scholarly journals Unsteady feeding and optimal strokes of model ciliates

2013 ◽  
Vol 715 ◽  
pp. 1-31 ◽  
Author(s):  
Sébastien Michelin ◽  
Eric Lauga
Keyword(s):  
Peclet Number ◽  
Feeding Rate ◽  
Diffusion Problem ◽  
Advective Transport ◽  
Péclet Number ◽  
Fixed Energy ◽  
Advection Diffusion ◽  
The Mean ◽  
Micro Organisms ◽  
Time Periodic

AbstractThe flow field created by swimming micro-organisms not only enables their locomotion but also leads to advective transport of nutrients. In this paper we address analytically and computationally the link between unsteady feeding and unsteady swimming on a model micro-organism, the spherical squirmer, actuating the fluid in a time-periodic manner. We start by performing asymptotic calculations at low Péclet number ($\mathit{Pe}$) on the advection–diffusion problem for the nutrients. We show that the mean rate of feeding as well as its fluctuations in time depend only on the swimming modes of the squirmer up to order ${\mathit{Pe}}^{3/ 2} $, even when no swimming occurs on average, while the influence of non-swimming modes comes in only at order ${\mathit{Pe}}^{2} $. We also show that generically we expect a phase delay between feeding and swimming of $1/ 8\mathrm{th} $ of a period. Numerical computations for illustrative strokes at finite $\mathit{Pe}$ confirm quantitatively our analytical results linking swimming and feeding. We finally derive, and use, an adjoint-based optimization algorithm to determine the optimal unsteady strokes maximizing feeding rate for a fixed energy budget. The overall optimal feeder is always the optimal steady swimmer. Within the set of time-periodic strokes, the optimal feeding strokes are found to be equivalent to those optimizing periodic swimming for all values of the Péclet number, and correspond to a regularization of the overall steady optimal.

Study of a Non-Conventional Aeration System

1987 ◽  
Vol 19 (5-6) ◽  
pp. 869-876
Author(s):  
L. Raschid-Sally ◽  
M. Roustan ◽  
H. Roques ◽  
G. M. Faup
Keyword(s):  
Oxygen Transfer ◽  
Peclet Number ◽  
Péclet Number ◽  
Theoretical Approaches ◽  
Aeration System ◽  
Circulation Velocity ◽  
The Mean ◽  
Oxygen Transfer Capacity ◽  
Mean Circulation ◽  
Conventional Systems

A non-conventional aeration system for oxidation ditches using jets has been developed. The principle of this system is based on the separation of the 2 actions: aeration and circulation. It was concluded that the flow of the liquid in the channel can be successfully modelled using various theoretical approaches. The mean circulation velocity VC, the power dissipated P, and the Peclet number Pe are the 3 important parameters governing the circulation. The oxygen transfer capacity of the system has been studied and compares favourably with that of conventional systems. The advantage of such systems over conventional ones has been discussed.

scholarly journals Transient residence and exposure times

2005 ◽  
Vol 2 (3) ◽  
pp. 247-265 ◽  
Author(s):  
E. J. M. Delhez
Keyword(s):  
Boundary Conditions ◽  
Residence Time ◽  
Mean Residence Time ◽  
Water Body ◽  
Adjoint Method ◽  
Diffusion Problem ◽  
Advection Diffusion ◽  
The Mean ◽  
Estuarine Coastal ◽  
Water Parcel

Abstract. The residence time measures the time spent by a water parcel or a pollutant in a given water body and is therefore widely used in environmental studies. The adjoint method introduced by Delhez et al. (Estuarine Coastal and Shelf Sciences, 2004) to compute this diagnostic is revised here to take into account the effect of the initialisation and of the boundary conditions. In addition to the equation for the mean residence time, it is suggested to solve a simple advection-diffusion problem to quantify the effect of the initialisation and clarify the interpretation of the results. Using the two same equations but with modified boundary conditions, the method can also be used to quantify the accumulated time spent by water/tracer parcels in a control domain. This diagnostic is called "exposure time".

