It is well known that the mixing of two or more species in flows at low Reynolds numbers cannot be easily achieved since inertial effects are essentially absent and molecular diffusion is slow. To achieve mixing in Newtonian fluids under these circumstances requires innovative new ideas such as the use of external body forces (eg, electromagnetic mixers) or the stretching and folding of fluid elements (eg, chaotic advection). For non‐Newtonian fluids with elasticity, mixing can be achieved by enabling the emergence of elastic instabilities that results in chaotic flows in which mixing is significantly enhanced. In this work, our goal is to demonstrate that clearly identifiable vortical structures (eg, vortex rings) can be generated in a viscoelastic fluid initially at rest by the release of elastic stresses. In turn, these vortex motions promote bulk mixing by transporting fluid elements from one location to another more efficiently than diffusion alone. We demonstrate this first theoretically by using the finitely extensible nonlinear elastic Peterlin (FENE‐P) model to show that elastic forces can generate torque. Using this model, we derive an expression for the time rate of change of vorticity in an elastic fluid initially at rest caused by a sudden release of stored elastic stress. This process can be thought of as the release of elastic energy from a stretched rubber band that is suddenly cut at its center. We confirm this ansatz by performing a series of direct numerical simulations based on an in‐house pseudo‐spectral code that couples the FENE‐P model to the equations of motion for an incompressible fluid. The simulations reveal that a pair of vortex rings traveling in opposite directions, with Reynolds numbers on the order of one, is generated from the sudden release of elastic stresses. Secondary vortical structures are also generated. In the concluding section of this work, we address the potential for vortex motions generated by elastic stresses to promote mixing in microflows, and we describe a possible experiment that may demonstrate this effect.
Inviscid damping of an elliptical vortex subject to an external strain flow
Inviscid spatial Landau damping is studied experimentally for the case of oscillatory motion of a two-dimensional vortex about its elliptical equilibrium in the presence of an applied strain flow. The experiments are performed using electron plasmas in a Penning–Malmberg trap. They exploit the isomorphism between the two-dimensional Euler equations for an ideal fluid and the drift-Poisson equations for the plasma, where plasma density is the analog of vorticity. Perturbed elliptical vortex states are created using [Formula: see text] strain flows, which are generated by applying voltages to electrodes surrounding the plasma. Measurements of spatial Landau damping (also called critical-layer damping) are in agreement with previous studies in the absence of an applied strain, where the damping is due to a resonance between the local fluid motion and the vortex oscillations. Interestingly, the damping rate does not change significantly over a wide range of applied strain rates. This can be accurately predicted from the initial vorticity profile, even though the resonant frequency is reduced substantially due to the applied strain. For higher amplitude perturbations, nonlinear trapping oscillations also exhibit behavior similar to the strain-free case. In principle, higher-order effects of the applied strain, such as separatrix crossing of peripheral vorticity and interactions with harmonics of the fundamental resonance, are expected to change the damping rate. However, this occurs only for conditions that are not realized in the experiments described here. Vortex-in-cell simulations are used to investigate the possible roles of these effects.
more »
« less
- Award ID(s):
- 2106332
- NSF-PAR ID:
- 10334018
- Date Published:
- Journal Name:
- Physics of Plasmas
- Volume:
- 29
- Issue:
- 5
- ISSN:
- 1070-664X
- Page Range / eLocation ID:
- 052107
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
more » « less
Abstract -
Flow-induced vibrations of flexible surfaces driven by coherent vortical structures are ubiquitous in engineering and biological flows; from the extraction of fluidic energy via oscillating electro-active polymers to vocal fold dynamics during voiced speech production. These scenarios may involve either discrete or periodic loading conditions due to the advection of vortices past the structure. This work considers, as a function of the vortex production frequency, the fluid-structure interaction that occurs as vortices are propagated tangentially over flexible plates with variable structural properties. Velocity fields are acquired with particle image velocimetry and used to compute the vorticity and pressure fields, while the plate energy is estimated from its kinematics. Primary and secondary peaks in plate deflection amplitude and the plate energy as a function of vortex production frequency are observed at integer fractions of the fundamental plate frequency. At resonance conditions, plate energy relative to discrete vortex loading is increased by approximately three orders of magnitude, while the sub-harmonics increase the plate energy by about two orders of magnitude. Additional physical influences on the energy exchange process, including vortex-to-plate spacing and Strouhal number, are also investigated, detailing the importance of spatial and temporal interactions. The magnitude of the initial plate deflection as the vortex ring approaches the plate, due to persistent vibrations from previous interactions, is shown to retard the time at which the maximum load is applied as the increased relative vortex-to-plate spacing weakens cross-sign vorticity interactions. Finally, plate properties are scaled to model the structural properties of the vocal folds and the effect of intra-glottal vortices on vocal fold dynamics is quantified, where a negligible influence is observed.more » « less
-
more » « less
Abstract We consider the full three‐dimensional Ginzburg–Landau model of superconductivity with applied magnetic field, in the regime where the intensity of the applied field is close to the ‘first critical field’ at which vortex filaments appear, and in the asymptotics of a small inverse Ginzburg–Landau parameter . This onset of vorticity is directly related to an ‘isoflux problem’ on curves (finding a curve that maximizes the ratio of a magnetic flux by its length), whose study was initiated in [22] and which we continue here. By assuming a nondegeneracy condition for this isoflux problem, which we show holds at least for instance in the case of a ball, we prove that if the intensity of the applied field remains below , the total vorticity remains bounded independently of , with vortex lines concentrating near the maximizer of the isoflux problem, thus extending to the three‐dimensional setting a two‐dimensional result of [28]. We finish by showing an improved estimate on the value of in some specific simple geometries.
