Fluid and Solid Mechanics

The following sections are included:

• Preface
• Contents

This chapter concerns mathematical modelling of large-scale fluid flows relative to a rotating frame of reference, for which the effects of rotation are dominant and to leading order there is a balance of horizontal pressure gradients and Coriolis forces. The principal application is to oceanic and atmospheric flows with horizontal scales of tens of kilometres or more, and timescales of days or more. A fundamental equation in the dynamics of such flows is that for quasigeostrophic potential vorticity, and this is derived in the first part of the chapter, with stratification effects included in the form of layers with constant density within each layer. Large-scale wave-like behaviour is supported in the form of Rossby waves, and some basic properties of these waves are presented. Simplified conceptual quasigeostrophic models provide understanding of dynamical processes, and two examples are described: ocean spin-up and multiple equilibria.

This chapter is an introduction to the asymptotic theory of laminar viscous flows at large Reynolds numbers. Starting with a derivation of the governing Navier–Stokes equations, the theory is presented as a sequence of worked examples of increasing complexity including the classical boundary layer, the triple-deck flow with viscous-inviscid interaction and the double-deck interactive flow in a wall jet, briefly touching upon supersonic and condensed flow regimes. The theory is used to study the phenomena of upstream influence, flow induced by a local roughness, short-scale boundary layer separation near an irregularity in the wall geometry, self-induced separation as part of a global picture, and instability in the flow.

This contribution gives an introduction to the description of time-averaged single-phase turbulent flows where the largeness of the typical Reynolds numbers at play is their essential characteristic. Hence, the full Navier–Stokes equations form the starting point, and the viewpoint is the most rigorous asymptotic one. Emphasis is placed on the aspects of modelling the unclosed terms in the accordingly Reynolds-averaged equations when governing slender shear flows. Such represent the natural manifestation of turbulence as triggered internally in laminar shear layers by the no-slip condition to be satisfied at rigid walls rather than by free-stream turbulence, neglected here. Given the inherent closure problem associated with the separation and interaction of the variety of spatial/temporal scales involved, this focus allows for a surprisingly deep understanding of turbulent flows resorting to formal asymptotic techniques under the premise of a minimum of reliable assumptions. These are motivated by physical intuition and/or based on classical findings of the statistical theory of locally isotropic turbulence. Intrinsic differences to the analysis of related problems dealing with laminar high-Reynolds-number flows are highlighted. Finally, the crucial aspects of numerical simulation of turbulent flows are considered for the staggered levels of filtering, ranging from a most complete resolution to full averaging.

This chapter is concerned with the computation of nonlinear free surface flows. Both the effects of surface tension and gravity are included in the dynamic boundary condition. Special attention is devoted to the singular behaviour at the points where free surfaces intersect rigid walls. Applications to bubbles rising in a fluid, flows emerging from a nozzle and cavitating flows are presented. It is shown how physical solutions are selected in the limit as the surface tension tends to zero.

The chapter is on constriction/distortion and branching in a vessel or network of vessels containing fluid flow. The central interest is in medium-to-high Reynolds numbers where asymptotic approaches and matching yield insight in symmetric and non-symmetric cases over short or long length scales with or without viscous-inviscid interactions. Attention is given to side-branching, large networks, viscous wall layers, flow reversal, eddies, upstream influence and three-dimensional (3D) effects. A final discussion includes exercises and outlook.

This chapter introduces the fundamental concepts of elasticity and presents the application of continuum mechanics to modelling biological soft tissue and bone. The microstructure of tissue is examined and commonly-used constitutive relations presented. The chapter then continues by examining wave propagation in biological tissue, as well as providing further details pertaining to ultrasound imaging techniques. Finally, extensions to standard elasticity models that incorporate viscoelastic effects and the coupling between a porous solid and an interstitial fluid are briefly presented.