Abstract
In a seminal paper published in 1951, Taylor studied the interactions between a viscous fluid and an immersed flat sheet which is subjected to a travelling wave of transversal displacement. The net reaction of the fluid over the sheet turned out to be a force in the direction of the wave phase-speed. This effect is a key mechanism for the swimming of micro-organisms in viscous fluids. Here, we study the interaction between a viscous fluid and a special class of nonlinear morphing shells. We consider pre-stressed shells showing a one-dimensional set of neutrally stable equilibria with almost cylindrical configurations. Their shape can be effectively controlled through embedded active materials, generating a large-amplitude shape-wave associated with precession of the axis of maximal curvature. We show that this shape-wave constitutes the rotational analogue of a Taylor's sheet, where the translational swimming velocity is replaced by an angular velocity. Despite the net force acting on the shell vanishes, the resultant torque does not. A similar mechanism can be used to manoeuver in viscous fluids.
Highlights
The problem of locomotion at low Reynolds numbers has been initiated by the groundbreaking paper of Taylor [1], which has had an enormous impact and continue to motivate a substantial amount of research (e.g. [2] and references cited therein, and the more than 1000 references that have cited this review paper since its publication 10 years ago)
Where U is the swimming speed of the sheet. With this choice of frame, the swimming speed appears as an unknown boundary condition for the problem
Given the assumption of small amplitude, Taylor shows that expanding the boundary condition (1.2) in powers of the dimensionless parameter bk, and knowing the form of the general solution for the twodimensional Stokes flow, one can solve the problem approximating U with an increasing order of accuracy
Summary
The problem of locomotion at low Reynolds numbers has been initiated by the groundbreaking paper of Taylor [1], which has had an enormous impact and continue to motivate a substantial amount of research (e.g. [2] and references cited therein, and the more than 1000 references that have cited this review paper since its publication 10 years ago). In [1], he considered the self-propulsion mechanism of a two-dimensional sheet, immersed in a viscous fluid, on which waves of transversal displacement propagate. Assuming these waves have small amplitude, a perturbative expansion of the boundary conditions can be used to compute the swimming speed of the oscillating sheet. Given the assumption of small amplitude, Taylor shows that expanding the boundary condition (1.2) in powers of the dimensionless parameter bk, and knowing the form of the general solution for the twodimensional Stokes flow, one can solve the problem approximating U with an increasing order of accuracy. Up to second-order terms in b k, this velocity is
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