Hydrodynamic model of fish orientation in a channel flow

For over a century, scientists have sought to understand how fish orient against an incoming flow, even without visual and flow cues. Here, we elucidate a potential hydrodynamic mechanism of rheotaxis through the study of the bidirectional coupling between fish and the surrounding fluid. By modeling a fish as a vortex dipole in an infinite channel with an imposed background flow, we establish a planar dynamical system for the cross-stream coordinate and orientation. The system dynamics captures the existence of a critical flow speed for fish to successfully orient while performing cross-stream, periodic sweeping movements. Model predictions are examined in the context of experimental observations in the literature on the rheotactic behavior of fish deprived of visual and lateral line cues. The crucial role of bidirectional hydrodynamic interactions unveiled by this model points at an overlooked limitation of existing experimental paradigms to study rheotaxis in the laboratory.

background flow. Just as fish motion influences the local flow field, so too does the local flow field A major contribution of the proposed model is the treatment of the fish as an invasive sensor 116 that both reacts to and influences the background flow, thereby establishing a coupled interaction 117 between the fish and the surrounding environment. A fish swimming in the vicinity of a wall will 118 induce rotational flow near the boundary. In the inviscid limit, this boundary layer is infinitesimally 119 thin and can be considered as wall-bounded vorticity (Batchelor, 2000). Employing where is a non-negative integer representing the -th set of images. Subscripts "<" and ">" cor-128 respond to position vectors of the images at < 0 and > ℎ, respectively. Likewise, superscript "±" 129 denotes the orientation of the image dipole as ± ; that is, a position vector with superscript "+" 130 indicates that the associated image has the same orientation as the fish.

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The potential function for a given image is found by replacing ⃗ in (1) with its position vector 132 from (2) and adjusting the sign of in (1) to match the superscript of its vector. The potential field 133 at ⃗ due to the image dipoles is fish and the walls in its proximity and creating long-range swirling patterns in the channel (Fig. 2).

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The presence of a background flow in the channel is modelled by superimposing a weakly ro- which has speed 0 at the channel centerline and 0 (1 − ) at the walls, being a small positive 142 parameter. As → 0, a uniform (irrotational) background flow is recovered: such a flow is indistin-143 guishable from the one in Fig. 2, provided that the observer is moving with the background flow.

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For ≪ 1, the imposed velocity profile approximates that of a turbulent channel flow, wherein 145 a modest degree of velocity profile curvature is present near the channel centerline. We note that 146 this velocity profile does not satisfy the no-slip boundary condition (zero velocity on the walls), and 147 the flow is entirely described by only two parameters ( 0 and ). For ≃ 1, the profile approaches 148 that of a laminar flow with parabolic dependence on the cross-stream coordinate. The overall fluid 149 flow in the channel is ultimately computed as ⃗ = ⃗ + ⃗ + ⃗ .

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The circulation in a region  in the flow field centered at some location is Γ = ∫  d , where 151 = (∇ × ⃗) ⋅̂is the local fluid vorticity (̂=̂×̂). For the considered flow field, we determine Stokes simulations remains elusive. The potential flow framework in which these models are 158 grounded neglects boundary layers and the resulting wakes that emerge from viscous effects.

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Quantifying the extent to which these effects influence the flow field generated by the fish is part 160 of this study. Information.

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The mean velocity field averaged over a tail beating cycle is displayed in Fig. 3(a). The predom- where is the feedback mechanism based on the circulation measurement through the lateral 190 line.

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The advective velocity is found by de-singularizing the total velocity field ⃗ at ⃗ = ⃗ , which is equivalent to calculating the sum of the velocity due to the walls and the background flow in correspondence of the fish (Milne-Thomson, 1996) ) 2 )̂. by the two constituent vortices comprising the dipole, namely, wherê⟂ =̂×̂; see Methods and Materials Section for the mathematical derivation. Equa-197 tion (8) indicates that interaction with the walls causes the fish to turn towards the nearest wall; 198 for example, a fish at = 3∕4ℎ, will experience a turn rate due to the wall of ( 3 2 0 )∕(2ℎ) cos , 199 such that it will be rotated counter-clockwise if swimming downstream and clockwise if swimming irrespective of fish orientation, so that a fish at = 3∕4ℎ will always be rotated counter-clockwise.

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As a result, the fish may turn towards or away from a wall, depending on model parameters and ( where we non-dimensionalized by the time needed for the fish to traverse the channel in the ab-229 sence of a background flow, that is, ℎ∕ 0 , and introduced = 0 ∕ 0 and = 0 (see Methods

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and Materials Section for estimation of these parameters from experimental observations).

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In search of the equilibria of the dynamical system, we note that swimming downstream or 232 upstream ( = 0 and , respectively) solves (10a) for any choice of the cross-stream coordinate, 233 the value of which is determined from the solution of (10b) for the corresponding orientation .  the walls as → ∞ (Fig. 4(a), see Methods and Materials Section for mathematical derivations).

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Local stability of these equilibria is determined by studying the eigenvalues of the state matrix always negative, such that the equilibrium ( = 0, = 0) is a saddle point (Fig. 4(b)). For upstream and Coombs, 2015) ( Fig. 1(b)); the other two equilibria located away from the centerline are always 248 unstable (Figs. 4(b,c)). Oscillations about the centerline during rheotaxis have a radian frequency water-motion cues can also be accessible to tactile or other cutaneous senses, beyond the lateral 284 line that is included in our model. In addition, body-motion cues are not limited to visual senses, whereby they can be accessed by tactile and vestibular senses. Hence, a one-to-one comparison 286 between experiments and theory is presently not possible.

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The model predicts the emergence of rheotaxis in the absence of any sensory information. robot is actuated by a built-in step motor to undertake a periodic tail beating with a predetermined 305 frequency. All its electronics is encased in the frontal section of the robot, so that its size and shape 306 can be readily adjusted though rapid prototyping. ments, which are not modelled in our work.

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Finally, the model anticipates the onset of periodic cross-stream sweeping, which has been 337 studied in some experiments on fish swimming in channels without vision (Coombs et al., 2020). 338 While there is not conclusive experimental evidence regarding the dependence of the frequency

Derivation of the turn rate equation for the fish dynamics 389
The expression for the turn rate in equation (8)  Re By carrying out the complex algebra in (11), we determine which supports the intuition that the dipole will turn counter-clockwise if the right vortex would  (Bakker, 1991). Hence, stability requires that 438 1 24 Since the first factor is always negative ( ≪ 1) and the second is positive, the inequality is never 439 fulfilled and the equilibrium is a saddle point (unstable) (Fig. 4(a,b)). .
Similar to the previous case, stability requires that det > 0, that is, 442 1 24 Due to the sign change in the first summand appearing in the second factor with respect to the 443 previous case, stability becomes possible. Specifically, the equilibrium is a neutral center (stable) 444 for > * = 4 ∕32, which is also the necessary condition for the existence of the two equilibria 445 symmetrically located with respect to the channel centerline ( Fig. 4(a,c)).

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The estimation of the non-dimensional parameter associated with the shear in the flow is more Data and materials availability 513 The authors declare that the data supporting the findings of this study are available within the 514 paper. The Mathematica notebook used to derive the governing equations, study the planar dy-515 namical, and generate associated figures, together with the CFD data discussed in the text, are also 516 available at https://github.com/dynamicalsystemslaboratory/Rheotaxis. Gazzola M, Argentina M, Mahadevan L. Scaling macroscopic aquatic locomotion. Nature Physics. 2014;