2 results
Boundary element methods for particles and microswimmers in a linear viscoelastic fluid
- Kenta Ishimoto, Eamonn A. Gaffney
-
- Journal:
- Journal of Fluid Mechanics / Volume 831 / 25 November 2017
- Published online by Cambridge University Press:
- 13 October 2017, pp. 228-251
-
- Article
- Export citation
-
The consideration of viscoelasticity within fluid dynamical boundary element methods has traditionally required meshing over the whole flow domain. In turn, a major advantage of the boundary element method is lost, namely the need to consider only surface boundary integrals. Here, using a generalised reciprocal relation and viscoelastic force singularities, a boundary element method is developed for linear viscoelastic flows. We proceed to explore finite-deformation microswimming in a linear Maxwell fluid. We firstly deduce a finite-amplitude generalisation of a previously reported result that the flow field is unchanged between a Newtonian and linear Maxwell fluid for prescribed small-amplitude deformations. Hence Purcell’s theorem holds for a linear Maxwell fluid. We proceed to consider deformation swimming in a linear Maxwell fluid given an external forcing. Boundary scattering trajectories for an exemplar squirmer approaching a surface are observed to exhibit a weak dependence on the Deborah number, while the trajectories of a sperm and monotrichous bacterium near a surface are predicted to be essentially unaffected at moderate Deborah number. In turn, the latter supports the common simplification of using Newtonian Stokes flows for studying flagellate swimming in linear Maxwell media. In addition, the motion of a magnetic helix under the influence of an external magnetic field is considered, and highlights that linear viscoelasticity can significantly impact the propagation of the helix, in turn demonstrating that even linear rheology is important to consider for forced swimmers. Finally, the presented framework requires minimalistic adjustments to Newtonian boundary element codes, enabling rapid implementation, and is more generally applicable, for instance to studies of particle interactions in active linear rheology on the microscale.
9 - Turing's Theory of Developmental Pattern Formation
- from Part Three - The Reverse Engineering Road to Computing Life
-
- By Philip K. Maini, Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK, Thomas E. Woolley, Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK, Eamonn A. Gaffney, Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK, Ruth E. Baker, Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
- Edited by S. Barry Cooper, University of Leeds, Andrew Hodges, University of Oxford
-
- Book:
- The Once and Future Turing
- Published online:
- 05 March 2016
- Print publication:
- 24 March 2016, pp 131-143
-
- Chapter
- Export citation
-
Summary
Introduction
Elucidating the mechanisms underlying the formation of structure and form is one of the great challenges in developmental biology. From an initial, seemingly spatially uniform mass of cells, emerge the spectacular patterns that characterise the animal kingdom – butterfly wing patterns, animal coat markings, skeletal structures, skin organs, horns etc. (Figure 9.1). Although genes obviously play a key role, the study of genetics alone does not tell us why certain genes are switched on or off in specific places and how the properties they impart to cells result in the highly coordinated emergence of pattern and form. Modern genomics has revealed remarkable molecular similarity among different animal species. Specifically, biological diversity typically emerges from differences in regulatory DNA rather than detailed protein coding sequences. This implicit universality highlights that many aspects of animal development can be understood from studies of exemplar species such as fruit flies and zebrafish while also motivating theoretical studies to explore and understand the underlying common mechanisms beyond a simply descriptive level.
However, when Alan Turing wrote his seminal paper, ‘The chemical basis of morphogenesis’ (Turing, 1952), such observations were many decades away. At that time biology was following a very traditional classification route of list-making activities. Indeed, there was very little theory regarding development other than D'Arcy Thompson's classic 1917 work (see Thompson, 1992, for the abridged version) exploring how biological forms arose, though even this was still very much at the descriptive rather than the mechanistic level.
Undeterred, Turing started exploring the question of how developmental systems might undertake symmetry-breaking and thus create and amplify structure from seeming uniformity. For example, if one looks at a cross-section of a tree trunk, it has circular symmetry which is broken when a branch starts to grow outwards. Turing proposed an underlying mechanism explaining how asymmetric structure could emerge dynamically, without innate hardwiring. In particular, he described how a symmetric pattern, for instance of a growth hormone, could break up so that more hormone was concentrated on one part of the circle, thus inducing extra growth there.
In order to achieve such behaviour Turing came up with a truly ingenious theory. He considered a system of chemicals reacting with each other and assumed that in the well-mixed case (no spatial heterogeneities) this system exhibited an equilibrium (steady) state which was stable.