skip to main content
Explore Scholarly Publications and Datasets in the NSF-PAR

  • NSF Public Access
  • Search Results
  • Accepted Manuscript: Slender body theory for particles with non-circular cross-sections with application to particle dynamics in shear flows
Title: Slender body theory for particles with non-circular cross-sections with application to particle dynamics in shear flows
This paper presents a theory to obtain the force per unit length acting on a slender filament with a non-circular cross-section moving in a fluid at low Reynolds number. Using a regular perturbation of the inner solution, we show that the force per unit length has $O(1/\ln (2A))+O(\unicode[STIX]{x1D6FC}/\ln ^{2}(2A))$ contributions driven by the relative motion of the particle and the local fluid velocity and an $O(\unicode[STIX]{x1D6FC}/(\ln (2A)A))$ contribution driven by the gradient in the imposed fluid velocity. Here, the aspect ratio ( $A=l/a_{0}$ ) is defined as the ratio of the particle size ( $l$ ) to the cross-sectional dimension ( $a_{0}$ ) and $\unicode[STIX]{x1D6FC}$ is the amplitude of the non-circular perturbation. Using thought experiments, we show that two-lobed and three-lobed cross-sections affect the response to relative motion and velocity gradients, respectively. A two-dimensional Stokes flow calculation is used to extend the perturbation analysis to cross-sections that deviate significantly from a circle (i.e. $\unicode[STIX]{x1D6FC}\sim O(1)$ ). We demonstrate the ability of our method to accurately compute the resistance to translation and rotation of a slender triaxial ellipsoid. Furthermore, we illustrate novel dynamics of straight rods in a simple shear flow that translate and rotate quasi-periodically if they have two-lobed cross-section, more » and rotate chaotically and translate diffusively if they have a combination of two- and three-lobed cross-sections. Finally, we show the remarkable ability of our theory to accurately predict the motion of rings, retaining great accuracy for moderate aspect ratios ( ${\sim}10$ ) and cross-sections that deviate significantly from a circle, thereby making our theory a computationally inexpensive alternative to other Stokes flow solvers. « less
Authors:
;
Award ID(s):
1803156
Publication Date:
NSF-PAR ID:
10162408
Journal Name:
Journal of Fluid Mechanics
Volume:
877
Page Range or eLocation-ID:
1098 to 1133
ISSN:
0022-1120
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ( , Journal of Fluid Mechanics)
    The hydrodynamic quantification of superhydrophobic slipperiness has traditionally employed two canonical problems – namely, shear flow about a single surface and pressure-driven channel flow. We here advocate the use of a new class of canonical problems, defined by the motion of a superhydrophobic particle through an otherwise quiescent liquid. In these problems the superhydrophobic effect is naturally measured by the enhancement of the Stokes mobility relative to the corresponding mobility of a homogeneous particle. We focus upon what may be the simplest problem in that class – the rotation of an infinite circular cylinder whose boundary is periodically decorated bymore »a finite number of infinite grooves – with the goal of calculating the rotational mobility (velocity-to-torque ratio). The associated two-dimensional flow problem is defined by two geometric parameters – namely, the number $N$ of grooves and the solid fraction $\unicode[STIX]{x1D719}$ . Using matched asymptotic expansions we analyse the large- $N$ limit, seeking the mobility enhancement from the respective homogeneous-cylinder mobility value. We thus find the two-term approximation, $$\begin{eqnarray}\displaystyle 1+{\displaystyle \frac{2}{N}}\ln \csc {\displaystyle \frac{\unicode[STIX]{x03C0}\unicode[STIX]{x1D719}}{2}}, & & \displaystyle \nonumber\end{eqnarray}$$ for the ratio of the enhanced mobility to the homogeneous-cylinder mobility. Making use of conformal-mapping techniques and inductive arguments we prove that the preceding approximation is actually exact for $N=1,2,4,8,\ldots$ . We conjecture that it is exact for all $N$ .« less
