This is an idiosyncratic survey of statistical fluid mechanics centering on the Hopf functional differential equation. Using the Burgers equation for illustration we review several functional integration approaches to theory of turbulence. We notice in particular that some important contributions have been brought about by researchers working on wave propagation in random media, among which Uriel Frisch is not an exception. We also discuss a particular finite-dimensional approximation for the Burgers equation.
We explore the relationship between mechanical systems describing the motion of a particle with the mechanical systems describing a continuous medium. More specifically, we will study how the so-called intermediate integrals or fields of solutions of a finite dimensional mechanical system (a second order differential equation) are simultaneously Euler's equations of fluids and conversely. This will be done both in the classical and relativistic context. A direct relationship will be established by means of the so-called time constraint (classical unsteady case, static or not) and the relativis…
These lecture notes have been prepared as a course in fluid mechanics up to the presentation of the millennium problem listed by the Clay Mathematical Institute. At the end, a very modern aspect of fluid mechanics is covered concerning the subtle issue of its application to high energetic hadronic collisions.
These notes are based on some lectures that the author gave at the University of Campinas - UNICAMP. The notes are in Portuguese, and deal with some methods of mathematics applied to Fluid Mechanics.
We review opportunities for stochastic geometric mechanics to incorporate observed data into variational principles, in order to derive data-driven nonlinear dynamical models of effects on the variability of computationally resolvable scales of fluid motion, due to unresolvable, small, rapid scales of fluid motion.
This is a collection of notes for part of a short course on modal methods in fluid mechanics held at DAMTP, University of Cambridge, in the summer of 2019. These notes introduce the reader to resolvent analysis as it is currently used in fluid mechanics. They are aimed at the average beginning PhD student, and will serve as a base from which to explore the literature.
This article attempts to use the ideas from the field of complexity sciences to revisit the classical field of fluid mechanics. For almost a century, the mathematical self-consistency of Navier-Stokes equations has remained elusive to the community of functional analysts, who posed the Navier-Stokes problem as one of the seven millennium problems in the dawn of 21st century. This article attempts to trace back the historical developments of fluid mechanics as a discipline and explain the consequences of not rationalising one of the commonly agreed upon tenets - continuum hypothesis - in the co…
Predicting how much mixing occurs when a given amount of energy is injected into a Boussinesq fluid is a longstanding problem in stratified turbulence. The huge number of degrees of freedom involved in those processes renders extremely difficult a deterministic approach to the problem. Here we present a statistical mechanics approach yielding prediction for a cumulative, global mixing efficiency as a function of a global Richardson number and the background buoyancy profile.
John Blake (1947--2016) was a leader in fluid mechanics, his two principal areas of expertise being biological fluid mechanics on microscopic scales and bubble dynamics. He produced leading research and mentored others in both Australia, his home country, and the UK, his adopted home. This article reviews John Blake's contributions in biological fluid mechanics, as well as giving the author's personal viewpoint as one of the many graduate students and researchers who benefitted from his supervision, guidance and inspiration. The key topics from biological mechanics discussed are: `squirmer' mo…
A description of thermodynamics for continuum mechanical systems is presented in the coordinate-free language of exterior calculus. First, a careful description of the mathematical tools that are needed to formulate the relevant conservation laws is given. Second, following an axiomatic approach, the two thermodynamic principles will be described, leading to a consistent description of entropy creation mechanisms on manifolds. Third, a specialisation to Fourier-Navier-Stokes fluids will be carried through.
We formulate the mapping between a large class of nonlinear wave equations and flow equations for barotropic fluid with internal surface tension and capillary effects. Motivated by statistical mechanics and multi-channel physics arguments, we focus on wave equations with logarithmic nonlinearity, and further generalize them to matrix equations. We map the resulting equation to flow equations of multi-channel or multi-component Korteweg-type materials. For some special cases, we analytically derive Gaussian-type matrix solutions and study them in the context of fluid mechanics.
