extended membrane. In contrast, membrane bending leads to a repulsive interaction whose effect becomes more pronounced upon increasing the system size, where the lift force is found to diverge logarithmically for an infinitely large membrane. The unphysical divergence of the bending-induced lift force can be rendered finite by regularizing the solution with a cutoff length beyond which the bending forces become subdominant to an external body force.

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Phys. Rev. Fluids 3, 084101– Published 17 August, 2018
  • Received 19 April 2018
Article part of CHORUS
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