On oscillatory oseenlets: the forces generated by them, the far field expansion obtained using them, and their relation to oscillatory stokeslets

Chadwick, Edmund; Elmazuzi, Rabea

Abstract

Consider uniform flow past an oscillating body. Assume that the resulting far-field flow consists of both steady and time periodic components. The time periodic components can be decomposed into a Fourier expansion series of time harmonic components. The form of the steady component in terms of the steady oseenlet is well-known. However, the time-harmonic components in terms of the oscillatory oseenlet do not yet appear to be in the literature. The oscillatory oseenlet solution is presented for the velocity and pressure, and the forces generated by them are calculated and shown to be oscillatory. The formulation is checked against known solutions in the following two limits: As the frequency of the oscillations tend to zero, it is shown that the steady oseenlet solution is recovered; Also, as the Reynolds number of the flow tends to zero, it is shown that the oscillatory stokeslet solution is recovered. Both the oscillatory stokeslet and oscillatory oseenlet solutions are represented by velocity potentials, and the oscillatory oseenlets are used to give a far-field velocity expansion. 1. INTRODUCTION The problem of uniform flow past an oscillating body is a general one, examples being the flapping flight of birds and insects, and the swimming of mammals, fish and microorganisms. Of particular interest to the authors is a feasibility study for the robotics centre of the university into the propulsion of miniaturised robot devices through fluid by means of an oscillatory swimming motion. The eventual goal is for devices at the milli-to nano-metre range that could be used within the blood stream for various medical purposes such as tissue or cell repair, and this is very much at the feasibility study stage. The steady forward velocity lends itself to the Oseen linearisation in the far-field, that the velocity perturbation is small compared to the uniform stream (or forward propulsive velocity). The literature on time dependent Oseen and associated Stokes flows subdivides into transient analysis and oscillatory analysis, with the majority of work on transient rather than oscillatory analysis. Price [16] uses transient oseenlets in order to model ship motions. Also, Chwang and co-workers [6] [13] describe the unsteady (transient) stokeslet and oseenlet and give applications related to acceleration and free surface waves. Childress [7] uses transient oseenlets to model the effect of flapping of a swimming mollusc as it thrusts forward. A numerical solution of the transient oseenlet analysis is employed. However, it is clear that if a steady oscillatory motion of the swimming mollusc is required instead, an oscillatory oseenlet would be preferable but is not currently available in the literature. Riley's and Amin's work [17] [1] [18] does employ an oscillatory rather than transient analysis to model the flow generated by fixed oscillating bodies. Here, the focus is on matching the inner Stokes-type flow to an outer flow. Clarke et. al [8] consider the problem of a MEMS device vibrating in a fluid at rest. The device is treated as a slender body and the Stokes approximation is used. The oscillatory stokeslet given by Pozrikidis [15] is used. However, there is no uniform stream for these problems and so the outer flow is not an Oseen flow and very different from it. The problem of a forward moving body in the fluid additionally requires a far-field analysis where the velocity is linearized to a uniform stream. Within such a development, the oscillatory stokeslet is an inner near-field description to be matched to an outer far-field oscillatory oseenlet. In order to enable this, there is a requirement for the oscillatory oseenlet solution. Iima [9] considers a butterfly flapping and whether it can sustain a hovering motion. He formulates a two-dimensional far-field periodic Oseen representation for a small steady uniform flow motion and then lets that motion tend to zero. This representation is not expressed in terms of oseenlets, and instead uses an approach based upon that of Imai [10]. Yet the representation by singular (stokeslet, oseenlet) solutions has many advantages, one being that a body can be represented in a straightforward way by a distributed superposition of them [2], and another being the additional insight into the physical understanding of the flow such a model provides. For example, Iima's result is for two-rather than three-dimensional flow and it is difficult to see how the result can be extended by using Iima's formulation. By finding the oseenlets, it is anticipated that this will enable us to investigate the hovering paradox to three-dimensional flow, but this will be left for future work. In the literature are currently available the steady, transient and oscillatory stokeslet as well as the steady and transient oseenlet. The omission of the oscillatory oseenlet representation within the literature is significant, and restricts the

Citation

Chadwick, E., & Elmazuzi, R. (2011). On oscillatory oseenlets: the forces generated by them, the far field expansion obtained using them, and their relation to oscillatory stokeslets. #Journal not on list, 9(1),

• Journal Article Type: Article
• Publication Date: 2011-10
• Deposit Date: Jul 18, 2024
• Journal: IJRRAS
• Peer Reviewed: Peer Reviewed
• Volume: 9
• Issue: 1
• Related Public URLs: https://www.researchgate.net/publication/267299098_ON_OSCILLATORY_OSEENLETS_THE_FORCES_GENERATED_BY_THEM_THE_FAR_FIELD_EXPANSION_OBTAINED_USING_THEM_AND_THEIR_RELATION_TO_OSCILLATORY_STOKESLETS