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Boundaries and Geometries

Gary A. Glatzmaier

in Introduction to Modeling Convection in Planets and Stars: Magnetic Field, Density Stratification, Rotation

This chapter examines how boundary and geometry affect convection. It begins with a discussion of how one can implement “absorbing” top and bottom boundaries, which reduce the large-amplitude ... More

This chapter examines how boundary and geometry affect convection. It begins with a discussion of how one can implement “absorbing” top and bottom boundaries, which reduce the large-amplitude convectively driven flows within shallow boundary layers or the reflection of internal gravity waves off these boundaries in a stable stratification. It then considers how to replace the impermeable side boundary conditions with permeable periodic side boundary conditions to allow fluid flow through these boundaries and nonzero mean flow. It also introduces “two and a half dimensional” geometry within a cartesian box geometry and describes how a fully 3D cartesian box model could be constructed. Finally, it presents a model of convection in a fully 3D spherical-shell and shows how it can be easily reduced to a 2.5D spherical-shell model. The horizontal structures are represented in terms of spherical harmonic expansions.Less

A Model of Rayleigh-Bénard Convection

Gary A. Glatzmaier

in Introduction to Modeling Convection in Planets and Stars: Magnetic Field, Density Stratification, Rotation

This chapter presents a model of Rayleigh–Bénard convection. It first describes the fundamental dynamics expected in a fluid that is convectively stable and in one that is convectively unstable, ... More

This chapter presents a model of Rayleigh–Bénard convection. It first describes the fundamental dynamics expected in a fluid that is convectively stable and in one that is convectively unstable, focusing on thermal convection and internal gravity waves. Thermal convection and internal gravity waves are the two basic types of fluid flows within planets and stars that are driven by thermally produced buoyancy forces. The chapter then reviews the equations that govern fluid dynamics based on conservation of mass, momentum, and energy. It also examines the conditions under which the Boussinesq approximation simplifies conservation equations to a form very similar to that of an incompressible fluid. Finally, it discusses the key characteristics of the model of Rayleigh–Bénard convection.Less

Pattern Formation and Spatiotemporal Chaos

Robert C. Hilborn

in Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers

When a dynamical system has significant spatial extent, its nonlinear dynamics can lead to the spontaneous formation of spatial patterns. Such systems provide models for how nature might have ... More

When a dynamical system has significant spatial extent, its nonlinear dynamics can lead to the spontaneous formation of spatial patterns. Such systems provide models for how nature might have developed ordered, spatial structures from disordered states. Examples are given from fluid flow, transport models, coupled-oscillator modes, cellular automata, transport models, and reaction-diffusion systems. Diffusion-limited aggregation, viscous fingering, and dielectric breakdown provide further examples of pattern formation. Fractal structures make another appearance in this new context. This chapter also explores the somewhat controversial topic of self-organized criticality which has been put forward as an explanation for the occurrence of fractal structures in nature.Less

Fundamentals of fluid dynamics with an introduction to the importance of interfaces

H. A. Stone

in Soft Interfaces: Lecture Notes of the Les Houches Summer School: Volume 98, July 2012

The topics discussed are all related to basic fluid mechanics. In these introductory notes I highlight some of the main features of fluid flows and their mathematical characterization. There is much ... More

The topics discussed are all related to basic fluid mechanics. In these introductory notes I highlight some of the main features of fluid flows and their mathematical characterization. There is much physical intuition encapsulated in the differential equations, and one of our goals is to gain more experience (i) understanding the governing equations and various related principles of kinematics, (ii) developing intuition with approximating the equations, (iii) applying the principles to a wide range of problems, which includes (iv) being able to rationalize scaling laws and quantitative trends, often without having a detailed solution in hand. Where possible we provide examples of the ideas with ‘soft interfaces’ in mind.Less

Introduction

Bijan Mohammadi and Olivier Pironneau

in Applied Shape Optimization for Fluids

This is a general introduction with prospective issues. This book discusses shape optimization problems for fluids and presents the state of the art in shape optimization for an extended range of ... More

This is a general introduction with prospective issues. This book discusses shape optimization problems for fluids and presents the state of the art in shape optimization for an extended range of applications involving fluid flows.Less

Models of Incompressible Fluid Flow

Howard C. Elman, David J. Silvester, and Andrew J. Wathen

in Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics

This preliminary chapter gives a brief introduction to the equations studied in depth in the book, as they are used to model incompressible fluids.

Weak-Form Finite Element Models of Flows of Viscous Incompressible Fluids

J. N. Reddy

in An Introduction to Nonlinear Finite Element Analysis, 2nd Edn: with applications to heat transfer, fluid mechanics, and solid mechanics

This chapter presents finite element models of flows of viscous incompressible fluids for Newtonian and non-Newtonian (power-law) fluids. It develops two different finite element models, namely, the ... More

This chapter presents finite element models of flows of viscous incompressible fluids for Newtonian and non-Newtonian (power-law) fluids. It develops two different finite element models, namely, the mixed finite element model involving velocities and pressure, and the penalty finite element model that involves only velocity degrees of freedom. The formulation of the equations of motion for transient analysis is also discussed. Finally, finite element models of coupled fluid flow and heat transfer, using the penalty method, accounting for buoyancy effects are presented and a number of numerical examples are provided.Less

Behaviour of Fluids (Liquids and Gases, Flow and Pressure)

Patrick Magee and Mark Tooley

in The Physics, Clinical Measurement and Equipment of Anaesthetic Practice for the FRCA

A fluid can be either a liquid or a gas. Fluids exhibit different flow behaviours depending on their physical properties, in particular viscosity and density. Flow characteristics also depend on ... More

