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Abstract We consider the incompressible 3D Navier–Stokes equations subject to a shear induced by noisy movement of part of the boundary. The effect of the noise is quantified by upper bounds on the first two moments of the dissipation rate. The expected value estimate is consistent with the Kolmogorov dissipation law, recovering an upper bound as in (Doering and Constantin 1992 Phys. Rev. Lett. 69 1648) for the deterministic case. The movement of the boundary is given by an Ornstein–Uhlenbeck process; a potential for over-dissipation is noted if the Ornstein–Uhlenbeck process were replaced by the Wiener process.
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Exact solution for the Anisotropic Ornstein–Uhlenbeck process
Active-Matter models commonly consider particles with overdamped dynamics subject to a force (speed) with constant modulus and random direction. Some models also include random noise in particle displacement (a Wiener process), resulting in diffusive motion at short time scales. On the other hand, Ornstein–Uhlenbeck processes apply Langevin dynamics to the particles’ velocity and predict motion that is not diffusive at short time scales. Experiments show that migrating cells have gradually varying speeds at intermediate and long time scales, with short-time diffusive behavior. While Ornstein–Uhlenbeck processes can describe the moderate-and long-time speed variation, Active-Matter models for over-damped particles can explain the short-time diffusive behavior. Isotropic models cannot explain both regimes, because short-time diffusion renders instantaneous velocity ill-defined, and prevents the use of dynamical equations that require velocity time-derivatives. On the other hand, both models correctly describe some of the different temporal regimes seen in migrating biological cells and must, in the appropriate limit, yield the same observable predictions. Here we propose and solve analytically an Anisotropic Ornstein–Uhlenbeck process for polarized particles, with Langevin dynamics governing the particle’s movement in the polarization direction and a Wiener process governing displacement in the orthogonal direction. Our characterization provides a theoretically robust way to compare movement in dimensionless simulations to movement in experiments in which measurements have meaningful space and time units. We also propose an approach to deal with inevitable finite-precision effects in experiments and simulations.
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- Award ID(s):
- 1720625
- PAR ID:
- 10478025
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Physica A: Statistical Mechanics and its Applications
- Volume:
- 587
- Issue:
- C
- ISSN:
- 0378-4371
- Page Range / eLocation ID:
- 126526
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.physa.2021.126526