scholarly journals Transient residence and exposure times

Ocean Science ◽  
2006 ◽  
Vol 2 (1) ◽  
pp. 1-9 ◽  
Author(s):  
E. J. M. Delhez
Keyword(s):  
Boundary Conditions ◽  
Residence Time ◽  
Mean Residence Time ◽  
Water Body ◽  
Adjoint Method ◽  
Realistic Model ◽  
Diffusion Problem ◽  
Advection Diffusion ◽  
The Mean ◽  
Water Parcel

Abstract. The residence time measures the time spent by a water parcel or a pollutant in a given water body and is therefore widely used in environmental studies. The adjoint method introduced by Delhez et al. (2004) to compute this diagnostic is revised here to take into account the effect of the initialization and of the boundary conditions. In addition to the equation for the mean residence time, it is suggested to solve a simple advection-diffusion problem to quantify the effect of the initialization and clarify the interpretation of the results. Using the two same equations but with modified boundary conditions, the method can also be used to quantify the accumulated time spent by water/tracer parcels in a control domain. This diagnostic is called "exposure time". Analytical and realistic model results are used to illustrate the concepts.

Sedimentation in a dilute polydisperse system of interacting spheres. Part 1. General theory

1982 ◽  
Vol 119 ◽  
pp. 379-408 ◽  
Author(s):  
G. K. Batchelor
Keyword(s):  
Distribution Function ◽  
Brownian Motion ◽  
Peclet Number ◽  
Pair Distribution Function ◽  
Brownian Diffusion ◽  
Mean Velocity ◽  
Péclet Number ◽  
Interparticle Force ◽  
Equal Sphere ◽  
The Mean

Small rigid spherical partials are settling under gravity through Newtonian fluid, and the volume fraction of the particles (ϕ) is small although sufficiently large for the effects of interactions between pairs of particles to be significant. Two neighbouring particles interact both hydrodynamically (with low-Reynolds-number flow about each particle) and through the exertion of a mutual force of molecular or electrical origin which is mainly repulsive; and they also diffuse relatively to each other by Brownian motion. The dispersion contains several species of particle which differ in radius and density.The purpose of the paper is to derive formulae for the mean velocity of the particles of each species correct to order ϕ, that is, with allowance for the effect of pair interactions. The method devised for the calculation of the mean velocity in a monodisperse system (Batchelor 1972) is first generalized to give the mean additional velocity of a particle of species i due to the presence of a particle of species j in terms of the pair mobility functions and the probability distribution pii(r) for the relative position of an i and a j particle. The second step is to determine pij(r) from a differential equation of Fokker-Planck type representing the effects of relative motion of the two particles due to gravity, the interparticle force, and Brownian diffusion. The solution of this equation is investigated for a range of special conditions, including large values of the Péclet number (negligible effect of Brownian motion); small values of the Ptclet number; and extreme values of the ratio of the radii of the two spheres. There are found to be three different limits for pij(r) corresponding to different ways of approaching the state of equal sphere radii, equal sphere densities, and zero Brownian relative diffusivity.Consideration of the effect of relative diffusion on the pair-distribution function shows the existence of an effective interactive force between the two particles and consequently a contribution to the mean velocity of the particles of each species. The direct contributions to the mean velocity of particles of one species due to Brownian diffusion and to the interparticle force are non-zero whenever the pair-distribution function is non-isotropic, that is, at all except large values of the Péclet number.The forms taken by the expression for the mean velocity of the particles of one species in the various cases listed above are examined. Numerical values will be presented in Part 2.

scholarly journals Diffusion of a passive scalar from a no-slip boundary into a two-dimensional chaotic advection field

