Here we give an introduction to topics in geometry that will be relevant to the mechanics of fluids. More specifically, we will consider elementary aspects of differential geometry. Geometry can be defined as the study of shape, and differential geometry connotes that methods of calculus will be used to study shape. It is well known that fluids in motion may transform location and shape, such as shown in Fig. 2.1.

In this chapter, we consider a variety of topics related to the governing equations as a system. We briefly discuss boundary and interface conditions, necessary for a complete system, summarize the partial differential equations in various forms, present some special cases of the governing equations, present the equations in a dimensionless form, and consider a few cases for which the linear momenta equation can be integrated once.

In this chapter we will consider the kinematics and dynamics of fluid elements rotating about their centers of mass. Such an element is often described as a vortex, and is a commonly seen in fluids. However, a precise definition of a vortex is difficult to formulate. Rotating fluids may be observed, among other places, in weather patterns and airfoil wakes.

Here we consider some basic problems in one-dimensional viscous flow. Application areas range from ordinary pipe flow to microscale fluid mechanics, such as found in micro-electronic or biological systems. A typical scenario is shown in Fig. 10.1. We will select this and various problems that illustrate the effects of advection, diffusion, and unsteady effects.

In this chapter, we briefly introduce how to apply methods from the discipline of nonlinear dynamical systems to fluid flow. The governing equations for a fluid are a nonlinear system of partial differential equations with space and time as independent variables. Here we will adopt methods to rationally reduce the system of nonlinear partial differential equations to a system of nonlinear ordinary differential equations.

Here we advance from geometry to consider kinematics, the study of motion in space. We will not yet consider forces that cause the motion. If we knew the position of every fluid particle as a function of time, we could also describe the velocity and acceleration of each particle. We could also make statements about how groups of particles translate, rotate, and deform. This is the essence of kinematics, the tool to describe the motion of an infinitesimally small fluid particle, as well as a continuum of such particles.

In this chapter, we turn to the problem of completing the set of equations presented in Section 4.8 by introducing specific constitutive equations. They are material-specific and thus depend upon the constitution of the material.

We close this book with a brief discussion of one of the most important and challenging unsolved problems in the mechanics of fluids: turbulence. As it remains as much descriptive art as predictive science, it is appropriate to call upon visual and poetic sources for inspiration to examine this daunting subject. In the visual realm, the subject has been illustrated with a well-known sketch from da Vinci, seen in Fig. 14.1.

Here we consider some standard problems in multi-dimensional viscous flow. As for one-dimensional viscous flow, application areas are widespread and can include ordinary pipe flows as well as microscale fluid mechanics. We will restrict attention to problems that are steady and laminar. Most of the problems will be incompressible, except for one dealing with a problem in natural convection, Section 11.2.6, and another in compressible boundary layers, Section 11.2.7.

Gaining expertise in marine floating systems typically requires access to multiple resources to obtain the knowledge required, but this book fills the long-felt need for a single cohesive source that brings together the mathematical methods and dynamic analysis techniques required for a meaningful analysis, primarily, of large and small bodies in oceans. You will be introduced to fundamentals such as vector calculus, Fourier analysis, and ordinary and partial differential equations. Then you will be taken through dimensional analysis of marine systems, viscous and inviscid flow around structures, surface waves, and floating bodies in waves. Real-life applications are discussed and end-of-chapter problems help ensure full understanding. Students and practicing engineers will find this book an invaluable resource for developing problem-solving and design skills in a challenging ocean environment through the use of engineering mathematics.

Gaining expertise in marine floating systems typically requires access to multiple resources to obtain the knowledge required, but this book fills the long-felt need for a single cohesive source that brings together the mathematical methods and dynamic analysis techniques required for a meaningful analysis, primarily, of large and small bodies in oceans. You will be introduced to fundamentals such as vector calculus, Fourier analysis, and ordinary and partial differential equations. Then you will be taken through dimensional analysis of marine systems, viscous and inviscid flow around structures, surface waves, and floating bodies in waves. Real-life applications are discussed and end-of-chapter problems help ensure full understanding. Students and practicing engineers will find this book an invaluable resource for developing problem-solving and design skills in a challenging ocean environment through the use of engineering mathematics.

Gaining expertise in marine floating systems typically requires access to multiple resources to obtain the knowledge required, but this book fills the long-felt need for a single cohesive source that brings together the mathematical methods and dynamic analysis techniques required for a meaningful analysis, primarily, of large and small bodies in oceans. You will be introduced to fundamentals such as vector calculus, Fourier analysis, and ordinary and partial differential equations. Then you will be taken through dimensional analysis of marine systems, viscous and inviscid flow around structures, surface waves, and floating bodies in waves. Real-life applications are discussed and end-of-chapter problems help ensure full understanding. Students and practicing engineers will find this book an invaluable resource for developing problem-solving and design skills in a challenging ocean environment through the use of engineering mathematics.

Gaining expertise in marine floating systems typically requires access to multiple resources to obtain the knowledge required, but this book fills the long-felt need for a single cohesive source that brings together the mathematical methods and dynamic analysis techniques required for a meaningful analysis, primarily, of large and small bodies in oceans. You will be introduced to fundamentals such as vector calculus, Fourier analysis, and ordinary and partial differential equations. Then you will be taken through dimensional analysis of marine systems, viscous and inviscid flow around structures, surface waves, and floating bodies in waves. Real-life applications are discussed and end-of-chapter problems help ensure full understanding. Students and practicing engineers will find this book an invaluable resource for developing problem-solving and design skills in a challenging ocean environment through the use of engineering mathematics.

Gaining expertise in marine floating systems typically requires access to multiple resources to obtain the knowledge required, but this book fills the long-felt need for a single cohesive source that brings together the mathematical methods and dynamic analysis techniques required for a meaningful analysis, primarily, of large and small bodies in oceans. You will be introduced to fundamentals such as vector calculus, Fourier analysis, and ordinary and partial differential equations. Then you will be taken through dimensional analysis of marine systems, viscous and inviscid flow around structures, surface waves, and floating bodies in waves. Real-life applications are discussed and end-of-chapter problems help ensure full understanding. Students and practicing engineers will find this book an invaluable resource for developing problem-solving and design skills in a challenging ocean environment through the use of engineering mathematics.