An official website of the United States government
We propose a novel integral model describing the motion of both flexible and rigid slender fibers in viscous flow and develop a numerical method for simulating dynamics of curved rigid fibers. The model is derived from nonlocal slender body theory (SBT), which approximates flow near the fiber using singular solutions of the Stokes equations integrated along the fiber centerline. In contrast to other models based on (singular) SBT, our model yields a smooth integral kernel which incorporates the (possibly varying) fiber radius naturally. The integral operator is provably negative definite in a nonphysical idealized geometry, as expected from the partial differential equation theory. This is numerically verified in physically relevant geometries. We discuss the convergence and stability of a numerical method for solving the integral equation. The accuracy of the model and method is verified against known models for ellipsoids. Finally, we develop an algorithm for computing dynamics of rigid fibers with complex geometries in the case where the fiber density is much greater than that of the fluid, for example, in turbulent gas-fiber suspensions.
more »
« less
Remarks on Regularized Stokeslets in Slender Body Theory
We remark on the use of regularized Stokeslets in the slender body theory (SBT) approximation of Stokes flow about a thin fiber of radius ϵ>0. Denoting the regularization parameter by δ, we consider regularized SBT based on the most common regularized Stokeslet plus a regularized doublet correction. Given sufficiently smooth force data along the filament, we derive L∞ bounds for the difference between regularized SBT and its classical counterpart in terms of δ, ϵ, and the force data. We show that the regularized and classical expressions for the velocity of the filament itself differ by a term proportional to log(δ/ϵ); in particular, δ=ϵ is necessary to avoid an O(1) discrepancy between the theories. However, the flow at the surface of the fiber differs by an expression proportional to log(1+δ2/ϵ2), and any choice of δ∝ϵ will result in an O(1) discrepancy as ϵ→0. Consequently, the flow around a slender fiber due to regularized SBT does not converge to the solution of the well-posed slender body PDE which classical SBT is known to approximate. Numerics verify this O(1) discrepancy but also indicate that the difference may have little impact in practice.
more »
« less
- Award ID(s):
- 2001959
- PAR ID:
- 10539693
- Publisher / Repository:
- MDPI
- Date Published:
- Journal Name:
- Fluids
- Volume:
- 6
- Issue:
- 8
- ISSN:
- 2311-5521
- Page Range / eLocation ID:
- 283
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
null (Ed.)Abstract We propose a heuristic algorithm to solve the underlying hard problem of the CSIDH cryptosystem (and other isogeny-based cryptosystems using elliptic curves with endomorphism ring isomorphic to an imaginary quadratic order 𝒪). Let Δ = Disc(𝒪) (in CSIDH, Δ = −4 p for p the security parameter). Let 0 < α < 1/2, our algorithm requires: A classical circuit of size 2 O ˜ log ( | Δ | ) 1 − α . $$2^{\tilde{O}\left(\log(|\Delta|)^{1-\alpha}\right)}.$$ A quantum circuit of size 2 O ˜ log ( | Δ | ) α . $$2^{\tilde{O}\left(\log(|\Delta|)^{\alpha}\right)}.$$ Polynomial classical and quantum memory. Essentially, we propose to reduce the size of the quantum circuit below the state-of-the-art complexity 2 O ˜ log ( | Δ | ) 1 / 2 $$2^{\tilde{O}\left(\log(|\Delta|)^{1/2}\right)}$$ at the cost of increasing the classical circuit-size required. The required classical circuit remains subexponential, which is a superpolynomial improvement over the classical state-of-the-art exponential solutions to these problems. Our method requires polynomial memory, both classical and quantum.more » « less
-
null (Ed.)Synopsis Many flying insects utilize a membranous structure for flight, which is known as a “wing.” However, some spiders use silk fibers for their aerial dispersal. It is well known that spiders can disperse over hundreds of kilometers and rise several kilometers above the ground in this way. However, little is known about the ballooning mechanisms of spiders, owing to the lack of quantitative data. Recently, Cho et al. discovered previously unknown information on the types and physical properties of spiders’ ballooning silks. According to the data, a crab spider weighing 20 mg spins 50–60 ballooning silks simultaneously, which are about 200 nm thick and 3.22 m long for their flight. Based on these physical dimensions of ballooning silks, the significance of these filament-like structures is explained by a theoretical analysis reviewing the fluid-dynamics of an anisotropic particle (like a filament or a high-slender body). (1) The filament-like structure is materially efficient geometry to produce (or harvest, in the case of passive flight) fluid-dynamic force in a low Reynolds number flow