Mediated interactions between rigid inclusions in two-dimensional elastic or fluid films

Richter, Sonja K.; Menzel, Andreas M.

doi:10.1103/physreve.105.014609

Cited by 5 publications

(8 citation statements)

References 42 publications

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“…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.…”

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confidence: 60%

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Thin elastic films and membranes under rectangular confinement

Sprenger,

Reinken,

Richter

et al. 2024

EPL

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We address the deformations within a thin elastic film or membrane in a two-dimensional rectangular confinement. To this end, analytical considerations of the Navier-Cauchy equations describing linear elasticity are performed in the presence of a localized force center, that is, a corresponding Green's function is determined, under no-slip conditions at the clamped boundaries. Specifically, we find resulting displacement fields for different positions of the force center. It turns out that clamping regularizes the solution when compared to an infinitely extended system. Increasing compressibility renders the displacement field more homogeneous under the given confinement. Moreover, varying aspect ratios of the rectangular confining frame qualitatively affect the symmetry and appearance of the displacement field. Our results are confirmed by comparison with corresponding finite-element simulations.

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confidence: 60%

“…The logarithmic divergence does not occur if the net forces on the system sum up to zero. 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].…”

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confidence: 99%

Thin elastic films and membranes under rectangular confinement

Sprenger,

Reinken,

Richter

et al. 2024

EPL

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“…Previously, it was argued that this signals a breakdown of the linear theory and that nonlinear contributions then become important [18]. In a previous work, we have taken a different point of view and demonstrated that the logarithmic divergence does not appear, if the net forces on the system sum up to zero [19]. An example are inclusions that pairwise exert forces onto each other according to Newton's third law, which they then transmit to the fluid or elastic two-dimensional environment, implying vanishing net force.…”

Section: Introductionmentioning

confidence: 81%

“…As described above, the logarithmic divergence cancels for force configurations in which no net force is exerted on the medium [19]. However, under free-slip boundary conditions, the no-net-force condition when combining real and image forces only applies if the real force is oriented perpendicular to the boundary, meaning that F = F ẑ.…”

Section: Free-slip Boundary Conditionsmentioning

confidence: 99%

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

2022

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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.

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“…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].…”

Section: Resistance Coefficientsmentioning

confidence: 99%

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

Daddi-Moussa-Ider,

Tjhung,

Richter

et al. 2024

J. Phys.: Condens. Matter

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To make progress towards the development of a theory on the motion of inclusions in thin structured films and membranes, we here consider as an initial step a circular disk in a two-dimensional, uniaxially anisotropic fluid layer. We assume overdamped dynamics, incompressibility of the fluid, and global alignment of the axis of anisotropy. Motion within this layer is affected by additional linear friction with the environment, for instance, a supporting substrate. We investigate the induced flows in the fluid when the disk is translated parallel or perpendicular to the direction of anisotropy. Moreover, expressions for corresponding mobilities and resistance coefficients of the disk are derived. Our results are obtained within the framework of a perturbative expansion in the parameters that quantify the anisotropy of the fluid. Good agreement is found for moderate anisotropy when compared to associated results from finite-element simulations. At pronounced anisotropy, the induced flow fields are still predicted qualitatively correctly by the perturbative theory, although quantitative deviations arise. We hope to stimulate with our investigations corresponding experimental analyses, for example, concerning fluid flows in anisotropic thin films on uniaxially rubbed supporting substrates.

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Mediated interactions between rigid inclusions in two-dimensional elastic or fluid films

Cited by 5 publications

References 42 publications

Thin elastic films and membranes under rectangular confinement

Thin elastic films and membranes under rectangular confinement

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

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

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