Elastic displacements and viscous flows in wedge-shaped geometries with a straight edge: Green's functions for perpendicular forces
arXiv:2504.10239 · doi:10.1007/s10659-025-10146-4
Abstract
Edges are abundant when elastic solids glide in guiding rails or fluids are contained in vessels. We here address induced displacements in elastic solids or small-scale flows in viscous fluids in the vicinity of one such edge. For this purpose, we solve the governing elasticity equations for linearly elastic, potentially compressible solids, as well as the low-Reynolds-number flow equations for incompressible fluids. Technically speaking, we derive the associated Green's functions under confinement by two planar boundaries that meet at a straight edge. The two boundaries both feature no-slip or free-slip conditions, or one of these two conditions per boundary. Previously, we solved the simpler case of the force being oriented parallel to the straight edge. Here, we complement this solution by the more challenging case of the force pointing into a direction perpendicular to the edge. Together, these two cases provide the general solution. Specific situations in which our analysis may find application in terms of quantitative theoretical descriptions are particle motion in confined colloidal suspensions, dynamics of active microswimmers near edges, or actuated distortions of elastic materials due to activated contained functionalized particles.
arXiv commentTo appear in J. Elast
References in corpus (23)
- The hydrodynamics of swimming microorganisms
- Physics of Microswimmers - Single Particle Motion and Collective Behavior
- Bacterial hydrodynamics
- Emergent behavior in active colloids
- Brownian motion near a partial-slip boundary: A local probe of the no-slip condition
- Hydrodynamics of Micro-swimmers in Films
- Buckling of paramagnetic chains in soft gels
- Forces on rigid inclusions in elastic media and resulting matrix-mediated interactions
- State diagram of a three-sphere microswimmer in a channel
- Magnetostriction in magnetic gels and elastomers as a function of the internal structure and particle distribution
- Towards an analytical description of active microswimmers in clean and in surfactant-covered drops
- Hydrodynamic mobility of a solid particle nearby a spherical elastic membrane. I. Axisymmetric motion
- Creeping motion of a solid particle inside a spherical elastic cavity
- Hydrodynamic mobility of a solid particle nearby a spherical elastic membrane. II. Asymmetric motion
- Force-induced elastic matrix-mediated interactions in the presence of a rigid wall
- Displacement field around a rigid sphere in a compressible elastic environment, corresponding higher-order Faxén relations, as well as higher-order displaceability and rotateability matrices
- Towards a soft magnetoelastic twist actuator
- Dynamics of a droplet driven by an internal active device
- Creeping motion of a solid particle inside a spherical elastic cavity. II. Asymmetric motion
- Magnetic elastomers as specific soft actuators -- predicting particular modes of deformation from selected configurations of magnetizable inclusions
- Alignment-Induced Self-Organization of Autonomously Steering Microswimmers: Turbulence, Vortices, and Jets
- Migration and deformation of a droplet enclosing an active particle
- Internal sites of actuation and activation in thin elastic films and membranes of finite thickness