Few words are used with so many different meanings as the term “model.” In everyday language the word “model” can be applied in a moral, fashion, economic, linguistic, or scientific context; in each case it means something completely different. Even if we restrict ourselves to the category of scientific models, the notion is ambiguous, because it could signify the reproduction in miniature of a certain physical phenomenon, and at the same time present a theoretical description of its nature that preserves the broad outline of its behavior. It is the theoretical aspect of models that we wish to consider; in order to emphasize this, we describe this type of model as “mathematical” (Tarski 1953). Formulating a mathematical model is a logical operation consisting in: (i) making a selection of variables relevant to the problem; (ii) postulating statements of a general law in precise mathematical form, establishing relations between some variables said to be data and others unknown; and (iii) carrying out the treatment of the mathematical problem to make the connections between these variables explicit.

The motivations underlying the use of mathematical models are of different types. Sometimes a model is the passage from a lesser known theoretical domain to another for which the theory is well established, as, for example, when we describe neurological processes by means of network theory. In other cases a model is simply a bridge between theory and observation (Aris 1978). The word “model” must be distinguished from “simulation.” The simulation of a phenomenon increases in usefulness with the quantity of specific details incorporated, as, for example, in trying to predict the circumstances under which an epidemic propagates. The mathematical model should instead include as few details as possible, but preserve the essential outline of the problem. The “simulation’ is concretely descriptive, but applies to only one case; the “mathematical model” is abstract and universal. Another special property of a good mathematical model is that it can isolate only some aspects of the physical fact, but not all. The merit of such a model is not of finding what is common to two groups of observed facts, but rather of indicating their diversities. A long-debated and important question is that of how to formulate a model in its most useful form.

The inclusion of viscosity in the modelling of the fluid requires that, at the free surface, the stresses there must be known (given) and, at the bottom, that there is no slip between the fluid and the bottom boundary. The surface stresses are resolved to produce the normal stress and any two (independent) tangential stresses. The normal stress is prescribed, predominantly, by the ambient pressure above the surface, but it may also contain a contribution from the surface tension (see Section 1.2.2). The tangential stresses describe the shearing action of the air at the surface, and therefore may be significant in the analysis of the motion of the surface which interacts with a surface wind. The bottom condition is the far simpler (and familiar) one which states that, for a viscous fluid, the fluid in contact with a solid boundary must move with that boundary.

The appropriate stress conditions are derived by considering the equilibrium of an element of the surface under the action of the forces generated by the stresses. The normal and shear stresses in the fluid (see Appendix A) produce forces that are resolved normal and tangential to the free surface, although the details of this calculation will not be reproduced here. It is sufficient for our purposes (and for general reference) to quote the results – in both rectangular Cartesian and cylindrical coordinates – for the three surface stresses.

Geometric Definitions and Mechanical Assumptions

A curved plate or shell may be described by means of its middle surface, its edge line, and its thickness 2h. We shall take the thickness to be constant, and consider the two surfaces of constant normal distance h from the middle surface and placed on opposite sides of it. These two surfaces constitute the faces of the shell. Let s denote a closed curve drawn on the strained middle surface, and consider the outer normal v to this curve at a point P1? drawn in the tangent plane to the surface at P1 and let s be the unit vector tangent to s, directed counterclockwise. Then let n denote the unit normal to the middle surface at Pi oriented positively so that the triad (v,s,n) is righthanded (Love 1927, Art. 328). We consider a point P1 on the curve s, at a small distance 8s from Pl5 and take the two segments constituted by those parts of the normals to the middle surface at Px and P1 which are included between the faces of the shell. These two pieces of normal at Pi and P1 mark out an element of area δA (Figure 54.1) belonging to a developable surface the generators of which are perpendicular to the middle surface. The contact tractions exerted across the area δA are made statically equivalent to a force and to a couple applied at Pj. We define the averaged components of this force and couple per unit length of s by dividing their components by δs. The limits of these averages for δs → 0 are the stress-resultants and the stress couples transmitted across the curve s at P1. We assume that all these limits exist, and that the component of the stress couple along n vanishes. On taking a Cartesian system x', y’ z along v, s, n, with its origin at P1, the stress-resultants are denoted by T, S, and N, respectively, and the nonzero couple resultants by H and G. These stress- and couple-resultants may be written in terms of the stress components referred to the axes x', y’ and z.

Constitutive Properties of Membranes

A curved membrane can be described geometrically by means of its middle surface, its contour, and its thickness. We shall take, for the moment, the thickness to be constant and equal to 2h, so that the upper and lower faces of the membrane are two surfaces each at a distance h from the middle surface and situated on opposite sides of it. We draw a closed curve s on the middle surface and consider the surface described by the normals to the middle surface drawn from the points of s and bounded by the two outer surfaces of the membrane with spacing 2h. We call this surface . The edge of the membrane is a surface like. If j1 is smooth, we can define the outer normal v at any point on s, so that v lies in the tangent plane to the middle surface, while s is the tangent to the line s, and n the normal to the middle surface, so that v, s, and n form a right-handed triad.

