Experiment and model for a Stokes layer in a strongly coupled dusty plasma

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Experiment and model for a Stokes layer in a strongly coupled dusty plasma

Jorge Berumen and J. Goree

Phys. Rev. E 104, 035208 – Published 30 September 2021

Abstract

A Stokes layer, which is a flow pattern that arises in a viscous fluid adjacent to an oscillatory boundary, was observed in an experiment using a two-dimensional strongly coupled dusty plasma. Liquid conditions were maintained using laser heating, while a separate laser manipulation applied an oscillatory shear that was localized and sinusoidal. The evolution of the resulting flow was analyzed using space-time diagrams. These figures provide an intuitive visualization of a Stokes layer, including features such as the depth of penetration and wavelength. Another feature, the characteristic speed for the penetration of the oscillatory flow, also appears prominently in space-time diagrams. To model the experiment, the Maxwell-fluid model of a Stokes layer was generalized to describe a two-phase liquid. In our experiment, the phases were gas and dust, where the dust cloud was viscoelastic due to strong Coulomb coupling. The model is found to agree with the experiment, in the appearance of the space-time diagrams, and in the values of the characteristic speed, depth of penetration, and wavelength.

Received 4 June 2021
Accepted 28 July 2021

DOI:https://doi.org/10.1103/PhysRevE.104.035208

Physics Subject Headings (PhySH)

Particle-laden flows Shear flows

Dusty or complex plasma Strongly-coupled plasmas

Plasma PhysicsFluid Dynamics

Authors & Affiliations

Jorge Berumen and J. Goree

Department of Physics and Astronomy, University of Iowa, Iowa City, Iowa 52242, USA

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Issue

Vol. 104, Iss. 3 — September 2021

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Images

Figure 1
Sketch of flow in a Stokes layer. An infinite planar boundary located at the bottom oscillates in the $\pm x$ direction. A flow develops in the adjoining fluid that fills the volume $y > 0$ . These sketches portray two different times, (a) $t ω = 0$ and (b) $t ω = 2$ . Over time, as it oscillates, the flow reverses direction, as indicated by the different colors. This reversal occurs periodically with distance $y$ from the boundary, as is characterized by the parameter $λ$ . The amplitude of the oscillation gradually weakens with increasing distance $y$ , as characterized by the depth of penetration $δ$ .
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Figure 2
Sketch of flow profile in a Stokes layer, at a particular time. In the Stokes layer literature, it is common to present the flow velocity $u_{x}$ in a one-dimensional graph, as sketched here.
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Figure 3
Space-time diagrams depicting the temporal evolution of the flow velocity $u_{x}$ in a planar Stokes layer. The boundary, located at $y = 0$ , oscillates in the $\pm x$ direction at frequency $ω$ , as described by Eq. (2), with a peak boundary velocity of $\pm \tilde{U}$ . All five diagrams are plots of Eq. (3), which is a solution of a differential equation, which is Eq. (4) for the purely viscous fluid in (a), Eq. (12) for a viscoelastic fluid in [(b), (c)], and Eq. (19) for a two-phase viscoelastic fluid in [(d), (e)]. The fading color, far from the boundary at $y = 0$ , shows how the flow decays with an $e$ -folding distance $δ$ . Along the $y$ axis, a reversal of velocity reveals the wavelength $λ$ . The dark line has a slope representing the characteristic speed $C$ , as explained in Sec. 4. Time axes are normalized by the oscillation frequency $ω$ , while $y$ axes are normalized by the theoretical depth of penetration $δ_{vis}$ for a purely viscous fluid, Eq. (5). Physical values for the experiment in Sec. 6e were used for these solutions: $ρ = (1.5 \pm 0.1) \times 10^{- 6} {kg/m}^{2}$ , $η_{0} = (3.5 \pm 0.4) \times 10^{- 12} kg s^{- 1}$ , $τ = 0.05$ s, and $ν_{g} = 0.97 s^{- 1}$ , with $ω / 2 π = 0.1$ Hz in (a), (b), and (d), but $ω / 2 π = 5$ Hz in (c) and (e).
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Figure 4
(a) Sketch of the optical setup used to heat and apply shear to the dust cloud. The two laser beams in the lower portion of this diagram provided the shear. Before the shear beam was divided by a beam splitter (BS2), it was shaped by a combination of cylindrical lenses (CL) and a galvanometer mirror. It was also modulated sinusoidally in time, using a rotating half-wave plate and a linear polarizer (P), to allow a study of a Stokes layer rather than a steady-shear layer. The two heating beams in the upper portion were rastered over the entire dust-particle cloud. The top-view camera was our main diagnostic. (b) A single still image from the top-view camera, showing its field of view. The placement and width of the two shear beams are drawn to scale. The region of interest (ROI) was the portion of the image for $y > 0$ ; this is where we measured particle velocities to yield the flow profile.
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Figure 5
Space-time diagrams obtained from experimental data (top row) and our two-phase fluid model (bottom row), for modulation frequencies of $0.5 Hz$ (left column) and $1.0 Hz$ (right column). Zero time corresponds to a trigger event in the experiment. The color bar scale for ${\tilde{u}}_{x}$ is the same for all panels. In the experimental space-time diagrams, the main features of a Stokes layer are apparent: the finite penetration of the flow can be seen as fading of the color with increasing distance; a reversal of the flow direction can be seen as a reversing color with distance from the boundary, for example, along the vertical dashed line in each diagram; the characteristic speed is indicated by the slope of the solid lines. In addition to obtaining $C$ , these experimental diagrams also allow obtaining values for $δ$ and $λ$ . The theoretical diagrams were obtained by combining Eqs. (3), (22), and (24), while the theoretical characteristic speed $C_{2ph}$ was calculated using Eq. (26), all with an input of experimental parameters listed in the paper.
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