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Introduction

This is an introduction to the mathematical equations of fluid dynamics, with an example of the fluid dynamics of the ocean. Fluid dynamics is a branch of applied mathematics or mathematical physics that deals with the flow of fluids. A fluid can be defined as anything that flows. Water, of course, is a fluid; the water in the oceans obeys the mathematical equations of fluid dynamics. Air flows; the air in the atmosphere also obeys the equations of fluid dynamics. But the application of fluid dynamics is much wider. The air flowing in your lungs, the blood in your veins, the air flowing over an aeroplane wing and the hot gases in a car's combustion cylinder all obey fluid-dynamical equations. Fluid dynamics is vital in many branches of science and engineering.

Here is a procedure that is typically followed in solving a fluid dynamics problem:

1. Get the mathematical equations describing your particular fluid-dynamical system (the ocean surface, say) by thinking about the physics of that system.

2. Scale the equations: Think about a particular condition that your system is in, that you would like to analyse. (Say the ocean surface is far from shore and is moving fairly gently). Scaling allows the equations to be simplified to make a solution easier.

3. Solve the equations. Usually there are standard mathematical techniques available to do this.

4. Compare the solution with reality - see where it can be put to practical use, and also where and why the solution breaks down.