A fluid can be either a liquid or a gas. Fluids exhibit different flow behaviours depending on their physical properties, in particular viscosity and density. Flow characteristics also depend on the geometry of the pipes or channels through which they flow, and on the driving pressure regimes. These principles can be applied to any fluid, and the complexity of the analysis depends on the flow regimes described in this section [Massey 1970]. Fluid flow is generally described as laminar or turbulent. Laminar flow, demonstrated by Osborne Reynolds in 1867, is flow in which laminae or layers of fluid run parallel to each other. In a circular pipe, such as a blood vessel or a bronchus, velocity within the layers nearest the wall of the pipe is least; in the layer immediately adjacent to the wall it is probably actually zero. In fully developed laminar flow, the velocity profile across the pipe is parabolic, as shown in Figure 7.1, and as discussed in Chapter 1. Peak velocity of the fluid occurs in the mid line of the pipe, and is twice the average velocity across the pipe at equilibrium, and layers equidistant from the wall have equal velocity. The importance of laminar flow is that there is minimum energy loss in the flow, i.e. it is an efficient transport mode. This is in contrast to turbulent flow, where eddies and vortices (flow in directions other than the predominant one) mean that energy in fluid transport is wasted in production of heat, additional friction and noise. The result is that the pressure drop required to drive a given flow from one end of the pipe to the other is greater in turbulent than in laminar flow. The shear stress τ, which is the mechanical stress between layers of fluid and between the fluid and the tube wall, is proportional to the velocity gradient across the tube (dv/dr) of the fluid layers. The constant of proportionality between these two variables is the dynamic viscosity, η. Less

Least-Squares Finite Element Models of Flows of Viscous Incompressible Fluids

J. N. Reddy

in An Introduction to Nonlinear Finite Element Analysis, 2nd Edn: with applications to heat transfer, fluid mechanics, and solid mechanics

This chapter develops finite element formulations for flows of viscous incompressible fluids using high-order spectral/hp finite element technology and least-squares finite element formulation. The ... More

This chapter develops finite element formulations for flows of viscous incompressible fluids using high-order spectral/hp finite element technology and least-squares finite element formulation. The primary objective has been to present novel mathematical models and innovative discretization procedures in the numerical simulation of fluid mechanics problems, wherein the additional benefits of employing high-order spectral/hp finite element technology and least-squares formulations are pronounced. As a result, ad-hoc tricks (e.g., reduced integration and/or mixed interpolations) required to stabilize low-order finite element formulations are unnecessary.Less

Why a Kinetic Theory of Fluids?

Sauro Succi

in The Lattice Boltzmann Equation: For Complex States of Flowing Matter

Fluid flows are a pervasive presence across most branches of human activity, including daily life. Although the basic equations governing the motion of fluid flows are known for two centuries (1822), ... More

Fluid flows are a pervasive presence across most branches of human activity, including daily life. Although the basic equations governing the motion of fluid flows are known for two centuries (1822), since the work of Claude–Louis Navier (1785–1836) and Gabriel Stokes (1819–1903), these equations still set a formidable challenge to the quantitative, and sometimes even qualitative, understanding of the way fluid matter flows in space and time. Meteorological phenomena are among the most popular examples in point, but the challenge extends to many otherinstances of collective fluid motion, both in classical and quantum physics. This Chapter presents the Navier–Stokes equations of fluid mechanics and discuss the main motivations behind the kinetic approach to computational fluid dynamics.Less

Ductile deformation and microstructures

Jean-Luc Bouchez and Adolphe Nicolas

in Principles of Rock Deformation and Tectonics

In contrast to the elastic deformation, which is reversible, usually neglected by field geologists but important for geophysicists working in seismology, ductile deformation is irreversible. This ... More

In contrast to the elastic deformation, which is reversible, usually neglected by field geologists but important for geophysicists working in seismology, ductile deformation is irreversible. This chapter is restricted to solid materials. Materials containing a melt fraction will be examined in Chapter 7. In the geological literature, ‘ductile’ is often used as a synonym for ‘plastic’. The latter is rather used, and will be used to specify deformation mechanisms that dominantly involve the action of dislocations. In contrast to brittle deformation, which by essence is discontinuous and highly localized (see Chapter 3), ductile deformation is generally continuous and affects large volumes of rock. However, ductile deformation may be concentrated into restricted rock volumes (or domains). Such localization is common in shear zones and/or when superplastic deformation mechanism is involved. Plastic deformation mechanisms naturally depend on temperature, magnitude of the applied stress, mineral nature and grain-size of the rocks. In upper parts of the crust, fluids are able to carry chemical elements over large distances and influence the deformation mechanisms. Micrographs of several microstructural types as well as deformation maps for olivine and calcite are given at the end of this chapter.Less

Hydrodynamics and black holes

Yaron Oz

in Theoretical Physics to Face the Challenge of LHC: Lecture Notes of the Les Houches Summer School: Volume 97, August 2011

This chapter describes how the AdS/CFT correspondence (the Holographic Principle) relates field theory hydrodynamics to perturbations of black hole (brane) gravitational backgrounds. The ... More

This chapter describes how the AdS/CFT correspondence (the Holographic Principle) relates field theory hydrodynamics to perturbations of black hole (brane) gravitational backgrounds. The hydrodynamics framework is first presented from the field theory point of view, after which the dual gravitational description is outlined, first for relativistic fluids and then for the nonrelativistic case. Further details of the fluid/gravity correspondence are then discussed, including the bulk geometry and the dynamics of the black hole horizon.Less