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- ItemOn the Stokes-Type Resolvent Problem Associated with Time-Periodic Flow Around a Rotating Obstacle(Dordrecht [u.a.] : Springer Science + Business Media B.V, 2022) Eiter, ThomasConsider the resolvent problem associated with the linearized viscous flow around a rotating body. Within a setting of classical Sobolev spaces, this problem is not well posed on the whole imaginary axis. Therefore, a framework of homogeneous Sobolev spaces is introduced where existence of a unique solution can be guaranteed for every purely imaginary resolvent parameter. For this purpose, the problem is reduced to an auxiliary problem, which is studied by means of Fourier analytic tools in a group setting. In the end, uniform resolvent estimates can be derived, which lead to the existence of solutions to the associated time-periodic linear problem.
- ItemA Light-Driven Microgel Rotor(Weinheim : Wiley-VCH, 2019) Zhang, Hang; Koens, Lyndon; Lauga, Eric; Mourran, Ahmed; Möller, MartinThe current understanding of motility through body shape deformation of micro-organisms and the knowledge of fluid flows at the microscale provides ample examples for mimicry and design of soft microrobots. In this work, a 2D spiral is presented that is capable of rotating by non-reciprocal curling deformations. The body of the microswimmer is a ribbon consisting of a thermoresponsive hydrogel bilayer with embedded plasmonic gold nanorods. Such a system allows fast local photothermal heating and nonreciprocal bending deformation of the hydrogel bilayer under nonequilibrium conditions. It is shown that the spiral acts as a spring capable of large deformations thanks to its low stiffness, which is tunable by the swelling degree of the hydrogel and the temperature. Tethering the ribbon to a freely rotating microsphere enables rotational motion of the spiral by stroboscopic irradiation. The efficiency of the rotor is estimated using resistive force theory for Stokes flow. This research demonstrates microscopic locomotion by the shape change of a spiral and may find applications in the field of microfluidics, or soft microrobotics.
- ItemStokes flows under random boundary velocity excitations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Sabelfeld, KarlA viscous Stokes flow over a disc under random fluctuations of the velocity on the boundary is studied. We give exact Karhunen-Loève (K-L) expansions for the velocity components, pressure, stress, and vorticity, and the series representations for the corresponding correlation tensors. Both the white noise fluctuations, and general homogeneous random excitations of the velocities prescribed on the boundary are studied. We analyze the decay of correlation functions in angular and radial directions, both for exterior and interior Stokes problems. Numerical experiments show the fast convergence of the K-L expansions. The results indicate that ignoring the boundary condition uncertainty dramatically underestimates the variance of the velocity and pressure in the interior/exterior of the domain.
- ItemOn the Stokes-type resolvent problem associated with time-periodic flow around a rotating obstacle(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Eiter, ThomasConsider the resolvent problem associated with the linearized viscous flow around a rotating body. Within a setting of classical Sobolev spaces, this problem is not well posed on the whole imaginary axis. Therefore, a framework of homogeneous Sobolev spaces is introduced where existence of a unique solution can be guaranteed for every purely imaginary resolvent parameter. For this purpose, the problem is reduced to an auxiliary problem, which is studied by means of Fourier analytic tools in a group setting. In the end, uniform resolvent estimates can be derived, which lead to the existence of solutions to the associated time-periodic linear problem.