  2. ; ; ( , Journal of Fluid Mechanics)
    The relative velocities and positions of monodisperse high-inertia particle pairs in isotropic turbulence are studied using direct numerical simulations (DNS), as well as Langevin simulations (LS) based on a probability density function (PDF) kinetic model for pair relative motion. In a prior study (Rani et al. , J. Fluid Mech. , vol. 756, 2014, pp. 870–902), the authors developed a stochastic theory that involved deriving closures in the limit of high Stokes number for the diffusivity tensor in the PDF equation for monodisperse particle pairs. The diffusivity contained the time integral of the Eulerian two-time correlation of fluid relative velocities seenmore »by pairs that are nearly stationary. The two-time correlation was analytically resolved through the approximation that the temporal change in the fluid relative velocities seen by a pair occurs principally due to the advection of smaller eddies past the pair by large-scale eddies. Accordingly, two diffusivity expressions were obtained based on whether the pair centre of mass remained fixed during flow time scales, or moved in response to integral-scale eddies. In the current study, a quantitative analysis of the (Rani et al. 2014) stochastic theory is performed through a comparison of the pair statistics obtained using LS with those from DNS. LS consist of evolving the Langevin equations for pair separation and relative velocity, which is statistically equivalent to solving the classical Fokker–Planck form of the pair PDF equation. Langevin simulations of particle-pair dispersion were performed using three closure forms of the diffusivity – i.e. the one containing the time integral of the Eulerian two-time correlation of the seen fluid relative velocities and the two analytical diffusivity expressions. In the first closure form, the two-time correlation was computed using DNS of forced isotropic turbulence laden with stationary particles. The two analytical closure forms have the advantage that they can be evaluated using a model for the turbulence energy spectrum that closely matched the DNS spectrum. The three diffusivities are analysed to quantify the effects of the approximations made in deriving them. Pair relative-motion statistics obtained from the three sets of Langevin simulations are compared with the results from the DNS of (moving) particle-laden forced isotropic turbulence for $St_{\unicode[STIX]{x1D702}}=10,20,40,80$ and $Re_{\unicode[STIX]{x1D706}}=76,131$ . Here, $St_{\unicode[STIX]{x1D702}}$ is the particle Stokes number based on the Kolmogorov time scale and $Re_{\unicode[STIX]{x1D706}}$  is the Taylor micro-scale Reynolds number. Statistics such as the radial distribution function (RDF), the variance and kurtosis of particle-pair relative velocities and the particle collision kernel were computed using both Langevin and DNS runs, and compared. The RDFs from the stochastic runs were in good agreement with those from the DNS. Also computed were the PDFs $\unicode[STIX]{x1D6FA}(U|r)$ and $\unicode[STIX]{x1D6FA}(U_{r}|r)$ of relative velocity $U$ and of the radial component of relative velocity $U_{r}$ respectively, both PDFs conditioned on separation $r$ . The first closure form, involving the Eulerian two-time correlation of fluid relative velocities, showed the best agreement with the DNS results for the PDFs.« less
  3. ; ; ; ; ( , Journal of Fluid Mechanics)
    We study the emergence of precessing vortex core (PVC) oscillations in a swirling jet experiment. We vary the swirl intensity while keeping the net mass flow rate fixed using a radial-entry swirler with movable blades upstream of the jet exit. The swirl intensity is quantified in terms of a swirl number $S$ . Time-resolved velocity measurements in a radial–axial plane anchored at the jet exit for various $S$ values are obtained using stereoscopic particle image velocimetry. Spectral proper orthogonal decomposition and spatial cross-spectral analysis reveal the simultaneous emergence of a bubble-type vortex breakdown and a strong helical limit-cycle oscillation