This chapter introduces influence of density change on a flow, i.e., the compressible flow theory. Strictly speaking, any gas flow is both viscous and compressible. In tradition the influence of viscosity and compressibility are dealt with separately to make things easy. In this book, the chapter 6 deals with viscosity, and the chapter 7 deals with compressibility. Sound speed and Mach number are introduced in the beginning, then the equations for steady isentropic flow are derived with statics and total parameters introduced. Some gas dynamic functions are derived that use coefficient of velocity in replace of Mach number. Propagation mode of pressure waves are discussed next, and expansion and compression waves are introduced. Shock wave, as a strong compression wave, is discussed in depth. In the end, transonic and supersonic flow in a variable cross-section pipe is discussed, especially the characteristics of the flow in a Laval nozzle.

In this chapter, basic concepts in fluid mechanics are introduced. Firstly, the definition of a fluid is discussed in depth with the conclusion that a fluid is such a substance that cannot generate internal shear stresses by static deformation alone. Secondly, some important properties of fluids are discussed, which includes viscosity of fluids, surface tension of liquids, equation of state for gases, compressibility of gases, and thermal conductivity of gases. Lastly, some important concepts in fluid mechanics are discussed, which includes the concept of continuum and forces in a fluid. Within these discussions, fluid is compared to solid in both microscopic and macroscopic to reveal the mechanism of its mechanical property. Viscosity of fluid is compared to friction and elasticity of solid to give readers a better idea how it works microscopically. Forces is classified as body force and surface force for further analysis. Finally, continuum hypothesis is introduced to deem the fluid as continuously separable, which tells the reader that fluid mechanics is a kind of macroscopic mechanics that conforms Newtonian mechanics and thermodynamics.

In this chapter, the basic equations of fluid dynamics are derived and their physical significances are discussed in depth and in examples. Both integral and differential forms of the continuity equation, momentum equation, and energy equation are derived. In addition, Bernoulli’s equation, angular momentum equation, enthalpy equation and entropy equation are also introduced. Finally, several analytical solutions of these governing equations are shown, and the mathematical properties of the equations are discussed. Besides the fundamental equations, some important concepts are explained in this chapter, such as the shaft work in integral energy equation and its origin in differential equations, the viscous dissipation term in the differential energy equation and its relation with stress and deformation, and the method to increase total enthalpy of a fluid isentropically.

This chapter introduces the description of fluid motion, that is, the fluid kinematics. At first, the Lagrangian and Eulerian method is compared to emphasize that most problems in fluid mechanics is more suitable for Eulerian method. Secondly, the concepts of pathlines and streamlines are introduced. Next, Acceleration equation and substantial derivative are derived in Eulerian coordinates and their physical significance is discussed in depth and in examples. Reynolds transport theorem is then introduced and compared with substantial derivative to demonstrate that they are the same relation in integral and differential form respectively. Deformation of a finite fluid element is discussed in the next. Linear deformation, rotation, angular deformation equations are derived individually with equations and illustrations. These knowledges are the key to derive the differential equations of a flow, which will be introduced in chapter 4.

In this chapter, twenty-five carefully selected flow-related phenomena are analyzed, with the purpose of consolidating the understanding of the subject and strengthening the ability to link theory with practice. These flow phenomena include some interesting examples such as the principle of lift, the thrust of a water rocket, the mechanism of a faucet et al. Another type of the examples includes those with controversial explains in some popular science books or websites, such as the pressure of jet flow, the pressure change by a passing train, the reason why the water does not spill when a cup is upside-down et al.

This chapter introduces the mechanical property of a fluid when it is at rest. In the absence of shear force, fluid is balanced between pressure force and body force. A universal differential equation is derived to describe the pressure distribution in a static fluid. This equation, can be called hydrostatic equilibrium equation, is the key to solve any fluid static problems. Two typical situations are then discussed as applications of the hydrostatic equilibrium equation, one is static fluid under the action of gravity, the other is fluid under the action of inertial forces. Differences and similarities of fluids and solids in the transfer of force are discussed in the end. Atmospheric pressure at different heights is calculated in the “Expanded Knowledge” section.

