Effect of boundaries on displacements and motion in two-dimensional fluid or elastic films and membranes

Abstract

Thin fluid or elastic films and membranes are found in nature and technology, for instance, as confinements of living cells or in loudspeakers. When applying a net force, resulting flows in an unbounded two-dimensional incompressible low-Reynolds-number fluid or displacements in a twodimensional linearly elastic solid seem to diverge logarithmically with the distance from the force center, which has led to some debate. Recently, we have demonstrated that such divergences cancel when the total (net) force vanishes. Here, we illustrate that, if a net force is present, the boundaries play a prominent role. Already a single no-slip boundary regulates the flow and displacement fields and leads to their decay to leading order inversely in distance from a force center and the boundary. In other words, it is the boundary that stabilizes the system in this situation, unlike the threedimensional case, where an unbounded medium by itself is able to absorb a net force. We quantify the mobility and displaceability of an inclusion as a function of the distance from the boundary, as well as interactions between different inclusions. In the case of free-slip boundary conditions, a kinked boundary is necessary to achieve stabilization.

Thin elastic films and membranes under rectangular confinement

“…2(a)-(d). A characteristic feature of infinitely extended two-dimensional systems is a logarithmic spatial divergence of the displacement field in response to a pointlike force center [15,18,19]. Remarkably, within the given rectangular no-slip confinement, that is, in the displacement near-field, we observe a clearly logarithmic behavior as well, now regularized through the boundaries.…”

“…This condition is met, for example, by inclusions that pairwise exert forces onto each other according to Newton's third law [18]. Similarly, clamped boundaries prevent overall displacement of the elastic medium [19]. In the latter situation, the boundaries absorb the net force by a corresponding counterforce density.…”

Hydrodynamics of a disk in a thin film of weakly nematic fluid subject to linear friction

“…This behavior is in line with the features observed already for an isotropic fluid in a two-dimensional setting without external damping. In this situation, the magnitude of the resulting velocity field diverges logarithmically with the distance from the position where a net force is imposed onto the fluid [91,92]. It manifests that the infinitely extended twodimensional sheet of fluid is not sufficient to stabilize the flow in that case.…”

“…In contrast to the situation of a three-dimensional bulk system, in which the infinite amount of fluid stabilizes the flow, in the two-dimensional case the whole fluid is set into motion under persistent forcing and a stationary situation does not exist. The flow in two dimensions can be stabilized by adding appropriate counterforces on the fluid so that the whole forcing sums up to zero [91] and/or by introducing regularizing boundaries [92].…”

“…One conceivable reason lies with the nature of two-dimensional fluid flows themselves. Generally, in an isotropic fluid, the Green's function quantifying the flows induced by a locally applied net force diverges logarithmically with the distance from the point of force application [91,92]. It is the linear friction with the environment that regularizes this divergence in Brinkman fluids so that the induced flows decay towards zero with infinite distance [22].…”