-
null (Ed.)The energetically independent linear wave and geostrophic (vortex) solutions are shown to be a complete basis for velocity and density variables $(u,v,w,\rho )$ in a rotating non-hydrostatic Boussinesq fluid with arbitrary stratification and non-periodic vertical boundaries. This work extends the familiar wave-vortex decomposition for triply periodic domains with constant stratification. As a consequence of the decomposition, the fluid can be unambiguously separated into decoupled linear wave and geostrophic components at each instant in time, without the need for temporal filtering. The fluid can then be diagnosed for temporal changes in wave and geostrophic coefficients at each unique wavenumber and mode, including those that inevitably occur due to nonlinear interactions. We demonstrate that this methodology can be used to determine which physical interactions cause the transfer of energy between modes by projecting the nonlinear equations of motion onto the wave-vortex basis. In the particular example given, we show that an eddy in geostrophic balance superimposed with inertial oscillations at the surface transfers energy from the inertial oscillations to internal gravity wave modes. This approach can be applied more generally to determine which mechanisms are involved in energy transfers between wave and vortices, including their respective scales. Finally, we show that the nonlinear equations of motion expressed in a wave-vortex basis are computationally efficient for certain problems. In cases where stratification profiles vary strongly with depth, this approach may be an attractive alternative to traditional spectral models for rotating Boussinesq flow.more » « less
-
In Part 2 of our guide to collisionless fluid models, we concentrate on Landau fluid closures. These closures were pioneered by Hammett and Perkins and allow for the rigorous incorporation of collisionless Landau damping into a fluid framework. It is Landau damping that sharply separates traditional fluid models and collisionless kinetic theory, and is the main reason why the usual fluid models do not converge to the kinetic description, even in the long-wavelength low-frequency limit. We start with a brief introduction to kinetic theory, where we discuss in detail the plasma dispersion function $Z(\unicode[STIX]{x1D701})$ , and the associated plasma response function $R(\unicode[STIX]{x1D701})=1+\unicode[STIX]{x1D701}Z(\unicode[STIX]{x1D701})=-Z^{\prime }(\unicode[STIX]{x1D701})/2$ . We then consider a one-dimensional (1-D) (electrostatic) geometry and make a significant effort to map all possible Landau fluid closures that can be constructed at the fourth-order moment level. These closures for parallel moments have general validity from the largest astrophysical scales down to the Debye length, and we verify their validity by considering examples of the (proton and electron) Landau damping of the ion-acoustic mode, and the electron Landau damping of the Langmuir mode. We proceed by considering 1-D closures at higher-order moments than the fourth order, and as was concluded in Part 1, this is not possible without Landau fluid closures. We show that it is possible to reproduce linear Landau damping in the fluid framework to any desired precision, thus showing the convergence of the fluid and collisionless kinetic descriptions. We then consider a 3-D (electromagnetic) geometry in the gyrotropic (long-wavelength low-frequency) limit and map all closures that are available at the fourth-order moment level. In appendix A, we provide comprehensive tables with Padé approximants of $R(\unicode[STIX]{x1D701})$ up to the eighth-pole order, with many given in an analytic form.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1063/5.0086227