This chapter briefly introduces the application of similarity theory and dimensional analysis in fluid mechanics. The concept of flow similarity is discussed at first to give the readers a brief idea what similarity means. Then some important dimensionless number is listed and discussed, which includes Reynolds number, Mach number, Strouhal number, Froude number, Euler number and Weber number. Next, the governing equations are transformed to a dimensionless form to shows how the dimensionless numbers act in the equations. In the end, some flow examples are provided to show the role of the dimensionless numbers.

In this chapter, viscous flow is discussed in detail. This kind of flow represents the most common flow in daily life and industrial production. Firstly, shearing motion and flow patterns of viscous Fluids is introduced, characteristics of laminar flow and turbulent flow is discussed. Secondly, Prandtl’s boundary-layer theory is introduced and boundary-layer equation is derived from the Navier-Stokes equation through dimensional analysis. Thirdly, some theory and facts for turbulent boundary layer are introduced. Fourthly, some shear flows other than boundary layer flow, such as pipe flow, jets, and wakes are briefly introduced. Boundary layer separation is the most important issue in engineering design, so it is introduced and discussed in a separate section in depth. The two top concerns, namely the flow drag and the flow losses are discussed in a separate section with examples and illustrations. Some further knowledge concerning turbulent flow is briefly discussed in the “expanded knowledge” section, such as the theory of homogeneous isotropic turbulent flow and the numerical computation of turbulent flows.

This chapter introduces inviscid flow and potential flow method. Characteristics of inviscid flow is introduced and the rationality of neglecting viscosity in many actual flow cases is discussed. Then the characteristics of rotational flow for inviscid flow is discussed. The three factors that may cause a fluid to change from irrotational to rotational are enumerated and explained, namely the viscous force, baroclinic flow, and non-conservative body force. For irrotational flow, velocity potential is introduced and several elementary flows are taken as an example to illustrate the computational methods for planar potential flow theory. In the end, complex potential is briefly introduced.

In this chapter, we will discuss barriers to purely advective transport in velocity fields that may have complex spatial features but a simple (recurrent) temporal structure: steady, periodic or quasiperiodic. Such velocity fields can be integrated for all times on bounded domains and hence their trajectories can be interrogated over infinite time intervals. While such exact recurrence is atypical in nature, mixing processes with precisely repeating stirring protocols are abundant in technological applications. Here, we survey classic results on temporally recurrentvelocity fields partly for motivation, partly for historical completeness and partly because their predictions in distinguished (recurrent) frames coincide with the predictions of Lagrangian coherent structure (LCS) methods to be discussed in the next chapter. For this reason, recurrent velocity fields are ideal benchmarks for LCS techniques because their transport barriers can be unambiguously identified. There are also a number of technological mixing processes in which the velocity field is engineered to be spatially recurrent, and hence the techniques discussed here apply directly to them.

Here, we take our first step to discover barriers to transport outside the idealized setting of temporally recurrent (steady, periodic or quasiperiodic) velocity fields. While we can no longer hope for even approximately recurring material surfaces in this general setting, we can certainly look for material surfaces that remain coherent. We perceive a material surface to be coherent if it preserves the spatial integrity without developing smaller scales. Those smaller scales would manifest themselves as protrusions from either side of the material surface without a break-up of that surface. In other words, using the terminology of the Introduction, we seek advective transport barriers in nonrecurrent flows as Lagrangian coherent structures (LCS). We will refer to this instantaneous limit of LCSs as objective Eulerian coherent structures (OECSs). These Eulerian structures act as LCSs over infinitesimally short time scales and hence their time-evolution is not material. Despite being nonmaterial, OECSs have advantages and important applications in unsteady flow analysis, as we will discuss separately.