Let δs be a short arc of the curve s and take two generating lines of drawn through the extremities of 8s so as to include an area δA of. The tractions on the area δA are statically equivalent to a force at the centroid of δA together with a couple. The components of the forces in the v, s, and n directions are denoted by δT, δS, and δN, and those of the couple are denoted by δH, δG, and δK. So far we have introduced no specific hypothesis that distinguishes membranes from other continua having the same geometrical definition, called shells. The distinction arises from the constitutive behavior of membranes of being unable to transmit couples or forces perpendicular to the middle surface. In mathematical terms, this property is equivalent to the fact that, when δs is diminished indefinitely, the limits δH/δs, δG/δs, and δK/δs are zero, as is the limit of δN/δs. Only the limits δT/δs and δS/δs can be finite. Expressed in another way, a membrane is the two-dimensional analog of a string, that is to say, it is a two-dimensional continuum such that the only forces interacting between its parts are tangential to it.

Equilibrium Equations

Strings are slender bodies like rods, but are characterized by the properties that they cannot withstand compression or bending and can adapt their shape to any form of loading. They are important because they have wide technical applications, such as those required for the construction of suspension bridges, wire meshes, musical instruments, and nets of textile material. Strings are the vehicle through which some of the methods of mathematical physics have found their simplest applications.

Strings were used by Stevin in 1586 to given an experimental demonstration of the law of the triangle of forces. It seems likely that, in 1615, Beeckman solved the problem of finding the shape of a cable modeled as a string under a load uniformly distributed in a plane, and found that the string hangs in a parabolic arc. The problem of finding the configuration of a chain hanging under its own weight was considered by Galilei (1638), in his Discorsi, who concluded, erroneously, that the chain assumes a parabolic form. Subsequently, Leibniz, Huygens and James Bernoulli, apparently independently, discovered the solution now known as the catenary, taking its name from the Latin for a suspended chain. In investigating the catenary, different approaches were employed: Huygens relied on geometrical considerations, while Leibniz and James Bernoulli relied on the calculus. However, at the same time, Hooke (1675) found that a moment-free arch supports its own weight with a curve which is an inverted catenary. Another long-debated problem was that of velaria, that is, the curve assumed by an inextensible weightless string subjected to a normal force of constant magnitude. This is the form of a cylindrical sail under the action of a uniform wind, the name having its origin in this example. Huygens incorrectly stated that the curve is a parabola, but James Bernoulli aptly proved that the velaria is a circle.

The vibration of taut strings was extensively studied by d'Alembert, Daniel Bernoulli and Lagrange in the early part of the eighteenth century. Daniel Bernoulli (1733) found a solution for the natural frequencies of a chain hanging from one end.

Introduction and Overview

As indicated in the Preface, the main objectives of this book are to provide the reader, in a unifying way, with classical material in fluid mechanics and convection heat transfer and to introduce him or her to basic techniques for modeling engineering fluid dynamics systems. Thus, studying the book in a formal graduate course setting may enhance the student's physical understanding, increase problem-solving skills, and build up confidence to solve other thermal flow problems not discussed in this text. The approach and objectives of problem-solving steps, or in more complex cases “model development” in the engineering sciences, are summarized in Fig. 1.1. This sequence will be highlighted throughout.

The material in Chapters 1 and 2 together with Appendices A and B may equalize readers' different entry levels in fluid dynamics, systems analysis, and engineering mathematics. Specifically, in Section 1.2.1, the two fundamental flow field descriptions (i.e., Lagrange vs. Euler) are reviewed; Sections 1.2.2–1.2.4 discuss the kinematics of shear flow (i.e., fluid element translation, rotation, and deformation), thermodynamic properties (e.g., pressure, temperature, density, and entropy), and transport properties (e.g., viscosity, conductivity, and diffusivity). Some basics of particle dynamics are extended to fluid particle dynamics in Section 1.3. Differential operators and cartesian tensor applications, useful for Sections 1.2.2 and 1.3 as well as for Chapter 2 are summarized in Appendix A. Fluid flow systems under consideration in Chapters 1–5 are restricted to single-phase flow, continuum mechanics, deterministic processes, and Eulerian flow descriptions (cf. Sect. 1.4).

Engineering fluid dynamics is considered (here) to be synonymous with fluid mechanics and convection heat transfer with engineering applications. The textbook is written for intermediate to advanced readers: for professionals as well as selected seniors and first-year graduate students in mechanical, biomedical, nuclear, and chemical engineering.

