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3653 results in ebooks in fluid mechanics

Page 23 of 183

  • 8 - Energy conservation in the 3D Euler equations on T2 × R+

  • Edited by Charles L. Fefferman, Princeton University, New Jersey, James C. Robinson, University of Warwick, José L. Rodrigo, University of Warwick
  • Book:
    Partial Differential Equations in Fluid Mechanics
    Published online:
    15 August 2019
    Print publication:
    27 September 2018, pp 224-251

  • Summary

    The aim of this paper is to prove energy conservation for the incompressible Euler equations in a domain with boundary. We work in the domain $$\TT^2\times\R_+$$, where the boundary is both flat and has finite measure; in this geometry we do not require any estimates on the pressure, unlike the proof in general bounded domains due to Bardos & Titi (2018). However, first we study the equations on domains without boundary (the whole space $$\R^3$$, the torus $$\mathbb{T}^3$$, and the hybrid space $$\TT^2\times\R$$). We make use of somearguments due to Duchon & Robert (2000) to prove energy conservation under the assumption that $$u\in L^3(0,T;L^3(\R^3))$$ and $${|y|\to 0}\frac{1}{|y|}\int^T_0\int_{\R^3} |u(x+y)-u(x)|^3\,\d x\,\d t=0$$ or $$\int_0^T\int_{\R^3}\int_{\R^3}\frac{|u(x)-u(y)|^3}{|x-y|^{4+\delta}}\,\d x\,\d y\,\d t<\infty,\qquad\delta>0$$, the second of which is equivalent to $$u\in L^3(0,T;W^{\alpha,3}(\R^3))$$, $$\alpha>1/3$$.

  • 6 - Leray’s fundamental work on the Navier–Stokes equations: a modern review of “Sur le mouvement d’un liquide visqueux emplissant l’espace”

  • Edited by Charles L. Fefferman, Princeton University, New Jersey, James C. Robinson, University of Warwick, José L. Rodrigo, University of Warwick
  • Book:
    Partial Differential Equations in Fluid Mechanics
    Published online:
    15 August 2019
    Print publication:
    27 September 2018, pp 113-203

  • Summary

    This article offers a modern perspective that exposes the many contributions of Leray in his celebrated work on the three-dimensional incompressible Navier-Stokes equations from 1934. Although the importance of his work is widely acknowledged, the precise contents of his paper are perhaps less well known. The purpose of this article is to fill this gap. We follow Leray's results in detail: we prove local existence of strong solutions starting from divergence-free initial data that is either smooth or belongs to $$H^1$$ or $$L^2 \cap L^p$$ (with $$p \in (3,\infty]$$), as well as lower bounds on the norms $$\| \nabla u (t) \|_2$$ and $$\| u(t) \|_p$$ ($$p\in(3,\infty]$$)as t approaches a putative blow-up time. We show global existence of a weak solution and weak-strong uniqueness. We present Leray's characterisation of the set of singular times for the weak solution, from which we deduce that its upper box-counting dimension is at most 1/2. Throughout the text we provide additional details and clarifications for the modern reader and we expand on all ideas left implicit in the original work, some of which we have not found in the literature. We use some modern mathematical tools to bypass some technical details in Leray's work, and thus expose the elegance of his approach.

Gas Turbines
  • Internal Flow Systems Modeling
  • Bijay Sultanian
  • Published online:
    25 September 2018
    Print publication:
    13 September 2018
  • This long-awaited, physics-first and design-oriented text describes and explains the underlying flow and heat transfer theory of secondary air systems. An applications-oriented focus throughout the book provides the reader with robust solution techniques, state-of-the-art three-dimensional computational fluid dynamics (CFD) methodologies, and examples of compressible flow network modeling. It clearly explains elusive concepts of windage, non-isentropic generalized vortex, Ekman boundary layer, rotor disk pumping, and centrifugally-driven buoyant convection associated with gas turbine secondary flow systems featuring rotation. The book employs physics-based, design-oriented methodology to compute windage and swirl distributions in a complex rotor cavity formed by surfaces with arbitrary rotation, counter-rotation, and no rotation. This text will be a valuable tool for aircraft engine and industrial gas turbine design engineers as well as graduate students enrolled in advanced special topics courses.

Theoretical Mantle Dynamics
  • Neil M. Ribe
  • Published online:
    20 September 2018
    Print publication:
    04 October 2018
  • Geodynamics is the study of the deformation and flow of the solid Earth and other planetary interiors. Focusing on the Earth's mantle, this book provides a comprehensive, mathematically advanced treatment of the continuum mechanics of mantle processes and the craft of formulating geodynamical models to approximate them. Topics covered include slow viscous flow, elasticity and viscoelasticity, boundary-layer theory, long-wave theories including lubrication theory and shell theory, two-phase flow, and hydrodynamic stability and thermal convection. A unifying theme is the utility of powerful general methods (dimensional analysis, scaling analysis, and asymptotic analysis) that can be applied in many specific contexts. Featuring abundant exercises with worked solutions for graduate students and researchers, this book will make a useful resource for Earth scientists and applied mathematicians with an interest in mantle dynamics and geodynamics more broadly.

Page 23 of 183