This paper provides a short overview of how to use machine learning to build data-driven models in fluid mechanics. The process of machine learning is broken down into five stages: (1) formulating a problem to model, (2) collecting and curating training data to inform the model, (3) choosing an architecture with which to represent the model, (4) designing a loss function to assess the performance of the model, and (5) selecting and implementing an optimization algorithm to train the model. At each stage, we discuss how prior physical knowledge may be embedding into the process, with specific e…
University of Washington（华盛顿大学）；CNRS（法国国家科学研究中心）；Université Paris-Saclay（巴黎萨克雷大学）；TU Berlin（柏林工业大学）；ETH Zurich（苏黎世联邦理工学院）
The field of fluid mechanics is rapidly advancing, driven by unprecedented volumes of data from field measurements, experiments and large-scale simulations at multiple spatiotemporal scales. Machine learning offers a wealth of techniques to extract information from data that could be translated into knowledge about the underlying fluid mechanics. Moreover, machine learning algorithms can augment domain knowledge and automate tasks related to flow control and optimization. This article presents an overview of past history, current developments, and emerging opportunities of machine learning for…
Transition and separation are difficult but important problems in the field of fluid mechanics. Hitherto, separation and transition problems have not been described accurately in mathematical terms, leading to design errors and prediction problems in fluid machine engineering. The nonlinear uncertainty involved in separation and transition makes it difficult to accurately analyze these phenomena using experimental methods. Thus, new ideas and methods are required for the mathematical prediction of fluid separation and transition. In this article, after an axiomatic treatment of fluid mechanics…
We develop a geometric formulation of fluid dynamics, valid on arbitrary Riemannian manifolds, that regards the momentum-flux and stress tensors as 1-form valued 2-forms, and their divergence as a covariant exterior derivative. We review the necessary tools of differential geometry and obtain the corresponding coordinate-free form of the equations of motion for a variety of inviscid fluid models -- compressible and incompressible Euler equations, Lagrangian-averaged Euler-$α$ equations, magnetohydrodynamics and shallow-water models -- using a variational derivation which automatically yields a…
KTH Royal Institute of Technology（KTH皇家理工学院）；University of Washington（华盛顿大学）；Stanford University（斯坦福大学）；Swedish e-Science Research Centre (SeRC)（瑞典电子科学研究中心（SeRC））
The field of machine learning has rapidly advanced the state of the art in many fields of science and engineering, including experimental fluid dynamics, which is one of the original big-data disciplines. This perspective will highlight several aspects of experimental fluid mechanics that stand to benefit from progress advances in machine learning, including: 1) augmenting the fidelity and quality of measurement techniques, 2) improving experimental design and surrogate digital-twin models and 3) enabling real-time estimation and control. In each case, we discuss recent success stories and ong…
Plants live in constantly moving fluid, whether air or water. In response to the loads associated with fluid motion, plants bend and twist, often with great amplitude. These large deformations are not found in traditional engineering application and thus necessitate new specialised scientific developments. Studying Fluid-Structure Interactions (FSI) in botany, forestry and agricultural science is crucial to the optimisation of biomass production for food, energy, and construction materials. FSI are also central in the study of the ecological adaptation of plants to their environment. This revi…
2019 is the bicentenary of George Gabriel Stokes, who in 1851 described the drag - Stokes drag - on a body moving immersed in a fluid, and 2020 is the centenary of Christopher Robin Milne, for whom the game of poohsticks was invented; his father A. A. Milne's "The House at Pooh Corner", in which it was first described in print, appeared in 1928. So this is an apt moment to review the state of the art of the fluid mechanics of a solid body in a complex fluid flow, and one floating at the interface between two fluids in motion. Poohsticks pertains to the latter category, when the two fluids are …
The conditions of existence of extra mass flux in single component dissipative non-relativistic fluids are clarified. By considering Galilean invariance we show that if total mass flux is equal to total momentum density, then mass, momentum, angular momentum and booster (center-of-mass) are conserved. However, these conservation laws may be fulfilled also by other means. We show an example of weakly non-local hydrodynamics where the conservation laws are satisfied as well although the total mass flux is different from momentum density.
A formulation of the boundary integral method for solving partial differential equations has been developed whereby the usual weakly singular integral and the Cauchy principal value integral can be removed analytically. The broad applicability of the approach is illustrated with a number of problems of practical interest to fluid and continuum mechanics including the solution of the Laplace equation for potential flow, the Helmholtz equation as well as the equations for Stokes flow and linear elasticity.