The main objective of the textbook is to provide the reader with sufficient background to enable him/her

1. (i) to bridge fluid mechanics and convection heat transfer material on an introductory graduate level with specialized advancements in hydrodynamic instability, turbulence, multiphase flows, or computational fluid dynamics; and

2. (ii) to tackle basic research projects in fluid mechanics and convection-heat-transfer related fields.

Although the text contains the basic engineering concepts, physical explanations, exercises, and mathematical aids necessary to succeed, it is the experience students will gain with the homework assignments, in-class discussions, journal article reviews, and course project reports that will move them to a deeper understanding and a higher level of proficiency. Specifically, the in-depth understanding of (the) basics in fluid mechanics/convection heat transfer and the skills to apply fundamental knowledge to the solution of interdisciplinary fluid dynamics problems are more valuable than presentation of large amounts of material within, typically, a very restricted time frame. Thus, in addition to the potentially unique learning experience provided with the text, the advanced student is given powerful tools, including computational fluid dynamics (CFD) software (cf. App. F), which may lead him/her to a level of maturity to solve challenging (industrial) fluid flow and heat transfer problems.

Fluid flows with temperature gradients due to internal heating or heated/cooled walls are examples of thermal flows, which form an integral part of convection heat transfer. Internal heating may result from fluid friction (viscous dissipation), irradiation (thermal radiation), and/or chemical reactions (reactive flows). When the fluid temperature differs from the wall temperature, the thermal wall conditions are commonly expressed as Tw = const (i.e., isothermal wall) or qw = const (i.e., constant wall heat flux). Convection heat transfer is heat conduction, that is, an energy diffusion process, in a moving fluid. Combining heat conduction in solids with thermal convection in a fluid is called a conjugate heat transfer problem as it might occur in heat pipes, heat exchangers, fin cooling, tribology, porous media flow, and so forth. Mixed thermal convection, that is, simultaneous free and forced convection heat transfer, has to be considered when the buoyancy force is of an order of magnitude comparable to the inertia force. In free and in mixed convection problems a heat source or sink affects the fluid density, typically near a heated or cooled wall, and thus the momentum equation depends via the body force term on the heat transfer equation. Such a two-way coupling may also occur when other fluid properties are temperature-dependent.

Nonisothermal flow problems are typically subdivided into external flows and internal flows where the flow regime is either laminar or turbulent, and the flow is single phase or multiphase.

In order to obtain the basic transport equations, a suitable computational domain is selected and the conservation laws for mass, momentum, and energy are applied. The form of the resulting equations, including constitutive equations to gain closure, depends largely on the type of approach taken in terms of “suitable model selection.” As discussed in Section 1.2.1 for the closed system approach, we consider a deformable volume moving with the fluid such that the individual fluid particles are always accounted for (Lagrange). In contrast, from the Eulerian point of view (open system), we consider a fixed control volume with the fluid moving through it. When these fixed or moving volumes are of finite extent, rate equations or integral equations can be directly obtained for the global flow quantities such as flow rates and forces (cf. Reynolds Transport Theorem). When such a system volume or control volume shrinks to an infinitesimally small fluid element of the flow field, differential equations can be directly obtained for the local flow quantities such as velocities and pressure. Alternatively, one could take a molecular approach (as done at supercomputing research centers) where the fundamental laws of nature are directly applied to the individual moving, and colliding molecules of a particular fluid in motion.

In summary, engineers prefer the Eulerian framework, that is, mass, momentum, and energy balances for an open system.

Chapter 5 attempts to extend the basic knowledge gained and (homework) problem solution skills acquired from studying Chapters 1 through 4 and using Appendices A to F. Section 5.1 deals with mathematical modeling aspects important for the development of computer simulation models, which are becoming more and more acceptable in solving complex fluid mechanics and convection heat transfer problems. The selected case studies begin with turbulent shear-layer flows applied to external curved surfaces (Sect. 5.2.1) and internal walls (Sect. 5.2.2). The basic idea of boundary-layer theory is then extended in Section 5.3 to non-Newtonian fluid flow with exothermal chemical reaction (Sect. 5.3.1) and with wall heat transfer (Sect. 5.3.2). The fifth and sixth case studies are more complicated, focusing on two-phase flows with moving gas–liquid interface due to droplet vaporization (Sect. 5.4.1) and moving liquid–solid interface due to ice formation (Sect. 5.4.2). The last two case studies represent the state of the art in (bio-) fluid dynamics applied to laminar pulsatile flow in branching blood vessels (Sect. 5.5.1) and temperature-driven flow in a local hyperthermia treatment device that is a concentric heated cylinder in a water-filled balloon (Sect. 5.5.2). Both biofluid flow systems are transient three-dimensional and hence require the numerical solution of the complete Navier–Stokes equations.

The open literature provides many additional case studies that encapsulate both basic knowledge in engineering fluid dynamics and computer solution steps for real-world (thermal) flow problems.