Macroscopic thin liquid films are entities that are important in biophysics, physics, and engineering, as well as in natural settings. They can be composed of common liquids such as water or oil, rheologically complex materials such as polymers solutions or melts, or complex mixtures of phases or components. When the films are subjected to the action of various mechanical, thermal, or structural factors, they display interesting dynamic phenomena such as wave propagation, wave steepening, and development of chaotic responses. Such films can display rupture phenomena creating holes, spreading of fronts, and the development of fingers. In this review a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations. As a result of this long-wave theory, a mathematical system is obtained that does not have the mathematical complexity of the original free-boundary problem but does preserve many of the important features of its physics. The basics of the long-wave theory are explained. If, in addition, the Reynolds number of the flow is not too large, the analogy with Reynolds's theory of lubrication can be drawn. A general nonlinear evolution equation or equations are then derived and various particular cases are considered. Each case contains a discussion of the linear stability properties of the base-state solutions and of the nonlinear spatiotemporal evolution of the interface (and other scalar variables, such as temperature or solute concentration). The cases reducing to a single highly nonlinear evolution equation are first examined. These include: (a) films with constant interfacial shear stress and constant surface tension, (b) films with constant surface tension and gravity only, (c) films with van der Waals (long-range molecular) forces and constant surface tension only, (d) films with thermocapillarity, surface tension, and body force only, (e) films with temperature-dependent physical properties, (f) evaporating/condensing films, (g) films on a thick substrate, (h) films on a horizontal cylinder, and (i) films on a rotating disc. The dynamics of the films with a spatial dependence of the base-state solution are then studied. These include the examples of nonuniform temperature or heat flux at liquid-solid boundaries. Problems which reduce to a set of nonlinear evolution equations are considered next. Those include (a) the dynamics of free liquid films, (b) bounded films with interfacial viscosity, and (c) dynamics of soluble and insoluble surfactants in bounded and free films. The spreading of drops on a solid surface and moving contact lines, including effects of heat and mass transport and van der Waals attractions, are then addressed. Several related topics such as falling films and sheets and Hele-Shaw flows are also briefly discussed. The results discussed give motivation for the development of careful experiments which can be used to test the theories and exhibit new phenomena.

Long-scale evolution of thin liquid films

Surface-tension-driven flows and, in particular, their tendency to decay spontaneously into drops have long fascinated naturalists, the earliest systematic experiments dating back to the beginning of the 19th century. Linear stability theory governs the onset of breakup and was developed by Rayleigh, Plateau, and Maxwell. However, only recently has attention turned to the nonlinear behavior in the vicinity of the singular point where a drop separates. The increased attention is due to a number of recent and increasingly refined experiments, as well as to a host of technological applications, ranging from printing to mixing and fiber spinning. The description of drop separation becomes possible because jet motion turns out to be effectively governed by one-dimensional equations, which still contain most of the richness of the original dynamics. In addition, an attraction for physicists lies in the fact that the separation singularity is governed by universal scaling laws, which constitute an asymptotic solution of the Navier-Stokes equation before and after breakup. The Navier-Stokes equation is thus continued uniquely through the singularity. At high viscosities, a series of noise-driven instabilities has been observed, which are a nested superposition of singularities of the same universal form. At low viscosities, there is rich scaling behavior in addition to aesthetically pleasing breakup patterns driven by capillary waves. The author reviews the theoretical development of this field alongside recent experimental work, and outlines unsolved problems.

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Nonlinear dynamics and breakup of free-surface flows

Algorithmic trading has sharply increased over the past decade Does it improve market quality, and should it be encouraged? We provide the first analysis of this question The NYSE automated quote dissemination in 2003, and we use this change in market structure that increases algorithmic trading as an exogenous instrument to measure the causal effect of algorithmic trading on liquidity For large stocks in particular, algorithmic trading narrows spreads, reduces adverse selection, and reduces trade-related price discovery The findings indicate that algorithmic trading improves liquidity and enhances the informativeness of quotes

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID1668523_code49904.pdf?abstractid=1100635&rulid=5239579&mirid=2

