The paper discusses the fundamental singularity of Stokes flow (the stokeslet) in the context of applications to locomotion and feeding currents in micro-organisms. The image system for a stokeslet in a rigid plane boundary may be derived from Lorentz's mirror image technique [1] or by an appropriate limit of Oseen's solution for a sphere near a plane boundary [2]. An alternative derivation using Fourier transform methods [3] leads to an immediate physical interpretation of the image system in terms of a stokeslet and its multipole derivatives. The schematic illustration of a stokeslet and its image system in a plane boundary are exploited to explain the fluid dynamical principles of ciliary propulsion. For a point force oriented normal to the plane boundary, the resulting axisymmetric motion leads to a Stokes stream function representation which illustrates the toroidal eddy structure of the flow field. A similar eddy structure is also obtained for the two-dimensional system, although in this case, the toroidal structure is replaced by two eddies. This closed streamline model is developed to model chaotic filtration through the concept of a 'blinking stokeslet', a stokeslet alternating its vertical position according to a specific protocol. The resulting behaviour is illustrated via Poincar6 sections, particle dispersion and length of particle path tracings. Sessile micro-organisms may exploit a similar process so they can filter as large a volume of liquid as possible in search of food and nutrients.
Mixing and transport processes associated with slow viscous flows are studied in the context of a blinking stokeslet above a plane rigid boundary. Whilst the motivation for this study comes from feeding currents due to cilia or flagella in sessile micro- organisms, other applications in physiological fluid mechanics where eddying motions occur include the enhanced mixing which may arise in ‘bolus’ flow between red blood cells, peristaltic motion and airflow in alveoli. There will also be further applications to micro-engineering flows at micron lengthscales. This study is therefore of generic interest because it analyses the opportunities for enhanced transport and mixing in a Stokes flow environment in which one or more eddies are a central feature.The central premise in this study is that the flow induced by the beating of microscopic flagella or cilia can be modelled by point forces. The resulting system is mimicked by using an implicit map, the introduction of which greatly aids the study of the system's dynamics. In an earlier study, Blake & Otto (1996), it was noticed that the blinking stokeslet system can have a chaotic structure. Poincaré sections and local Lyapunov exponents are used here to explore the structure of the system and to give quantitative descriptions of mixing; calculations of the barriers to diffusion are also presented. Comparisons are made between the results of these approaches. We consider the trajectories of tracer particles whose density may differ from the ambient fluid; this implies that the motion of the particles is influenced by inertia. The smoothing effect of molecular diffusion can be incorporated via the direct solution of an advection–diffusion equation or equivalently the inclusion of white noise in the map. The enhancement to mixing, and the consequent ramifications for filter feeding due to chaotic advection are demonstrated.
It is known that certain configurations which possess curvature are prone to a class of instabilities which their ‘flat’ counterparts will not support. The main thrust of the study of these centrifugal instabilities has concentrated on curved solid boundaries and their effect on the fluid motion. In this article attention is shifted towards a fluid-fluid interface observed within a curved incompressible mixing layer. Experimental evidence is available to support the conjecture that this situation may be subject to centrifugal instabilities. The evolution of modes with wavelengths comparable with the layer's thickness is considered within moderately curved mixing layers. The high Taylor/Görtler number régime is also discussed which characterizes the ultimate fate of the modes.
Finding and interpreting the solutions of differential equations is a central and essential part of applied mathematics. This book aims to enable the reader to develop the required skills needed for a thorough understanding of the subject. The authors focus on the business of constructing solutions analytically, and interpreting their meaning, using rigorous analysis where needed. MATLAB is used extensively to illustrate the material. There are many worked examples based on interesting and unusual real world problems. A large selection of exercises is provided, including several lengthier projects, some of which involve the use of MATLAB. The coverage is broad, ranging from basic second-order ODEs and PDEs, through to techniques for nonlinear differential equations, chaos, asymptotics and control theory. This broad coverage, the authors' clear presentation and the fact that the book has been thoroughly class-tested will increase its attraction to undergraduates at each stage of their studies.
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