TY - JOUR

T1 - Reciprocal theorem for the prediction of the normal force induced on a particle translating parallel to an elastic membrane

AU - Daddi-Moussa-Ider, Abdallah

AU - Rallabandi, Bhargav

AU - Gekle, Stephan

AU - Stone, Howard A.

N1 - Funding Information:
A.D.M.I. thanks H.A.S., B.R., and the Complex Fluids Group at the Princeton University for their hospitality during a visit when this work was initiated. H.A.S. and B.R. are grateful to the National Foundation of the USA for funding via Grant No. DMS-1614907. S.G. and A.D.M.I. thank the Volkswagen Foundation for funding and acknowledge support from the Elite Study Program Biological Physics. Funding from the DFG (Deutsche Forschungsgemeinschaft) within DA 2107/1-1 (ADMI) is gratefully acknowledged. This work is supported by the COST Action MP1305, supported by COST (European Cooperation in Science and Technology).

PY - 2018/8

Y1 - 2018/8

N2 - 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.

AB - 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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U2 - 10.1103/PhysRevFluids.3.084101

DO - 10.1103/PhysRevFluids.3.084101

M3 - Article

AN - SCOPUS:85052973351

VL - 3

JO - Physical Review Fluids

JF - Physical Review Fluids

SN - 2469-990X

IS - 8

M1 - 084101

ER -