The many-body reciprocal theorem and swimmer hydrodynamics

We present a reinterpretation and extension of the reciprocal theorem for swimmers, extending its application from the motion of a single swimmer in an unbounded domain to the general setting, giving results for both swimmer interactions and general hydrodynamics. We illustrate the method for a squirmer near a planar surface, recovering standard literature results and extending them to a general squirming set, to motion in the presence of a ciliated surface, and expressions for the flow field throughout the domain. Finally, we present exact results for the hydrodynamics in two dimensions which shed light on the near-field behaviour.Comment: 6 pages, 6 figure

Exact solutions for hydrodynamic interactions of two squirming spheres

We provide exact solutions of the Stokes equations for a squirming sphere close to a no-slip surface, both planar and spherical, and for the interactions between two squirmers, in three dimensions. These allow the hydrodynamic interactions of swimming microscopic organisms with confining boundaries, or each other, to be determined for arbitrary separation and, in particular, in the close proximity regime where approximate methods based on point singularity descriptions cease to be valid. We give a detailed description of the circular motion of an arbitrary squirmer moving parallel to a no-slip spherical boundary or flat free surface at close separation, finding that the circling generically has opposite sense at free surfaces and at solid boundaries. While the asymptotic interaction is symmetric under head-tail reversal of the swimmer, in the near field microscopic structure can result in significant asymmetry. We also find the translational velocity towards the surface for a simple model with only the lowest two squirming modes. By comparing these to asymptotic approximations of the interaction we find that the transition from near- to far-field behaviour occurs at a separation of about two swimmer diameters. These solutions are for the rotational velocity about the wall normal, or common diameter of two spheres, and the translational speed along that same direction, and are obtained using the Lorentz reciprocal theorem for Stokes flows in conjunction with known solutions for the conjugate Stokes drag problems, the derivations of which are demonstrated here for completeness. The analogous motions in the perpendicular directions, i.e. parallel to the wall, currently cannot be calculated exactly since the relevant Stokes drag solutions needed for the reciprocal theorem are not available.Comment: 27 pages, 7 figure

The reciprocal theorem and swimmer interactions

We present a number of solutions for the hydrodynamic interaction between microscopic swimmers in a viscous fluid and confining geometries. The reciprocal theorem is adapted for this use, allowing existing solutions for Stokes drag problems to be used to calculate the motion and rotation of force-free swimmers as well as other aspects of the hydrodynamics, such as flow fields. We outline the general procedure for approximating the reciprocal theorem to calculate motion for an arbitrary slip velocity by exploiting existing solutions for point forces and point torques in Stokes flows. This is demonstrated with two examples: firstly, the commonly studied case of a swimmer in the presence of an infinite wall, where we find the reported circling of certain bacteria near a surface, and reproduce the equations of motion for a swimmer in the presence of a wall found by other means; and secondly, a calculation giving the leading contributions to the motion of a swimmer between two infinite parallel plates, representing a strongly confining geometry, and relying upon the derivation of the flow due to a point torque in this geometry, a new result. We then derive exact solutions in two and three dimensions. In two dimensions we find the equations of motion for a circular squirmer with arbitrary axisymmetric slip velocity near a plane wall or inside a circular cavity, and discuss the extension to the case of two squirmers interacting with each other, which presents some additional mathematical difficulties. In three dimensions we provide exact solutions for the axisymmetric motion of a squirming sphere close to a no-slip surface, both planar and spherical. These allow the hydrodynamic interactions of swimming microscopic organisms with confining boundaries, or each other, to be determined for arbitrary separation and, in particular, in the close proximity regime where approximate methods based on point singularity descriptions cease to be valid. We find that the circling motion of flagellated bacteria generically has opposite sense at free surfaces and at solid boundaries, as seen experimentally. By comparing these to asymptotic approximations of the interaction we find that the transition from near-to far-field behaviour occurs at a separation of about two swimmer diameters. Finally we discuss possible extensions to this work, and limitations of the approach used

Universality in ant behaviour

© 2014 The Authors. Prediction for social systems is a major challenge. Universality at the social level has inspired a unified theory for urban living but individual variation makes predicting relationships within societies difficult. Here, we show that in ant societies individual average speed is higher when event duration is longer. Expressed as a single scaling function, this relationship is universal because for any event duration an ant, on average, moves at the corresponding average speed except for a short acceleration and deceleration at the beginning and end. This establishes cause and effect within a social system and may inform engineering and control of artificial ones

Optimisation of Car Park Designs

The problem presented by ARUP to the UK Study Group 2014 was to investigate methods for maximising the number of car parking spaces that can be placed within a car park. This is particularly important for basement car parks in residential apartment blocks or offices where parking spaces command a high value. Currently the job of allocating spaces is done manually and is very time intensive. The Study Group working on this problem split into teams examining different aspects of the car park design process There were three approaches taken. These approaches include a so-called "tile-and-trim" method in which an optimal layout of cars from an `infinite car park' are overlaid onto the actual car park domain; adjustments are then made to accommodate access from one lane to the next. A second approach seeks to develop an algorithm for optimising the road within a car park on the assumption that car parking spaces should fill the space and that any space needs to be adjacent to the network. A third similar approach focused on schemes for assessing the potential capacity of a small selection of specified road networks within the car park to assist the architect in selecting the optimal road network(s). The problem is a variant of the "bin packing" problem, well known in computer science. It is further complicated by the fact that two different classes of item need to be packed (roads and cars), with both local (immediate access to a road) and global (connectivity of the road network) constraints. Bin-packing is known to be NP-hard, and hence the problem at hand has at least this level of computational complexity. None of the approaches produced a complete solution to the problem posed. Indeed, it was quickly determined by the group that this was a very hard problem (a view reinforced by the many different possible approaches considered) requiring far longer than a week to really make significant progress. All approaches rely to differing degrees on optimisation algorithms which are inherently unreliable unless designed specifically for the intended purpose. It is also not clear whether a relatively simple automated computer algorithm will be able to "beat the eye of the architect"; additional sophistication may be required due to subtle constraints. Apart from determining that the problem is hard, positive outcomes have included: Determining that parking perpendicular to the road in long aisles provides the most efficient packing of cars. Provision of code which "tiles and trims" from an infinite car park onto the given car park with interactive feedback on the number of cars in the packing. Provision of code for optimal packing in a parallel-walled car park. Methods for optimising a road within a given domain based on developing cost functions ensuring that cars fill the car park and have access to the road. Provision of code for optimising a single road in a given (square) space. Description of methods for assessing the capacity of a car park for a set of given road network in order to select optimal road networks. Some ideas for developing possible solutions further

Potentiation effect of the AMPK activator A-769662 on cardiac myocytes metabolism and survival

Abstract 286 van Poster session 2 Frontiers in CardioVascular Biology, London 30th March – 1st April 2012 Second Congress of the ESC Council on Basic Cardiovascular Science