OUTLINE

In this chapter we consider the application of the methods of displacement potential and demonstrate the implementation of these methods in Mathematica.

The fundamental expression for the Papkovich–Neuber representation of the elastic displacement fields is introduced first. Papkovich representations of the simple strain states are next considered, followed by the fundamental singular solutions for the centres of dilatation and rotation and the Kelvin solution for the concentrated force in an infinite solid. From the Kelvin solution the momentless force doublet and the force doublet with moment are derived by differentiation. The combination of three mutually perpendicular momentless force doublets is considered and is shown to be equivalent to the centre of dilatation. This example is used to demonstrate the nonuniqueness of the Papkovich description of elastic solutions. The combination of the centre of dilatation with a force doublet is also shown to correspond to a point eigenstrain solution. The point shear eigenstrain is compared with the combination of two force doublets.

Boussinesq and Cerruti solutions for the concentrated force applied at the boundary of a semi-infinite elastic solid are presented next. Solutions for concentrated forces applied at the vertex of an infinite cone are derived using the same principles from superpositions of known solutions for concentrated forces and lines of centres of rotation and dilatation.

MOTIVATION

The idea for this book arose when the authors discovered, working together on a particular problem in elastic contact mechanics, that they were making extensive and repeated use of Mathematica™ as a powerful, convenient, and versatile tool. Critically, the usefulness of this tool was not limited to its ability to compute and display complex two-and three-dimensional fields, but rather it helped in understanding the relationships between different vector and tensor quantities and the way these quantities transformed with changes of coordinate systems, orientation of surfaces, and representation.

We could still remember our own experiences of learning about classical elasticity and tensor analysis, in which grasping the complex nature of the objects being manipulated was only part of the challenge, the other part being the ability to carry out rather long, laborious, and therefore error-prone algebraic manipulations.

It was then natural to ask the question: Would it be possible to develop a set of algebraic instruments, within Mathematica, that would carry out these laborious manipulations in a way that was transparent, invariant of the coordinate system, and error-free? We started the project by reviewing the existing Mathematica packages, in particular the VectorAnalysis package, to assess what tools had been already developed by others before us, and what additions and modifications would be required to enable the manipulation of second-rank tensor field quantities, which are of central importance in classical elasticity.

OUTLINE

This chapter is devoted to the solution of elastic problems using the stress function approach. The Beltrami potential has already been introduced previously as a convenient form of representation for self-equilibrated stress fields. However, the main emphasis in the chapter is placed on the analysis of the Airy stress function formulation, even though it represents only a particular case of the Beltrami representation. The reason for this is the particular importance of this approach in the context of plane elasticity.

The Airy stress function approach is introduced taking particular care to ensure that conditions of strain compatibility are properly satisfied. The approximate nature of the plane stress formulation is elucidated.

The properties of Airy stress functions in cylindrical polar coordinates are then addressed. Particular care is taken to analyse some important fundamental solutions that serve as nuclei of strain within the elasticity theory, namely the solutions for a disclination, dislocations and dislocation dipoles, and concentrated forces.

Williams eigenfunction analysis of the stress state in an elastic wedge under homogeneous loading is presented next, and the elastic stress fields found around the tip of a sharp crack subjected either to opening or shear mode loading. Finally, two further important problems are treated, namely the Kirsch problem of remote loading of a circular hole in an infinite plate, and the Inglis problem of remote loading of an elliptical hole in an infinite plate.

OUTLINE

This chapter is devoted to the introduction of the fundamental concepts used to describe continuum deformation. This is probably most naturally done using examples from fluid dynamics, by considering the description of particle motion either with reference to the initial particle positions, or with reference to the current (actual) configuration. The relationship between the two approaches is illustrated using examples, and further illustrations are provided in the exercises at the end of the chapter. Some methods of flow visualisation (streamlines and streaklines) are described and are illustrated using simple examples. The concepts are then clarified further using the example of inviscid potential flow.

Placing the focus on the description of deformation, the fundamental concept of deformation gradient is introduced. The polar decomposition theorem is used to separate deformation into rotation and stretch using appropriate tensor forms, with particular attention being devoted to the analysis of the stretch tensor and the principal stretches, using pure shear as an illustrative example. Trigonometric representation of stretch and rotation is discussed briefly.

Discussion is further specialised to the consideration of small strains. Analysis of integrability of strain fields then leads to the identification of the invariant form of compatibility conditions. This subject is important for many applications within elastic theory and is therefore dwelt on in some detail.

Boundary layers are an elemental concept of high Reynolds number flow. They are a framework for discussing viscous fluid dynamics by separating the flow into distinct regions. That is the essential nature of boundary layer theory. The seminal ideas were described by Prandtl in 1904. He recognized that viscous flow along surfaces could be divided into two regions: a vortical layer next to the wall and a potential flow farther from the surface. Modern theories have expanded that to multilayered structure; but the basic notion always is of a thin, vortical layer next to the surface and an inviscid outer flow.

