Instytut Podstawowych Problemów Techniki

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We analyze the three-dimensional (3D) buckling of an elastic filament in a shear flow of a viscous fluid at low Reynolds number and high Péclet number. We apply the Euler-Bernoulli beam (elastica) theoretical model. We show the universal character of the full 3D spectral problem for a small perturbation of a thin filament from a straight position of arbitrary orientation. We use the eigenvalues and eigenfunctions for the linearized elastica equation in the shear plane, found earlier by Liu et al. [Phys. Rev. Fluids 9, 014101 (2024)] with the Chebyshev spectral collocation method, to solve the full 3D eigenproblem. We provide a simple analytic approximation of the eigenfunctions, represented as Gaussian wave packets. As the main result of the paper, we derive the square-root dependence of the eigenfunction wave number on the parameter χ˜ = −η sin 2φ sin2 θ, where

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The long-time behavior of highly elastic fibers in a shear flow is investigated experimentally and numerically. Characteristic attractors of the dynamics are found. It is shown that for a small ratio of bending to hydrodynamic forces, most fibers form a spinning elongated double helix, performing an effective Jeffery orbit very close to the vorticity direction. Recognition of these oriented shapes, and how they form in time, may prove useful in the future for understanding the time history of complex microstructures in fluid flows and considering processing steps for their synthesis.

Słowa kluczowe:

Stokes equations, shear flow, elastic fibers

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We present the spectral analysis of three-dimensional dynamics of an elastic filament in a shear flow of a viscous fluid at a low Reynolds number in the absence of Brownian motion. The elastica model is used. The fiber initially is almost straight at an arbitrary orientation, with small perpendicular perturbations in the shear plane and out-of-plane. To analyze the stability of both perturbations, equations for the eigenvalues and eigenfunctions are derived and solved by the Chebyshev spectral collocation method. It is shown that their crucial features are the same as in the case of the two-dimensional elastica dynamics in shear flow [Becker and Shelley, Phys. Rev. Lett. 87, 198301 (2001)] and the three-dimensional elastica dynamics in the compressional flow [Chakrabarti et al., Nat. Phys. 16, 689 (2020)]. We find a similar dependence of the buckled shapes on the ratio of bending to hydrodynamic forces as in the simulations for elastic fibers of a nonzero thickness [Słowicka et al., New J. Phys. 24, 013013 (2022)].

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Three-dimensional dynamics of flexible fibers in shear flow are studied numerically, with a qualitative comparison to experiments. Initially, the fibers are straight, with different orientations with respect to the flow. By changing the rotation speed of a shear rheometer, we change the ratio A of bending to shear forces. We observe fibers in the flow-vorticity plane, which gives insight into the motion out of the shear plane. The numerical simulations of moderately flexible fibers show that they rotate along effective Jeffery orbits, and therefore the fiber orientation rapidly becomes very close to the flow-vorticity plane, on average close to the flow direction, and the fiber remains in an almost straight configuration for a long time. This ``ordering'' of fibers is temporary since they alternately bend and straighten out while tumbling. We observe numerically and experimentally that if the fibers are initially in the compressional region of the shear flow, they can undergo a compressional buckling, with a pronounced deformation of shape along their whole length during a short time, which is in contrast to the typical local bending that originates over a long time from the fiber ends. We identify differences between local and compressional bending and discuss their competition, which depends on the initial orientation of the fiber and the bending stiffness ratio A. There are two main finding. First, the compressional buckling is limited to a certain small range of the initial orientations, excluding those from the flow-vorticity plane. Second, since fibers straighten out in the flow-vorticity plane while tumbling, the compressional buckling is transient - it does not appear for times longer than 1/4 of the Jeffery period. For larger times, bending of fibers is always driven by their ends.

Słowa kluczowe:

Stokes flow, flexible fibers, bending, buckling, orientational order

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We investigate qualitatively a uniform non-Brownian sedimenting suspension in a stationary state. As a base of our analysis we take the BBGKY hierarchy derived from the Liouville equation. We then show that assumption of the plasma-like screening relations can cancel some long-range terms in the hierarchy but it does not provide integrable solutions for correlation functions. This suggests breaking the translational symmetry of the system. Therefore a non-uniform structure can develop to suppress velocity fluctuations and make the range of correlations finite.

Słowa kluczowe:

non-Brownian sedimentation, stability, BBGKY hierarchy, hydrodynamic screening, correlation functions, low-Reynolds-number hydrodynamics

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Optical properties of a composite material made of ferromagnetic metal nanoparticles embedded in a dielectric host are studied. We constructed an effective dielectric tensor of the composite material taking into account the orientational distribution of nanoparticle magnetic moments in external magnetic field. A nonlinear dependence of the optical rotation on magnetic field resulting from the reorientation of nanoparticles is demonstrated. The theoretical findings were applied to the magneto-optical experimental data of cobalt ferromagnetic nanoparticles embedded in a dielectric liquid host. The dependence of the Faraday rotation on Co-based ferromagnetic nanoparticles was measured as a function of the external magnetic field, varying the size of nanoparticles and the wavelength of light. The proposed approach enables quantitative determination of the magnetic moment and the plasma frequency of a single nanoparticle, and from this the size of the nonmagnetic shell of magnetic nanoparticles.

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In an open competition, students were to determine the mass of a metal cylinder hanging on a spring inside a transparent enclosure. With the time for experiments limited to 24 h due to the unexpectedly large number of participants, a few surprisingly accurate results were submitted, the best of them differing by no more than 0.5% from the true value with a relative uncertainty of less than 1%.

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We report the theoretical study of the optical response of a periodically modulated two-dimensional electron gas. The density of states is calculated within the first order of the perturbation theory and the effects of the short-range disorder are explained and discussed. We demonstrate that the magnetic field values corresponding to the characteristic narrowing of the density of states width are given by the zeros of the subsequent Laguerre polynomials. The observed increase of the density of states at the edges are interpreted as van Hove singularities. The broadening effects are shown to modify and smear out the observed effects with increasing temperature above 2 K. The plasmon dispersion relation is discussed in terms of the random phase approximation. Small changes in plasmon dispersion relation related to the periodic modulation were predicted.

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Słowa kluczowe:

Stokes equations, Euler-Bernoulli beam, elastica, elastic filament, shear flow, buckling

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Streszczenie:

Stokes equations, Euler-Bernoulli beam, elastica, elastic filament, shear flow, buckling

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Słowa kluczowe:

fluid dynamics, Stokes flows, particulate flows, hydrodynamic interactions, bending

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