inmore »the flow for $S>S_{c}$ where $S_{c}=0.61$ . The oscillation frequency, $f_{PVC}$ , and the square of the flow oscillation amplitudes vary linearly with $S-S_{c}$ . A solution for the coherent unsteady field accurate up to $O(\unicode[STIX]{x1D716}^{3})$ ( $\unicode[STIX]{x1D716}\sim O((S-S_{c})^{1/2})$ ) is determined from the nonlinear Navier–Stokes equations, using the method of multiple scales. We show that onset of bubble type vortex breakdown at $S_{c}$ , results in a marginally stable, helical linear global hydrodynamic mode. This results in the stable limit-cycle precession of the breakdown bubble. The variation of $f_{LC}$ with $S-S_{c}$ is determined from the Stuart–Landau equation associated with the PVC. Reasonable agreement with the corresponding experimental result is observed, despite the highly turbulent nature of the flow in the present experiment. Further, amplitude saturation results from the time-averaged distortion imposed on the flow by the PVC, suggesting that linear stability analysis may predict PVC characteristics for $S>S_{c}$ .« less
  4. ; ; ; ; ( , Journal of Fluid Mechanics)
    We consider the unbounded settling dynamics of a circular disk of diameter $d$ and finite thickness $h$ evolving with a vertical speed $U$ in a linearly stratified fluid of kinematic viscosity $\unicode[STIX]{x1D708}$ and diffusivity $\unicode[STIX]{x1D705}$ of the stratifying agent, at moderate Reynolds numbers ( $Re=Ud/\unicode[STIX]{x1D708}$ ). The influence of the disk geometry (diameter $d$ and aspect ratio $\unicode[STIX]{x1D712}=d/h$ ) and of the stratified environment (buoyancy frequency $N$ , viscosity and diffusivity) are experimentally and numerically investigated. Three regimes for the settling dynamics have been identified for a disk reaching its gravitational equilibrium level. The disk first falls broadside-on, experiencing anmore »enhanced drag force that can be linked to the stratification. A second regime corresponds to a change of stability for the disk orientation, from broadside-on to edgewise settling. This occurs when the non-dimensional velocity $U/\sqrt{\unicode[STIX]{x1D708}N}$ becomes smaller than some threshold value. Uncertainties in identifying the threshold value is discussed in terms of disk quality. It differs from the same problem in a homogeneous fluid which is associated with a fixed orientation (at its initial value) in the Stokes regime and a broadside-on settling orientation at low, but finite Reynolds numbers. Finally, the third regime corresponds to the disk returning to its broadside orientation after stopping at its neutrally buoyant level.« less
  5. ( , Ergodic Theory and Dynamical Systems)
    Let $\unicode[STIX]{x1D6FC}\in \mathbb{R}\backslash \mathbb{Q}$ and $\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})=\limsup _{n\rightarrow \infty }(\ln q_{n+1})/q_{n}<\infty$ , where $p_{n}/q_{n}$ is the continued fraction approximation to $\unicode[STIX]{x1D6FC}$ . Let $(H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}u)(n)=u(n+1)+u(n-1)+2\unicode[STIX]{x1D706}\cos 2\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D703}+n\unicode[STIX]{x1D6FC})u(n)$ be the almost Mathieu operator on $\ell ^{2}(\mathbb{Z})$ , where $\unicode[STIX]{x1D706},\unicode[STIX]{x1D703}\in \mathbb{R}$ . Avila and Jitomirskaya [The ten Martini problem. Ann. of Math. (2), 170 (1) (2009), 303–342] conjectured that, for $2\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6FC}\mathbb{Z}+\mathbb{Z}$ , $H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}$ satisfies Anderson localization if $|\unicode[STIX]{x1D706}|>e^{2\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})}$ . In this paper, we develop a method to treat simultaneous frequency and phase resonances and obtain that, for $2\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6FC}\mathbb{Z}+\mathbb{Z}$ , $H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}$ satisfies Anderson localization if $|\unicode[STIX]{x1D706}|>e^{3\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})}$ .
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email