The boundary layer concept brought clarity to the puzzle of the high Reynolds number limit. High Reynolds number can be interpreted as low viscosity. Is inviscid flow the correct limit? Without viscosity, fluid flows freely over a surface, slipping relative to the wall. Hence, the tangential velocity is discontinuous between the stationary wall and the flowing fluid. The shear is infinite. Adding the smallest amount of viscosity would cause an infinite stress. Inviscid flow cannot be the correct high Re limit of viscous flow.

Any amount of viscosity will diffuse the velocity discontinuity: the fluid velocity will be brought smoothly to zero at the stationary wall. Even at the highest Reynolds numbers, viscous stresses cannot be ignored. How, then, is high Reynolds number flow to be constructed?

This puzzle is solved by recognizing that viscous influence is confined to a very thin layer next to the wall. The layer of viscous influence becomes increasingly thin as Re becomes increasingly large.

Separation, Vorticity, and Vortex Shedding

Most classical theory of viscous flow is based on approximations valid at low or high Reynolds number. Although there are a few exact solutions that illustrate aspects of the intermediate range, they are rather limited. The range of phenomena that fall under this heading is commonly illustrated by photographs taken in laboratories. Computational fluid dynamics makes the intermediate Reynolds number range quite accessible.

Inertia is now comparable to viscous stress, and all terms in the Navier–Stokes equations must be retained. Convection destroys the upstream–downstream symmetry of creeping flow (Chapter 3). A distinct wake can be identified leaving the downstream side of a body in an incident flow. Forces on blunt bodies become increasingly due to pressure rather than to viscous stress.

Vorticity is increasingly confined to regions near to walls on the upstream portions of a body and to wakes on the downstream side. The upstream vortical regions become boundary layers in the high Reynolds number limit. As vorticity diffuses away from the surface it is convected downstream, ultimately to form the wake. The upstream–downstream asymmetry leads to another important idea, that of separation. For example, flows into and out of a nozzle are quite different. The flow into a trumpet shaped orifice, say, will follow the walls. As the opening narrows, the flow accelerates to conserve mass [loosely, ρUA = constant, per Eq. (1.33), implies U increases as A decreases]. The accelerating flow convects vorticity toward the wall, keeping it confined near the surface.

Why Study Fluid Dynamics?

Fluid dynamics is a branch of classical physics. It is an instance of continuum mechanics. A fluid is a continuous, deformable material. It is a material that flows in response to imposed forces. This is embodied in the everyday experience of draining water from a sink. The water flows under the action of gravity. It does not have a fixed shape; it fills the sink, conforming to its shape. The water flows with variable velocity, depending on its distance from the drain. All these distinguish fluid motion from solid dynamics. As another example, a pump propels water through a pipe or through the cooling system of a car. How does the reciprocating movement of the pump produce directed flow, extending to distant parts of the cooling circuit? One way or the other, the pump must be exerting forces on the fluid; one way or the other, these forces are communicated to distant portions of the fluid and sets them in motion. It is far from obvious what the nature of that flow will be, especially in a complex geometry. It may be laminar, it may be turbulent; it may be unidirectional, it may be recirculating.

Recirculation is the occurrence of backflow, opposite to the direction of the primary stream. This can be seen behind the pedestals supporting a bridge in a swift river. Despite the strong current, the flow direction reverses, and a circulating eddy forms in a region behind the pedestal. How is such behavior understood and predicted? An understanding requires knowledge of viscous action, of vorticity, of turbulence, and of the governing equations.

Two principles distinguish compressible flow: gases heat when compressed and cool when expanded; disturbances propagate at the speed of sound. The first alludes to thermodynamics. The second alludes to gas dynamics.

Thermodynamics

Heating by compression converts work into thermal energy. This is a reversible conversion in the sense that the thermal energy can be converted back into work. Heating also occurs by frictional dissipation of fluid kinetic energy into thermal energy. That is an irreversible process; viscosity cannot convert the thermal energy back into ordered flow. Friction increases entropy.

Compression and expansion occur in the course of the motion of a gas. For instance, on approaching a blunt body, the flow will slow, and fluid elements will be compressed. That is the ultimate motive for reviewing basic thermodynamics: the governing equations of compressible flow must be consistent with thermodynamics, extended to a spatially distributed system. However, we start with the thermodynamic description of compression and expansion of a homogeneous gas and then proceed to discuss compressible fluid dynamics. Comprehensive texts (Saad, 1997) can be consulted if the reader desires a thorough treatment of thermodynamics. The following is an informal treatment that provides background to compressible flow analysis.

Define a fluid element as a fixed mass, M, of gas. This occupies a volume element, V, which contains that mass. The volume defined in this way is termed specific volume – specific properties are those associated with a given quantity of mass. The mass of the fluid element is invariant, because that is how the element is defined: its volume can change. Indeed, compressibility is the property of volume change in consequence of pressure variations.

When two fluids occupy the domain, with a sharp boundary between them, we speak of fluid–fluid interfaces or just interfaces. To the extent that the fluids are immiscible, their interface is a type of boundary. The governing laws are unchanged; the new features are boundary conditions. They are of a different nature from those at fixed, solid walls. They depend on the flow on either side of the interface; indeed, the position of the interface is itself a variable. Interface conditions are alternatively described as matching conditions: velocities and stresses on either side must properly match at the interface. Despite this complicating aspect, the view that only boundary conditions are at issue provides some clarity.

