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

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 mechanism 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 induced 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 finitesized membrane exhibiting a resistance toward both shear and bending. Our analytical approach proceeds through the application of 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 cut-off length beyond which the bending forces become subdominant to an external body force. F H ⊥,B = −ηaV P Λ B 2π 0 b ⋆ 0 σ ⋆ zz ∂u 0 ⋆ z ∂x ⋆ + a ⋆σ⋆ zr u 0 ⋆ z ∂v 0 ⋆ r ∂z ⋆ r dr dφ ,

Rotation of a submerged finite cylinder moving down a soft incline

“…For symmetrical objects, the results show that elastic deformations lead to a non-symmetric pressure field and to the emergence of a friction-reducing lift force. Of particular importance in nature are cases of freely moving particles close to soft surfaces as seen in flows of cells in vessels [25] or microfluidic devices [26,27], the mobility of suspended or falling objects near elastic membranes [28][29][30][31], the behavior of vesicles near walls [32] or the collisions between suspended particles [33]. It is only very recently that a theoretical work [34] addressed freely moving objects and showed how a free falling cylinder can sediment, slide and spin along a soft incline.…”

Direct Measurement of the Elastohydrodynamic Lift Force at the Nanoscale

“…In such a soft-matter context, a novel elastohydrodynamic lift force was theoretically introduced 16 , and further explored and generalized through: the motion of vesicles 17 , a zoology of elastic media and geometries [18][19][20] , added effects of intermolecular interactions 21 , self-similar properties of the soft lubricated contact 22 , the inertial-like motion of a free particle 23 , viscoelastic effects 24 , an equivalent emerging torque 25 , and the case of membranes 26,27 . Essentially, any symmetric rigid object moving within a viscous fluid and along a nearby soft surface is repelled from the latter by a normal force.…”

Multi-species dynamical density functional theory for microswimmers: Derivation, orientational ordering, trapping potentials, and shear cells

“…[106][107][108] The theory is based on a discrete force-dipole minimal microswimmer model, which has already been used successfully in several previous works. 58,94,[106][107][108] We then applied our theory to three illustrative example situations of planar swimmer configurations inside a three-dimensional bulk fluid.…”

“…6 For example, the complexity can arise from steric confinement of the swimmers [48][49][50][51][52][53] or be induced by a complex dispersion medium. [54][55][56][57][58][59] Here, we consider the complementing case of complexity caused by interactions between different swimmer species, as can occur in a diverse set of situations.…”