We analyse a simple 'Stokesian squirmer' model for the enhanced mixing due to swimming micro-organisms. The model is based on a calculation of Thiffeault & Childress [Physics Letters A, 374, 3487 (2010), arXiv:0911.5511], where fluid particle displacements due to inviscid swimmers are added to produce an effective diffusivity. Here we show that for the viscous case the swimmers cannot be assumed to swim an infinite distance, even though their total mass displacement is finite. Instead, the largest contributions to particle displacement, and hence to mixing, arise from random changes of direction of swimming and are dominated by the far-field stresslet term in our simple model. We validate the results by numerical simulation. We also calculate nonzero Reynolds number corrections to the effective diffusivity. Finally, we show that displacements due to randomly-swimming squirmers exhibit PDFs with exponential tails and a short-time superdiffusive regime, as found previously by several authors. In our case, the exponential tails are due to 'sticking' near the stagnation points on the squirmer's surface.Comment: 10 pages, 12 figures. PDFLaTeX with JFM style (included). Accepted for publication in Journal of Fluid Mechanic
We address the challenge of optimal incompressible stirring to mix an initially inhomogeneous distribution of passive tracers. As a quantitative measure of mixing we adopt the $H^{-1}$ norm of the scalar fluctuation field, equivalent to the (square-root of the) variance of a low-pass filtered image of the tracer concentration field. First we establish that this is a useful gauge even in the absence of molecular diffusion: its vanishing as $t --> \infty$ is evidence of the stirring flow's mixing properties in the sense of ergodic theory. Then we derive absolute limits on the total amount of mixing, as a function of time, on a periodic spatial domain with a prescribed instantaneous stirring energy or stirring power budget. We subsequently determine the flow field that instantaneously maximizes the decay of this mixing measure---when such a flow exists. When no such `steepest descent' flow exists (a possible but non-generic situation) we determine the flow that maximizes the growth rate of the $H^{-1}$ norm's decay rate. This local-in-time optimal stirring strategy is implemented numerically on a benchmark problem and compared to an optimal control approach using a restricted set of flows. Some significant challenges for analysis are outlined.Comment: 10 pages, 3 figures. PDFLaTeX with JFM style (included
This work reviews the present position of and surveys future perspectives in the physics of chaotic advection: the field that emerged three decades ago at the intersection of fluid mechanics and nonlinear dynamics, which encompasses a range of applications with length scales ranging from micrometers to hundreds of kilometers, including systems as diverse as mixing and thermal processing of viscous fluids, microfluidics, biological flows, and oceanographic and atmospheric flows.
Mixing is relevant to many areas of science and engineering, including the pharmaceutical and food industries, oceanography, atmospheric sciences, and civil engineering. In all these situations one goal is to quantify and often then to improve the degree of homogenisation of a substance being stirred, referred to as a passive scalar or tracer. A classical measure of mixing is the variance of the concentration of the scalar, which can be related to the L 2 norm of the concentration field. Recently other norms have been used to quantify mixing, in particular the mix-norm as well as negative Sobolev norms. These norms have the advantage that unlike variance they decay even in the absence of diffusion, and their decay corresponds to the flow being mixing in the sense of ergodic theory. General Sobolev norms weigh scalar gradients differently, and are known as multiscale norms for mixing. We review the applications of such norms to mixing and transport, and show how they can be used to optimise the stirring and mixing of a decaying passive scalar. We then review recent work on the less-studied case of a continuously-replenished scalar field -the source-sink problem. In that case the flows that optimally reduce the norms are associated with transport rather than mixing: they push sources onto sinks, and vice versa.
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