Far-field flow and drift due to particles and organisms in density-stratified fluids

Vaseem A. Shaik and Arezoo M. Ardekani

Phys. Rev. E 102, 063106 – Published 21 December 2020

Abstract

In the limit of small inertia, stratification, and advection of density, Ardekani and Stocker [Phys. Rev. Lett. 105, 084502 (2010)] derived the flow due to a point-force and force-dipole placed in a linearly density-stratified fluid. In this limit, these flows also represent the far-field flow due to a towed particle and a neutrally buoyant swimming organism in a stratified fluid. Here, we derive these two far-field flows in the limit of small inertia, stratification but at large advection of density. In both these limits, the flow in a stratified fluid decays rapidly and has closed streamlines but certain symmetries present at small advection are lost at large advection. To illustrate the application of these flows, we use them to calculate the drift induced by a towed drop and a swimming organism, as a means to quantify the mixing caused by them. The drift induced in a stratified fluid is less than that in the homogeneous fluid. A towed drop induces a large drift relative to its own volume at small advection while it induces at least an order of magnitude smaller drift at large advection. On the other hand, a swimming organism induces a large partial drift as compared with its own volume irrespective of the magnitude of advection, unless the stresslet exerted by the swimmer is small. These results are useful in understanding the stratification effects on the drift-based contributions to mixing.

Received 5 September 2020
Accepted 24 November 2020

DOI:https://doi.org/10.1103/PhysRevE.102.063106

Physics Subject Headings (PhySH)

Biological fluid dynamics Drops & bubbles Fluid-particle interactions Stratified geophysical flows Swimming

Fluid DynamicsPhysics of Living Systems

Authors & Affiliations

Vaseem A. Shaik and Arezoo M. Ardekani ^*

School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 47907, USA

^*ardekani@purdue.edu

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Vol. 102, Iss. 6 — December 2020

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Images

Figure 1
Flow due to a towed sphere in the far field or that due to a point-force. The sphere is towed and the point-force is oriented in the vertically upward direction. (a) Filled contours of $ln | \bar{w}_{3}^{'} |$ along with the streamlines in a stratified fluid at high Pe. (b) The spatial variation of the vertical velocity along horizontal (blue) and vertical (red) directions. The data in the homogeneous fluids are shown by dotted lines whereas those in stratified fluids at low Pe are shown by symbols and those at high Pe are shown by dashed, solid, and dash-dotted lines. At high Pe, dash-dotted and solid lines show the vertical variation along the upstream and the downstream directions, respectively. The data at low Pe is taken from Fig. 2(a) of Ref. [15]. (c) Streamlines associated with the flow caused by a point-force located asymmetrically between two horizontal walls in a homogeneous fluid. One wall is at $r_{3} = - 1$ while the other is at $r_{3} = 8$ . This flow is found by adding the image flow due to the walls [35, 36] to the Stokeslet flow. Such a simple calculation yields inaccurate flow far from the point-force, which justifies the nonzero velocities seen near the bottom wall.
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Figure 2
Far-field flow due to a vertically oriented $(u_{3} > 0)$ pusher swimmer $(b = 1)$ . (a) Filled contours of $ln | \bar{w}_{3}^{'} |$ along with the streamlines in a stratified fluid at high Pe. (b) The vertical variation of the vertical velocity. The line style and color follow the same notation as that given in Fig. 1. The data at low Pe are taken from Fig. 2(a) of Ref. [15].
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Figure 3
Schematic showing the definition of the partial drift volume $D_{p}$ . In the comoving frame, the drop or organism is stationary and the marked fluid is initially at a distance $x_{d}$ upstream of the drop or swimmer. At time $t$ , the marked fluid deforms as it passes the drop or swimmer. The volume enclosed between the deformed (blue curve) and the undeformed (green line) marked fluids at time $t$ determines the partial drift at that time (light blue region).
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Figure 4
The variation of the normalized partial drift induced by a towed drop $D_{p} / R h^{2}$ with (a) $τ, ε_{h}$ at low Pe and (b) $R i_{h}^{1 / 3}$ at high Pe. To obtain general results that do not depend on $R, h$ , we normalized $D_{p}$ by $R h^{2}$ . Here, we choose $τ_{0} = - 10$ . The low Pe limit $Pe ≪ ε$ is the same as $l / l_{o} ≪ {Pr}^{- 1}$ while the high-Pe and low-Re limit $Pe ≫ R i^{1 / 3}, Re ≪ R i^{1 / 3}$ is same as ${Pr}^{- 1} ≪ l / l_{o} ≪ {Pr}^{- 1 / 4}$ , where $Pr = ν / κ$ is the Prandtl number. Panel (a) is adapted from Ref. [17] with permission from the American Physical Society.
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Figure 5
The variation of the normalized partial drift induced by a swimming organism $D_{p} / b h$ with (a) $τ, ε_{h}$ at low Pe and (b) $R i_{h}^{1 / 3}$ at high Pe. Here, we choose $τ_{0} \to - \infty$ . To find $D_{p}$ for any value of $τ_{0}, τ_{f}$ , subtract $D_{p} (τ_{0})$ from $D_{p} (τ_{f})$ on any specific curve.
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