1998 ◽  
Vol 372 ◽  
pp. 119-163 ◽  
Author(s):  
S. GHOSH ◽  
A. LEONARD ◽  
S. WIGGINS
Keyword(s):  
Passive Scalar ◽  
Diffusion Problem ◽  
Chaotic Advection ◽  
Two Dimensional ◽  
Shadowing Property ◽  
Slip Boundary ◽  
Advection Diffusion ◽  
Scalar Diffusion ◽  
Time Periodic ◽  
Large Peclet Number

Using a time-periodic perturbation of a two-dimensional steady separation bubble on a plane no-slip boundary to generate chaotic particle trajectories in a localized region of an unbounded boundary layer flow, we study the impact of various geometrical structures that arise naturally in chaotic advection fields on the transport of a passive scalar from a local ‘hot spot’ on the no-slip boundary. We consider here the full advection-diffusion problem, though attention is restricted to the case of small scalar diffusion, or large Péclet number. In this regime, a certain one-dimensional unstable manifold is shown to be the dominant organizing structure in the distribution of the passive scalar. In general, it is found that the chaotic structures in the flow strongly influence the scalar distribution while, in contrast, the flux of passive scalar from the localized active no-slip surface is, to dominant order, independent of the overlying chaotic advection. Increasing the intensity of the chaotic advection by perturbing the velocity field further away from integrability results in more non-uniform scalar distributions, unlike the case in bounded flows where the chaotic advection leads to rapid homogenization of diffusive tracer. In the region of chaotic particle motion the scalar distribution attains an asymptotic state which is time-periodic, with the period the same as that of the time-dependent advection field. Some of these results are understood by using the shadowing property from dynamical systems theory. The shadowing property allows us to relate the advection-diffusion solution at large Péclet numbers to a fictitious zero-diffusivity or frozen-field solution, corresponding to infinitely large Péclet number. The zero-diffusivity solution is an unphysical quantity, but is found to be a powerful heuristic tool in understanding the role of small scalar diffusion. A novel feature in this problem is that the chaotic advection field is adjacent to a no-slip boundary. The interaction between the necessarily non-hyperbolic particle dynamics in a thin near-wall region and the strongly hyperbolic dynamics in the overlying chaotic advection field is found to have important consequences on the scalar distribution; that this is indeed the case is shown using shadowing. Comparisons are made throughout with the flux and the distributions of the passive scalar for the advection-diffusion problem corresponding to the steady, unperturbed, integrable advection field.

NUMERICAL SCHEMES OBTAINED FROM LATTICE BOLTZMANN EQUATIONS FOR ADVECTION DIFFUSION EQUATIONS

2006 ◽  
Vol 17 (11) ◽  
pp. 1563-1577 ◽  
Author(s):  
SHINSUKE SUGA
Keyword(s):  
Lattice Boltzmann ◽  
Peclet Number ◽  
Eigenvalue Problems ◽  
Velocity Model ◽  
Diffusion Equations ◽  
Numerical Schemes ◽  
Péclet Number ◽  
Velocity Models ◽  
Advection Diffusion ◽  
The Stability

Stability and accuracy of the numerical schemes obtained from the lattice Boltzmann equation (LBE) used for numerical solutions of two-dimensional advection-diffusion equations are presented. Three kinds of velocity models are used to determine the moving velocity of particles on a squre lattice. A system of explicit finite difference equations are derived from the LBE based on the Bhatnagar, Gross and Krook (BGK) model for individual velocity model. In order to approximate the advecting velocity field, a linear equilibrium distribution function is used for each of the moving directions. The stability regions of the schemes in the special case of the relaxation parameter ω in the LBE being set to ω=1 are found by analytically solving the eigenvalue problems of the amplification matrices corresponding to each scheme. As for the cases of general relaxation parameters, the eigenvalue problems are solved numerically. A benchmark problem is solved in order to investigate the relationship between the accuracy of the numerical schemes and the order of the Peclet number. The numerical experiments result in indicating that for the scheme based on a 9-velocity model we can find the parameters depending on the order of the given Peclet number, which generate accurate solutions in the stability region.

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