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1680 results in Nonlinear Science and Fluid Dynamics

Page 6 of 84

Physics of Wave Turbulence
  • Sébastien Galtier
  • Published online:
    08 December 2022
    Print publication:
    22 December 2022
  • A century ago, Lewis Fry Richardson introduced the concept of energy cascades in turbulence. Since this conceptual breakthrough, turbulence has been studied in diverse systems and our knowledge has increased considerably through theoretical, numerical, experimental and observational advances. Eddy turbulence and wave turbulence are the two regimes we can find in nature. So far, most attention has been devoted to the former regime, eddy turbulence, which is often observed in water. However, physicists are often interested in systems for which wave turbulence is relevant. This textbook deals with wave turbulence and systems composed of a sea of weak waves interacting non-linearly. After a general introduction which includes a brief history of the field, the theory of wave turbulence is introduced rigorously for surface waves. The theory is then applied to examples in hydrodynamics, plasma physics, astrophysics and cosmology, giving the reader a modern and interdisciplinary view of the subject.

Nanoscale Hydrodynamics of Simple Systems
  • Jesper Schmidt Hansen
  • Published online:
    22 September 2022
    Print publication:
    29 September 2022
  • Written for graduate students and researchers, Nanoscale Hydrodynamics of Simple Systems covers fundamental aspects of nanoscale hydrodynamics and extends this basis to examples. Covering classical, generalised and extended hydrodynamic theories, the title also discusses their limitations. It introduces the reader to nanoscale fluid phenomena and explores how fluid dynamics on this extreme length scale can be understood using hydrodynamic theory and detailed atomistic simulations. It also comes with additional resources including a series of explanatory videos on the installation of the code package, as well as discussion, analysis and visualisations of simulations. This title primarily focusses on training the reader to identify when classical theory breaks down, how to extend and generalise the theory, as well as assimilate how simulations and theory together can be used to gain fundamental knowledge about the fluid dynamics of small-scale systems.

Introductory Incompressible Fluid Mechanics
  • Frank H. Berkshire, Simon J. A. Malham, J. Trevor Stuart
  • Published online:
    05 January 2022
    Print publication:
    02 December 2021
  • This introduction to the mathematics of incompressible fluid mechanics and its applications keeps prerequisites to a minimum – only a background knowledge in multivariable calculus and differential equations is required. Part One covers inviscid fluid mechanics, guiding readers from the very basics of how to represent fluid flows through to the incompressible Euler equations and many real-world applications. Part Two covers viscous fluid mechanics, from the stress/rate of strain relation to deriving the incompressible Navier-Stokes equations, through to Beltrami flows, the Reynolds number, Stokes flows, lubrication theory and boundary layers. Also included is a self-contained guide on the global existence of solutions to the incompressible Navier-Stokes equations. Students can test their understanding on 100 progressively structured exercises and look beyond the scope of the text with carefully selected mini-projects. Based on the authors' extensive teaching experience, this is a valuable resource for undergraduate and graduate students across mathematics, science, and engineering.

  • 4 - Navier–Stokes Flow

  • from Part II - Viscous Flow
  • Frank H. Berkshire, Imperial College of Science, Technology and Medicine, London, Simon J. A. Malham, Heriot-Watt University, Edinburgh, J. Trevor Stuart, Imperial College of Science, Technology and Medicine, London
  • Book:
    Introductory Incompressible Fluid Mechanics
    Published online:
    05 January 2022
    Print publication:
    02 December 2021, pp 143-181

  • Summary

    Recall our discussion on internal fluid forces in . Here and in the next two sections we consider the explicit form of the shear stresses and in particular the deviatoric stress matrix. This is necessary if we want to consider/model any real fluid, i.e. non-ideal fluid. We explain shear stresses as follows – see , p. 31).

  • 7 - Navier–Stokes Regularity

  • from Part II - Viscous Flow
  • Frank H. Berkshire, Imperial College of Science, Technology and Medicine, London, Simon J. A. Malham, Heriot-Watt University, Edinburgh, J. Trevor Stuart, Imperial College of Science, Technology and Medicine, London
  • Book:
    Introductory Incompressible Fluid Mechanics
    Published online:
    05 January 2022
    Print publication:
    02 December 2021, pp 253-296

  • Summary

    The well-posedness of smooth solutions to the three-dimensional incompressible Navier–Stokes equations globally in time is a major open mathematical problem. Our goal in this chapter is to provide a succinct though comprehensive introduction to the main well-posedness results that are known. In the three-dimensional case, we indicate how smooth solutions may develop singularities in finite time. At the same time we establish classical regularity assumptions/conditions that guarantee well-posedness globally in time, i.e. if we could prove three-dimensional incompressible Navier–Stokes solutions satisfied those assumptions/conditions, then we would establish global regularity.

Page 6 of 84