In the previous chapter, the forces acting on a moving fluid element were exhaustively studied. Using Newtons second law of motion, the Navier–Stokes equations for both compressible and incompressible flows were obtained. This chapter uses an alternative approach to developing the Navier–Stokes equations. Namely, by starting from a Eulerian description (as opposed to a Lagrangian description), the integral form and conservation form of the Navier–Stokes equations are developed. The continuity and Navier–Stokes equations in its various forms are tabulated and reviewed in this chapter. This chapter ends by solving some very simple, yet common, problems involving the incompressible Navier–Stokes equations.

This chapter develops the Navier–Stokes equations using a Lagrangian description. In doing so, the concept of a stress tensor and its role in the overall force balance on a fluid element is discussed. In addition, the various terms in the stress tensor as well as the individual force terms in the Navier–Stokes equations are investigated. The chapter ends with a discussion on the incompressible Navier–Stokes equations.

This chapter serves as an introduction to the concept of conservation and how conservation principles are used in fluid mechanics. The conservation principle is then applied to mass and an equation known as the continuity equation is developed. Various mathematical operations such as the dot product, the divergence, and the divergence theorem are introduced along the way. The continuity equation is discussed and the idea of an incompressible flow is introduced. Some examples using mass conservation are also given.

In this chapter, a concept known as scaling is introduced. Scaling (also known as nondimensionalization) is essentially a form of dimensional analysis. Dimensional analysis is a general term used to describe a means of analyzing a system based off the units of the problem (e.g. kilogram for mass, kelvin for temperature, meter for length, coulomb for electric change, etc.). The concepts of this chapter, while not entirely about the fluid equations per se, is arguably the most useful in understanding the various concepts of fluid mechanics. In addition, the concepts discussed within this chapter can be extended to other areas of physics, particularly areas that are heavily reliant on differential equations (which is most of physics and engineering).

In addition to the continuity equation, there is another very important equation that is often employed alongside the Navier–Stokes equations: the energy equation. The energy equation is required to fully describe compressible flows. This chapter guides the student through the development of the energy equation, which can be an intimidating equation. A discussion on diffusion and its interplay with advection is also included, leading to the idea of a boundary layer. The chapter ends with the addition of the energy equation in shear-driven and pressure-driven flows.

In every aspect of life, during all ages, energy has been one of the major essentials for human beings. When we eat, the carbohydrates, proteins, and fats in food furnish the calories needed by our bodies. Livestock utilize energy from the food that they eat in terms of producing labor and products such as meat, milk, or eggs. There has always been a need for energy to carry objects from one point to another. Invention of the wheel made this easier. But the task still required energy, and it still does.

Hydropower is one of the renewable technologies that has been utilized for centuries. In the early stages of the use of hydropower, energy from water was harnessed to do mechanical work in agriculture and forestry. In ancient Greece, water wheels were used for grinding wheat. The Egyptians used the Archimedes screw for irrigation purposes. The foundation for transitioning to electrical energy from mechanical energy dates back to the 1750s when the French engineer Bernard Forest de Bélidor published L’architecture hydraulique (Hydraulic architecture) [].

Over millions of years, buried organisms such as prehistoric animals and plants have decomposed under pressure and heat from the Earth’s crust. The pressure from the rocks and the heat from the crust transformed the decaying organic material into fossil fuels. This transformation resulted in three different kinds of fuels: oil (petroleum), coal, and natural gas. Fossil fuels are the world’s primary energy source.

Energy from the Sun can be utilized in many ways. The hydrologic cycle is driven by the Sun. It is this cycle that provides water to gain potential energy and be used in harvesting energy at the dams built on rivers for hydropower generation. It is again the sunlight that is absorbed by chlorophyll pigments in plants to initiate photosynthesis which yields energy in the form of carbohydrates. The plants eventually become biomass which is another source of energy. Other forms of energy, which are directly or indirectly related to the energy from the Sun, also exist.

The word geothermal is a combination of two words: geo meaning earth, and thermal meaning heat related, with both words originating from Greek. Geothermal energy, therefore, is the energy that is extracted from the Earth. Geology is the science that studies the physical structure of the Earth, the substances that the structural features are formed of, and the processes that act on the formations. Besides the Earth, the scope of geology can include study of other planets such as Mars, or natural satellites such as the Moon. To better understand geothermal energy, one would first need to have a good understanding of the geological structure of the Earth. The Earth is made up of three layers, which are the core, the mantle, and the crust, from the center to the surface.