Shear and Bulk Acceleration Viscosities in Simple Fluids

Johannes Renner, Matthias Schmidt, and Daniel de las Heras

Phys. Rev. Lett. 128, 094502 – Published 4 March 2022

Abstract

Inhomogeneities in the velocity field of a moving fluid are dampened by the inherent viscous behavior of the system. Both bulk and shear effects, related to the divergence and the curl of the velocity field, are relevant. On molecular time scales, beyond the Navier-Stokes description, memory plays an important role. Using molecular and overdamped Brownian dynamics many-body simulations, we demonstrate that analogous viscous effects act on the acceleration field. This acceleration viscous behavior is associated with the divergence and the curl of the acceleration field, and it can be quantitatively described using simple exponentially decaying memory kernels. The simultaneous use of velocity and acceleration fields enables the description of fast dynamics on molecular scales.

Received 30 October 2021
Accepted 9 February 2022

DOI:https://doi.org/10.1103/PhysRevLett.128.094502

Physics Subject Headings (PhySH)

Collective behavior Compressible flows Interactions in fluids Microfluidics Nonequilibrium statistical mechanics Rheology Stable compressible flows Turbulence

Classical fluids Nonequilibrium systems

Viscosity

Brownian dynamics Many-body techniques Molecular dynamics

Fluid DynamicsNonlinear DynamicsStatistical Physics & Thermodynamics

Authors & Affiliations

Johannes Renner , Matthias Schmidt , and Daniel de las Heras ^*

Theoretische Physik II, Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany

^*Corresponding author. https://www.danieldelasheras.com delasheras.daniel@gmail.com

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Issue

Vol. 128, Iss. 9 — 4 March 2022

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Images

Figure 1
(a) Temporal part of the current $J_{t}$ vs time $t$ common to the bulk (b) and shear (c) flows. Four times $t_{i}$ with $i = 1$ , 2, 3, and 4 are highlighted with colored circles. The vertical dotted lines indicate the times $t_{↑}, t_{c}$ , and $t_{↓}$ . (b),(c) The external force $f_{ext}$ , density $ρ$ , velocity $v$ , acceleration $a$ , and viscous force $f_{vis}$ profiles as a function of $x$ for the bulk and shear flows, respectively. To improve the visualization, the external force has been smoothed by eliminating high-frequency Fourier modes (see details and raw data in the Supplemental Material [20]). The thin black solid lines are the target fields that coincide (up to numerical accuracy) with the sampled fields. The color of the profiles indicates the time $t_{1} = 0.5 τ$ (red), $t_{2} = 4 τ$ (blue), $t_{3} = 5.5 τ$ (yellow), and $t_{4} = 6.05 τ$ (purple), as indicated in (a). The arrows indicate the direction of the vector field at specific locations (arrow position) and times (arrow color).
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Figure 2
(a) Temporal dependency of the bulk viscous force $C_{b}$ as a function of time $t$ in molecular dynamics simulations (thick black line) and theoretically (violet) for the bulk flow. The vertical dotted lines indicate the times $t_{↑}, t_{c}$ , and $t_{↓}$ . The time $t_{3} = 5.5 τ$ is highlighted with a yellow circle. The light gray line fluctuating around the horizontal line is the difference between simulation (thick black) and theory (violet). (b) Bulk viscous force $f_{vis}$ as a function of $x$ at time $t_{3} = 5.5 τ$ according to MD (yellow) and theory (violet). The force points along the $x$ axis. The colored arrows indicate the direction of the corresponding force at selected positions. The contributions of the velocity (green) and of the acceleration (blue) to the total signal (violet) are also shown in (a) and (b). The bottom panels (c) and (d) show the same data as the top panels, but using overdamped Brownian dynamics instead of MD. In BD only the velocity field contributes to the viscosity.
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Figure 3
(a) Temporal dependency of the shear viscous force $C_{s}$ as a function of time $t$ in MD simulations (thick black line) and theoretically (violet) for the shear flow. The light gray line fluctuating around the horizontal line is the difference between simulation (thick black) and theory (violet). (b) Shear viscous force $f_{vis}$ as a function of $x$ at time $t_{3} = 5.5 τ$ according to MD (yellow) and theory (violet). The force points along the $y$ axis. (c) Illustrative velocity (green) and acceleration (blue) profiles vs $x$ for the traveling shear wave ( $t = 2.7 τ$ ). Note that $a$ and $v$ are not in phase. (d) Viscous force vs $x$ for the traveling shear wave according to MD (thick black) and theory (violet) ( $t = 2.7 τ$ ). The colored arrows indicate the direction of the corresponding field at the selected positions. The theoretical contributions of $v$ (green) and $a$ (blue) to the total signal (violet) are also shown in panels (a), (b), and (d) together with the theoretical predictions using only the velocity field (dashed green line). The colored circles over the $x$ axis in (d) indicate the position of the minimum of $f_{vis}$ according to MD (gray), and theory using both contributions (violet) or only the velocity contribution (green).
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