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Semi-flexible filaments interacting with molecular motors and immersed in rheologically complex and viscoelastic media constitute a common motif in biology. Synthetic mimics of filament-motor systems also feature active or field-activated filaments. A feature common to these active assemblies is the spontaneous emergence of stable oscillations as a collective dynamic response. In nature, the frequency of these emergent oscillations is seen to depend strongly on the viscoelastic characteristics of the ambient medium. Motivated by these observations, we study the instabilities and dynamics of a minimal filament-motor system immersed in model viscoelastic fluids. Using a combination of linear stability analysis and full non-linear numerical solutions, we identify steady states, test the linear stability of these states, derive analytical stability boundaries, and investigate emergent oscillatory solutions. We show that the interplay between motor activity, filament and motor elasticity, and fluid viscoelasticity allows for stable oscillations or limit cycles to bifurcate from steady states. When the ambient fluid is Newtonian, frequencies are controlled by motor kinetics at low viscosities, but decay monotonically with viscosity at high viscosities. In viscoelastic fluids that have the same viscosity as the Newtonian fluid, but additionally allow for elastic energy storage, emergent limit cycles are associated with higher frequencies. The increase in frequency depends on the competition between fluid relaxation time-scales and time-scales associated with motor binding and unbinding. Our results suggest that both the stability and oscillatory properties of active systems may be controlled by tailoring the rheological properties and relaxation times of ambient fluidic environments.
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Identification of Characteristic Spatial Scales to Improve the Performance of Analytical Spectral Solutions to the Groundwater Flow Equation
Abstract Analytical solutions for the three‐dimensional groundwater flow equation have been widely used to gain insight about subsurface flow structure and as an alternative to computationally expensive numerical models. Of particular interest are solutions that decompose prescribed hydraulic head boundaries (e.g., Dirichlet boundary condition) into a collection of harmonic functions. Previous studies estimate the frequencies and amplitudes of these harmonics with a least‐square approach where the amplitudes are fitted given a pre‐assigned set of frequencies. In these studies, an ad hoc and structured discretization of the frequency domain is typically used, excluding dominant frequencies while assigning importance to spurious frequencies, with significant consequences for estimating the fluxes and residence times. This study demonstrates the advantages of using a pre‐assigned frequency spectrum that targets the dominant frequencies based on rigorous statistical analysis with predefined significance levels. The new approach is tested for three hydrologic conceptualizations: (a) a synthetic periodic basin, (b) synthetic bedforms, and (c) a natural mountainous watershed. The performance of the frequency spectrum selection is compared with exact analytical or approximate numerical solutions. We found that the new approach better describes the fluxes and residence times for Dirichlet boundaries with well‐defined characteristics spatial scales (e.g., periodic basins and bedforms). For more complex scenarios, such as natural mountainous watersheds, both pre‐assigned frequency spectrums present similar performance. The spectral solutions presented here can play a central role in developing reduced‐complexity models for assessing regional water and solute fluxes within mountain watersheds and hyporheic zones.
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- PAR ID:
- 10370856
- Publisher / Repository:
- DOI PREFIX: 10.1029
- Date Published:
- Journal Name:
- Water Resources Research
- Volume:
- 57
- Issue:
- 12
- ISSN:
- 0043-1397
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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