25. Introduction¶
Fluid dynamics is the study of flow. Since anything that flows can be thought of as a fluid, this field of study covers a tremendous range of applications, including aerodynamics, meteorology, hydraulics, oceanography, astrophysical dynamics, acoustics, hemodynamics (blood flow) and mucus flows. Principles of fluid dynamics have even been applied to the study of granular flows, such as dry sand, vehicular traffic and crowd dynamics.
Despite the vast range of applications, most of these phenomena can be examined by just three unifying principles:
Conservation of mass
Conservation of momentum
Conservation of energy
For many flows (this course), we can neglect thermodynamic effects, and in that case we can also get by without conservation of energy.
The continuum hypothesis
So that we can use the tools of calculus, we treat the fluid as if it were continuous in structure and we regard physical quantities as locally uniform. This also means we can treat gases and liquids in the same way, since the equations of motion are independent of the particle structure.
An illustration of the idea is shown below, applied to measurements of density. We ignore molecular fluctuations, and consider a locally averaged measure of density based on the mean free path between atoms.
In actuality, the concept of “density” does not apply very meaningfully at the microscopic level, since the density at each point is defined by either being “inside” or “outside” an atom.
About this book:
In chapters 26-29 we introduce some of the mathematical ideas that are required to discuss and mathematically analyse fluid behaviours. We consider different ways of describing fluid motion and distinguish between streamlines and particle paths. We introduce the important concepts of divergence, vorticity, potential flow.
In chapters 30-31 the equations of motion are introduced from first principles. The concept of viscosity is given a detailed treatment, through which we start to consider the interesting behaviours of some fluids such as ketchup, paint, and blood.
In chapters 32-33 we look at the steady form of the Navier-Stokes equations. For the viscous case we derive flow profiles for some simple geometries. For the inviscid (Euler) case we derive Bernoulli’s equation, which relates pressure and velocity along a streamline.
In chapter 34 we begin to consider how scaling arguments can be used to compare the results of different experiments and to simplify the theory by establishing rational ways to make approximations. Here, we introduce the famous Reynolds number, as well as some other non-dimensional parameters of interest.
In chapter 35 we briefly introduce perturbation techniques that can be used to investigate nearly steady flows and develop models of laminar-turbulent transition. We also introduce the important concept of a boundary layer, and touch upon a mathematical technique that is used to identify and study different layers in the fluid.
In chapter 36 it is shown that vorticity can only arise in a viscous flow, and that by neglecting viscosity we encounter the unphysical result that an aeroplane can fly without experiencing any drag. Nevertheless, the inviscid result is useful here and helps us to understand where lift comes from.
In chapters 37-38 we take a brief look at water waves and sound waves. These two types of waves are generated by very different mechanisms. Surface water waves are gravity driven and can be treated by assuming that the fluid is both incompressible and irrotational. Sound waves are pressure driven and require compressibility to be taken into account.
In chapters 39-40 we look at some geophysical phenomena that relate to rotating fluids such as hurricanes, tornadoes and ocean gyres. We also explain the famous “tea leaf paradox”. The material in these two chapters it is not strictly necessary for this course, and could be skimmed or skipped entirely, though it will be discussed in lecture.
Please kindly note
These notes on fluid dynamics are new (2022). Therefore it is highly likely that they will contain some errors and omissions. If you spot anything in the notes that seems suspect, please let me know so that it can be corrected or explained.
–Ella.