The interface may be between liquid and gas – say, water and air. Often the matching conditions are simplified in this case. The density of air is three orders of magnitude smaller than that of water. For many purposes, the forces exerted by the air on the water can be neglected; then the interface is a force-free surface, insofar as the hydrodynamics are concerned. It nevertheless is a moveable surface, whose position must be solved as part of the analysis.

Or the interface could be between two viscous fluids – say, oil and water. The viscosity jumps across their common boundary. Conditions of stress continuity then determine the interaction between the fluid motions.

Oil and water might be placed in a vertical tube. The interface then curves in consequence of surface tension and the angle of contact with the tube. The line of contact is a three-phase boundary, among water, oil, and solid wall.

The basic laws of fluid dynamics are the Navier–Stokes momentum equations described in Chapter 1. Computational fluid dynamics (CFD) is the practice of solving those equations,* along with the mass conservation equation, by numerical algorithms. The ability of such seemingly simple governing equations to describe a wealth of complex fluid motions is quite remarkable. That remarkable capability is revealed most notably by computer simulation.

Numerical solution of Navier–Stokes equations nowadays has become almost routine. A variety of algorithms and solution methods for both incompressible and compressible flow have been developed over time and successfully implemented in a large number of computational codes (Ferziger and Peric, 2002; Fletcher, 1991; Tannehill et al., 1997). Initially, this software was primarily for research, mainly in academic institutions, government labs, and corporate research centers. But the appearance (and disappearance) of a number of general purpose, commercial CFD codes has been seen since the early 1990s. These were developed for use by nonexperts, as well as by those experienced in the practice of computation. Some of these codes have matured over time, becoming increasingly powerful as the latest techniques, methods, and analytical models were adapted to their requirements, and as high-speed computing power became increasingly available. Computational capabilities, previously mastered only in the research environment (higher-order numerical schemes, multigrid methods, advanced modeling capabilities, parallel processing) are now being used widely, through the medium of software packages. The engineer, student, or scientist no longer needs to have an intimate familiarity with computational methods to make productive use of CFD. Other technologies that have facilitated CFD include the graphic user interface, software for geometry and mesh creation, and techniques for plotting and visualization.

This is a book on fluid dynamics. It is not a book on computation. Many excellent books on fluid dynamics are available: why is another needed?

In recent decades, numerical algorithms and computer power have advanced to the point that computer simulations of the Navier–Stokes equations have become routine. This vastly expands our ability to solve these equations, further extending our understanding of fluid flow and providing a tool for engineering analysis. Computer simulations are solutions of a different nature from classical exact and approximate solutions. They are numerical data rather than formulas. One of our objectives in this text is to relate computer solutions to theoretical fluid dynamics. Indeed, it is this goal, rather than computation as a tool for complex engineering analysis, that provides the guideline for this text. Computer solutions can reproduce closed-form and approximate solutions; they can illuminate the merits and limits of simple analyses; and they can provide entirely new solutions of varying degrees of complexity. The time is ripe to integrate computer solutions into fluid dynamics education.

From a pedagogical perspective, readily available, commercial computational fluid dynamics (CFD) software provides a new resource for teaching fluid dynamics. This software converts CFD from a technique used by researchers and engineers in industry into a readily accessible facility. It is a challenge to integrate such software packages into the educational structure. Most of the examples in this book have been computed with commercial software, and exercises to be solved with such software have been suggested. How far to go in this direction was a true quandary.

The terminologies creeping flow, Stokes flow, or low Reynolds number hydrodynamics are used synonymously to refer to flows in which inertia is negligible compared to viscous and pressure forces. The formal requirement is that the Reynolds number be small: Re ≪ 1. However, in practice the low Reynolds approximation often remains satisfactory for Reynolds of order unity: Re ∼ 1.

With inertia neglected, momentum is transported by viscous diffusion but not by convection. Some ideas about fluid dynamics must be rethought in this limit. Without convection, there is no wake on the downstream side of an object; pressure scales on viscosity not on kinetic energy; when they occur, eddies are as likely upstream as downstream of a blunt body.

Low Reynolds number can mean highly viscous; hence, one can imagine objects moving through syrupy fluid or syrupy fluid being pumped through a conduit. The dominant forces are frictional in origin. Of course, low Reynolds number can also mean very low velocity or very small scale. One application of creeping flow is to locomotion of microorganisms through a fluid. These animals are a few microns in size. They do not move by propulsion; they drag themselves through the fluid, pushing or pulling by frictional forces. In some cases, they use spiral flagella to corkscrew themselves along. On their relative scale, the fluid appears to be very viscous. One can imagine pushing against a very thick fluid to move forward. To do so, the frictional force pushing forward must be greater than the frictional force resisting motion. Swimming is possible if the organism can produce motions that create more pushing friction than impeding friction.