In conventional fluid mechanics, the chemical composition and thermodynamic state of a fluid-solid interface are not considered when establishing velocity-field boundary conditions. As a consequence, fluid simulations are usually not able to generate different outputs when interfacial materials are varied. By considering an atomistic description of matter, theoretical determination of material-specific boundary conditions becomes possible, thereby providing an improved alternative to the completely-invariant no-slip condition. Such a scheme constitutes a multiscale approach to fluid dynamics i…
In recent years, Artificial Neural Networks (ANNs) and Deep Learning have become increasingly popular across a wide range of scientific and technical fields, including Fluid Mechanics. While it will take time to fully grasp the potentialities as well as the limitations of these methods, evidence is starting to accumulate that point to their potential in helping solve problems for which no theoretically optimal solution method is known. This is particularly true in Fluid Mechanics, where problems involving optimal control and optimal design are involved. Indeed, such problems are famously diffi…
Bubbly drinks are surprisingly attractive. There is something about the nature of the these beverages that make them preferable among other choices. In this article we explore the physics involved in this particular kind of two-phase, mass-transfer-driven flows.
The design and performance of liquid metal batteries, a new technology for grid-scale energy storage, depend on fluid mechanics because the battery electrodes and electrolytes are entirely liquid. Here we review prior and current research on the fluid mechanics of liquid metal batteries, pointing out opportunities for future studies. Because the technology in its present form is just a few years old, only a small number of publications have so far considered liquid metal batteries specifically. We hope to encourage collaboration and conversation by referencing as many of those publications as …
Discrete mechanics makes it possible to formulate any problem of fluid mechanics or fluid-structure interaction in velocity and potentials of acceleration; the equation system consists of a single vector equation and potentials updates. The scalar potential of the acceleration represents the pressure stress and the vector potential is related to the rotational-shear stress. The formulation of the equation of motion can be expressed in the form of a splitting which leads to an exact projection method; the application of the divergence operator to the discrete motion equation exhibits, without a…
The mechanism for singularity formation in an inviscid wall-bounded fluid flow is investigated. The incompressible Euler equations are numerically simulated in a cylindrical container. The flow is axisymmetric with swirl. The simulations reproduce and corroborate aspects of prior studies reporting strong evidence for a finite-time singularity. The analysis here focuses on the interplay between inertia and pressure, rather than on vorticity. Linearity of the pressure Poisson equation is exploited to decompose the pressure field into independent contributions arising from the meridional flow and…
Dimensionality reduction is the essence of many data processing problems, including filtering, data compression, reduced-order modeling and pattern analysis. While traditionally tackled using linear tools in the fluid dynamics community, nonlinear tools from machine learning are becoming increasingly popular. This article, halfway between a review and a tutorial, introduces a general framework for linear and nonlinear dimensionality reduction techniques. Differences and links between autoencoders and manifold learning methods are highlighted, and popular nonlinear techniques such as kernel Pri…
Analysis of surface tension in terms of force gradient per unit area -- Part I : a thought experiment using the principle of equivalence in fluid mechanics
Nanofluidics has firmly established itself as a new field in fluid mechanics, as novel properties have been shown to emerge in fluids at the nanometric scale. Thanks to recent developments in fabrication technology, artificial nanofluidic systems are now being designed at the scale of biological nanopores. This ultimate step in scale reduction has pushed the development of new experimental techniques and new theoretical tools, bridging fluid mechanics, statistical mechanics and condensed matter physics. This review is intended as a toolbox for fluids at the nanometre scale. After presenting th…
Numerical simulators are essential tools in the study of natural fluid-systems, but their performance often limits application in practice. Recent machine-learning approaches have demonstrated their ability to accelerate spatio-temporal predictions, although, with only moderate accuracy in comparison. Here we introduce MultiScaleGNN, a novel multi-scale graph neural network model for learning to infer unsteady continuum mechanics in problems encompassing a range of length scales and complex boundary geometries. We demonstrate this method on advection problems and incompressible fluid dynamics,…
