The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

There is a special reason for reviewing this book at this time: it is the 50th edition of a compendium that is known and used frequently in most chemical and physical laboratories in many parts of the world. Surely, a publication that has been published for 56 years, withstanding the vagaries of science in this century, must have had something to offer. There is another reason: while the book is a standard fixture in most chemical and physical laboratories, including those in medical centers, it is not as frequently seen in the laboratories of physician's offices (those either in solo or group practice). I believe that the Handbook can be useful in those laboratories. One of the reasons, among others, is that the various basic items of information it offers may be helpful in new tests, either physical or chemical, which are continuously being published. The basic information may relate

Handbook of Chemistry and Physics

Jets, ie collimated streams of matter, occur from the microscale up to the large-scale structure of the universe Our focus will be mostly on surface tension effects, which result from the cohesive properties of liquids Paradoxically, cohesive forces promote the breakup of jets, widely encountered in nature, technology and basic science, for example in nuclear fission, DNA sampling, medical diagnostics, sprays, agricultural irrigation and jet engine technology Liquid jets thus serve as a paradigm for free-surface motion, hydrodynamic instability and singularity formation leading to drop breakup In addition to their practical usefulness, jets are an ideal probe for liquid properties, such as surface tension, viscosity or non-Newtonian rheology They also arise from the last but one topology change of liquid masses bursting into sprays Jet dynamics are sensitive to the turbulent or thermal excitation of the fluid, as well as to the surrounding gas or fluid medium The aim of this review is to provide a unified description of the fundamental and the technological aspects of these subjects

Physics of liquid jets

Chemotactic bacteria are guided towards the source of a nutrient by local concentration gradients. That works on the microscopic scale, but at larger scales such local cues are unreliable pointers — for example, wind or water currents may disperse odours sought by foraging animals. Using statistical techniques, Vergassola et al. have developed a general search algorithm for movement strategies based on the detection of sporadic cues and partial information. The strategy, termed 'infotaxis' as it maximizes the expected rate of information gain, could find application in the design of 'sniffer' robots. A computational model of odour plume propagation and experimental data are used to devise a general search algorithm for movement strategies in chemotaxis, based on sporadic cues and partial information. The strategy is termed 'infotaxis' as it locally maximizes the expected rate of information gain. Chemotactic bacteria rely on local concentration gradients to guide them towards the source of a nutrient1. Such local cues pointing towards the location of the source are not always available at macroscopic scales because mixing in a flowing medium breaks up regions of high concentration into random and disconnected patches. Thus, animals sensing odours in air or water detect them only intermittently as patches sweep by on the wind or currents2,3,4,5,6. A macroscopic searcher must devise a strategy of movement based on sporadic cues and partial information. Here we propose a search algorithm, which we call ‘infotaxis’, designed to work under such conditions. Any search process can be thought of as acquisition of information on source location; for infotaxis, information plays a role similar to concentration in chemotaxis. The infotaxis strategy locally maximizes the expected rate of information gain. We demonstrate its efficiency using a computational model of odour plume propagation and experimental data on mixing flows7. Infotactic trajectories feature ‘zigzagging’ and ‘casting’ paths similar to those observed in the flight of moths8. The proposed search algorithm is relevant to the design of olfactory robots9,10,11, but the general idea of infotaxis can be applied more broadly in the context of searching with sparse information.

"Infotaxis" as a strategy for searching without gradients

We depict and analyse the successive steps of atomization of a liquid jet when a fast gas stream blows parallel to its surface. Experiments performed with various liquids in a fast air flow show that the liquid destabilization proceeds from a two-stage mechanism: a shear instability first forms waves on the liquid. The transient acceleration experienced by the liquid suggests that a Rayleigh–Taylor type of instability is triggered at the wave crests, producing liquid ligaments which further stretch in the air stream and break into droplets.

On spray formation

The spray formed when a fast gas stream blows over a liquid volume presents a wide distribution of fragment sizes. The process involves a succession of changes of the liquid topology, the last being the elongation and capillary breakup of ligaments torn off from the liquid surface. The coalescence of the liquid volumes constitutive of a ligament at the very moment it detaches from the liquid bulk produces larger drops. This aggregation process has its counterpart on the shape of the size distribution associated with the ligament breakup, found to be very well represented by gamma distributions. The exponential shape of the overall distribution in the spray coincides with the large excursion wing of these elementary distributions, underlying the crucial role played by the ligament dynamics in building up the broad statistics of sprays.

/pdf/ligament-mediated-spray-formation-4mupu3kwwo.pdf

Ligament-mediated spray formation.

Experiments show how a stirred scalar mixture relaxes towards uniformity through an aggregation process. The elementary bricks are stretched sheets whose rates of diffusive smoothing and coalescence build up the overall mixture concentration distribution. The cases studied, in particular, include mixtures in two and three dimensions, with different stirring protocols and Reynolds numbers which all lead to a unique family of concentration distributions stable by self-convolution, the signature of the aggregation mechanism from which they originate.

Mixing as an aggregation process

Mixing as an aggregation process.

We study the relaxation of initially segregated scalar mixtures in randomly stirred media, aiming to describe the overall concentration distribution of the mixture, its shape and rate of deformation as it evolves towards uniformity. A stirred scalar mixture can be viewed as a collection of stretched sheets, possibly interacting with each other. We consider a situation in which the interaction between the sheets is enforced by confinement and is the key factor ruling its evolution. It consists of following a mixture relaxing towards uniformity around a fixed average concentration while flowing along a constant cross-section channel. The interaction between the sheets is found to be of a random addition nature in concentration space, leading to concentration distributions that are stable by self-convolution. The resulting scalar field is naturally coarsened at a scale much larger than the dissipation scale. Consequences on the mixture entropy and scalar rate of dissipation are also examined.

Mixing by random stirring in confined mixtures

Experiments show that a blob of scalar released in the inertial range of scales of a turbulent medium is rapidly converted into a set of disjointed sheets whose spectrum exhibits a k−1 shape for wave numbers larger than the injection wave number 1/d. The sheets diffusive uniformization onsets at a time ts=(d/u)f(Sc) with f(Sc)∼ln(Sc) function of the injection time of the blob in the medium d/u, of the Schmidt number Sc, independently of the Reynolds number. This time is appreciably smaller than the time needed to cross the Kolmogorov cascade, which is “bypassed” by a strong and constant stretching rate acting at the injection scale.

/pdf/short-circuits-in-the-corrsin-obukhov-cascade-12jejrxxnu.pdf

Jérôme Duplat

Papers

Ligament-mediated spray formation.

Mixing as an aggregation process

Mixing as an aggregation process.

Mixing by random stirring in confined mixtures

Short circuits in the Corrsin–Obukhov cascade