Does Algorithmic Trading Improve Liquidity

Algorithmic trading (AT) has increased sharply over the past decade. Does it improve market quality, and should it be encouraged? We provide the first analysis of this question. The New York Stock Exchange automated quote dissemination in 2003, and we use this change in market structure that increases AT as an exogenous instrument to measure the causal effect of AT on liquidity. For large stocks in particular, AT narrows spreads, reduces adverse selection, and reduces trade-related price discovery. The findings indicate that AT improves liquidity and enhances the informativeness of quotes. TECHNOLOGICAL CHANGE HAS REVOLUTIONIZED the way financial assets are traded. Every step of the trading process, from order entry to trading venue to back office, is now highly automated, dramatically reducing the costs incurred by intermediaries. By reducing the frictions and costs of trading, technology has the potential to enable more efficient risk sharing, facilitate hedging, improve liquidity, and make prices more efficient. This could ultimately reduce firms’ cost of capital. Algorithmic trading (AT) is a dramatic example of this far-reaching technological change. Many market participants now employ AT, commonly defined as the use of computer algorithms to automatically make certain trading decisions, submit orders, and manage those orders after submission. From a starting point near zero in the mid-1990s, AT is thought to be responsible for

/pdf/does-algorithmic-trading-improve-liquidity-j10t9o2syx.pdf

Topology optimization of continuum structures: A review*

We consider the execution of portfolio transactions with the aim of minimizing a combination of volatility risk and transaction costs arising from permanent and temporary market impact. For a simple linear cost model, we explicitly construct the efficient frontier in the space of time-dependent liquidation strategies, which have minimum expected cost for a given level of uncertainty. We may then select optimal strategies either by minimizing a quadratic utility function, or by minimizing Value at Risk. The latter choice leads to the concept of Liquidity-adjusted VAR, or L-VaR, that explicitly considers the best tradeoff between volatility risk and liquidation costs. ∗We thank Andrew Alford, Alix Baudin, Mark Carhart, Ray Iwanowski, and Giorgio De Santis (Goldman Sachs Asset Management), Robert Ferstenberg (ITG), Michael Weber (Merrill Lynch), Andrew Lo (Sloan School, MIT), and George Constaninides (Graduate School of Business, University of Chicago) for helpful conversations. This paper was begun while the first author was at the University of Chicago, and the second author was first at Morgan Stanley Dean Witter and then at Goldman Sachs Asset Management. †University of Toronto, Departments of Mathematics and Computer Science; almgren@math.toronto.edu ‡ICor Brokerage and Courant Institute of Mathematical Sciences; Neil.Chriss@ICorBroker.com

Optimal execution of portfolio transactions

Optimal trading strategies are determined for liquidation of a large single-asset portfolio to minimize a combination of volatility risk and market impact costs. The market impact cost per share is taken to be a power law function of the trading rate, with an arbitrary positive exponent. This includes, for example, the square root law that has been proposed based on market microstructure theory. In analogy to the linear model, a ‘characteristic time’ for optimal trading is defined, which now depends on the initial portfolio size and decreases as execution proceeds. A model is also considered in which uncertainty of the realized price is increased by demanding rapid execution; it is shown that optimal trajectories are described by a ‘critical portfolio size’ above which this effect is dominant and below which it may be neglected.

Optimal execution with nonlinear impact functions and trading-enhanced risk

The impact of large trades on market prices is a widely discussed but rarely measured phenomenon, of essential importance to selland buy-side participants. We analyse a large data set from the Citigroup US equity trading desks, using a simple but realistic theoretical framework. We fit the model across a wide range of stocks, determining the dependence of the coecients on parameters such as volatility, average daily volume, and turnover. We reject the common square-root model for temporary impact as function of trade rate, in favor of a 3/5 power law across the range of order sizes considered. Our results can be directly incorporated into optimal trade scheduling algorithms and pre- and post-trade cost estimation.

/pdf/direct-estimation-of-equity-market-impact-40xgl1xjq4.pdf

Direct Estimation of Equity Market Impact

Etude d'une structure de barre, charniere et ressort, isotrope dans ses proprietes elastiques macroscopiques mais anisotrope dans les details microscopiques de sa structure

An isotropic three-dimensional structure with Poisson's ratio=−1

We extend Karma and Rappel's improved asymptotic analysis of the phase field model to different diffusivities in solid and liquid. We consider both second-order "classical" asymptotics, in which the interface thickness is taken much smaller than the capillary length, and the new "isothermal" asymptotics, in which the two lengths are considered comparable. In the first case, if the phase field model is required to be gradient flow for an entropy functional, then for unequal diffusivities it is impossible to construct a phase equation with finite kinetics which converges with second-order accuracy to a Gibbs--Thomson equilibrium condition with infinitely fast kinetics. In the second case, some error terms are pushed to higher orders, and it is easy to eliminate the remaining errors with finite phase kinetics.

Robert Almgren

Papers

Optimal execution of portfolio transactions

Optimal execution with nonlinear impact functions and trading-enhanced risk

Direct Estimation of Equity Market Impact

An isotropic three-dimensional structure with Poisson's ratio=−1

Second-order phase field asymptotics for unequal conductivities