regime. (2) Multiple nanoscale fibers are the result of the physical characteristics of a thin fiber, the drag of which is proportional to its length but not to its diameter. Because of this nonlinear characteristic of a fiber, spinning multiple thin ballooning fibers is, for spiders, a better way to produce drag forces than spinning a single thick spider silk, because spiders can maximize their drag on the ballooning fibers using the same amount of silk dope. (3) The mean thickness of fibers, 200 nm, is constrained by the mechanical strength of the ballooning fibers and the rarefaction effect of air molecules on a nanoscale fiber, because the slip condition on a fiber could predominate if the thickness of the fiber becomes thinner than 100 nm.more » « less
-
Slender fibres are ubiquitous in biology, physics and engineering, with prominent examples including bacterial flagella and cytoskeletal fibres. In this setting, slender body theories (SBTs), which give the resistance on the fibre asymptotically in its slenderness $$\epsilon$$ , are useful tools for both analysis and computations. However, a difficulty arises when accounting for twist and cross-sectional rotation: because the angular velocity of a filament can vary depending on the order of magnitude of the applied torque, asymptotic theories must give accurate results for rotational dynamics over a range of angular velocities. In this paper, we first survey the challenges in applying existing SBTs, which are based on either singularity or full boundary integral representations, to rotating filaments, showing in particular that they fail to consistently treat rotation–translation coupling in curved filaments. We then provide an alternative approach which approximates the three-dimensional dynamics via a one-dimensional line integral of Rotne–Prager–Yamakawa regularized singularities. While unable to accurately resolve the flow field near the filament, this approach gives a grand mobility with symmetric rotation–translation and translation–rotation coupling, making it applicable to a broad range of angular velocities. To restore fidelity to the three-dimensional filament geometry, we use our regularized singularity model to inform a simple empirical equation which relates the mean force and torque along the filament centreline to the translational and rotational velocity of the cross-section. The single unknown coefficient in the model is estimated numerically from three-dimensional boundary integral calculations on a rotating, curved filament.more » « less
-
Mikołaj Bojańczyk and Emanuela Merelli and David P. Woodruff (Ed.)The classical coding theorem in Kolmogorov complexity states that if an n-bit string x is sampled with probability δ by an algorithm with prefix-free domain then K(x) ≤ log(1/δ) + O(1). In a recent work, Lu and Oliveira [31] established an unconditional time-bounded version of this result, by showing that if x can be efficiently sampled with probability δ then rKt(x) = O(log(1/δ)) + O(log n), where rKt denotes the randomized analogue of Levin’s Kt complexity. Unfortunately, this result is often insufficient when transferring applications of the classical coding theorem to the time-bounded setting, as it achieves a O(log(1/δ)) bound instead of the information-theoretic optimal log(1/δ). Motivated by this discrepancy, we investigate optimal coding theorems in the time-bounded setting. Our main contributions can be summarised as follows. • Efficient coding theorem for rKt with a factor of 2. Addressing a question from [31], we show that if x can be efficiently sampled with probability at least δ then rKt(x) ≤ (2 + o(1)) · log(1/δ) +O(log n). As in previous work, our coding theorem is efficient in the sense that it provides a polynomial-time probabilistic algorithm that, when given x, the code of the sampler, and δ, it outputs, with probability ≥ 0.99, a probabilistic representation of x that certifies this rKt complexity bound. • Optimality under a cryptographic assumption. Under a hypothesis about the security of cryptographic pseudorandom generators, we show that no efficient coding theorem can achieve a bound of the form rKt(x) ≤ (2 − o(1)) · log(1/δ) + poly(log n). Under a weaker assumption, we exhibit a gap between efficient coding theorems and existential coding theorems with near-optimal parameters. • Optimal coding theorem for pKt and unconditional Antunes-Fortnow. We consider pKt complexity [17], a variant of rKt where the randomness is public and the time bound is fixed. We observe the existence of an optimal coding theorem for pKt, and employ this result to establish an unconditional version of a theorem of Antunes and Fortnow [5] which characterizes the worst-case running times of languages that are in average polynomial-time over all P-samplable distributions.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript
- Journal Article:
- https://doi.org/10.3390/fluids6080283
