Take anything in the universe, put it in a box, and heat it up. Regardless of what you start with, the motion of the substance will be described by the equations of fluid mechanics. This remarkable universality is the reason why fluid mechanics is important.

The key equation of fluid mechanics is the Navier-Stokes equation. This textbook starts with the basics of fluid flows, building to the Navier-Stokes equation while explaining the physics behind the various terms and exploring the astonishingly rich landscape of solutions. The book then progresses to more advanced topics, including waves, fluid instabilities, and turbulence, before concluding by turning inwards and describing the atomic constituents of fluids. It introduces ideas of kinetic theory, including the Boltzmann equation, to explain why the collective motion of 1023 atoms is, under the right circumstances, always governed by the laws of fluid mechanics.

We all know what a wave is. But you may not know just how many different kinds of waves there are and what strange and interesting properties they have. We start this chapter with something very familiar from everyday life: waves on the surface of an ocean. While they may be familiar, their mathematical description is surprisingly subtle. This can be traced, like so many other things in fluid mechanics, to the boundary conditions.

The chapter then goes on to explore many other different kinds of waves that arise in different situations, from the atmosphere, to supersonic aircraft to traffic jams.

There are two great equations of classical physics: one is Einstein’s equation of general relativity, the other the Navier-Stokes equation that describes how fluids flow. In this chapter, we meet Navier-Stokes.

This equation differs from the Euler equation by the addition of a viscosity term. This is not a small change and makes solutions to the Navier-Stokes equation much richer and more subtle than those of the Euler equation. In this chapter, we begin our exploration of these solutions.

Drop some ink in a glass of water. It will slowly spread through the whole glass, moving in a manner known as diffusion. This process is so common that it gets its own chapter. We will describe the basics of diffusion, as captured by the heat equation, before understanding how diffusion comes about from an underlying randomness. We will see this through the eyes of the Langevin and Fokker-Planck equations.

Take anything in the universe, put it in a box, and heat it up. Regardless of what you start with, the motion of the substance will be described by the equations of fluid mechanics. This remarkable universality is the reason why fluid mechanics is important.

The key equation of fluid mechanics is the Navier-Stokes equation. This textbook starts with the basics of fluid flows, building to the Navier-Stokes equation while explaining the physics behind the various terms and exploring the astonishingly rich landscape of solutions. The book then progresses to more advanced topics, including waves, fluid instabilities, and turbulence, before concluding by turning inwards and describing the atomic constituents of fluids. It introduces ideas of kinetic theory, including the Boltzmann equation, to explain why the collective motion of 1023 atoms is, under the right circumstances, always governed by the laws of fluid mechanics.

Take water and push it through a pipe. If the flow is slow, then everything proceeds in a nice, orderly fashion. But as you force the water to move faster and faster, it starts to wobble. And then those wobbles get bigger until, at some point the flow loses all coherence as it tumbles and turn, tripping over itself in an attempt to push forwards. This is turbulent flow.

Understanding turbulence remains one of the great outstanding questions of classical physics. Why does it occur? How does it occur? How should we characterise such turbulent flows? The purpose of this chapter is to take the first tiny steps towards addressing these questions.

Many of the most interesting things in fluid mechanics occur because simple flows are unstable. If they get knocked a little bit, the fluid curls up into interesting shapes, or dissolves into some messy turbulent flow. In this chapter, we start to understand how these processes can happen.

Three models of a partially ionised fluid are considered by examining together three sets of (M)HD equations for the neutral, ionised, and electron components of a fluid. The first assumes low ionisation and isothermality leading to the one-fluid, isothermal model where all three non-ideal terms–resistance, the Hall effect, ambipolar diffusion–appear in the induction equation. New quantities introduced include: the ambipolar force density; coupling, rate, and ambipolar coefficients; and resistivity, all helping to determine the relative role of each non-ideal term. For resistive MHD, the Sweet–Parker model for magnetic reconnection, and dynamo theory are discussed. For the Hall effect, a two-fluid, isothermal model is introduced that refines the Sweet–Parker model to give a reconnection time scale in better keeping with observations of solar flares. Finally, the section on ambipolar diffusion derives the full two-fluid, non-isothermal model applicable for a fluid with arbitrary ionisation. Here, exchange terms are introduced to account for mass, momentum, and energy transfers when neutrals ionise or ions recombine.

This chapter looks at four important fluid instabilities – Kelvin–Helmholtz, Rayleigh–Taylor, magneto-rotational, and Parker–where normal mode analysis of the lin-earised equations is taught using each instability as an exemplar. All are examined from the linear regime in which conditions for instability and rates of growth of the fastest mode are developed from first principles. For the KHI, RTI, and MRI, numerical simulations are presented which recover the results of linear analysis from the early stages of a non-linear calculation. For the KHI and RTI, numerical simulations well into the non-linear regime are presented where the onset of fluid turbulence is noted. For the MRI, a section describing how it solved the angular momentum transport problem for accretion discs is included. For the Parker instability, an account is given how this purely astrophysical phenomenon explains the clumpy structure of the interstellar medium.

After some historical perspective on the subject, the introduction attempts to define, distinguish, and link in the broadest terms the various areas of physics related to fluid dynamics. These include fluid mechanics, hydrodynamics, gas dynamics, magnetohydrodynamics, and plasma physics. In particular, the link between ordinary hydrodynamics and magnetohydro-dynamics is made, and the approach this text takes in teaching both, namely wave mechanics, is revealed.

This chapter returns to the zero-field limit of MHD replacing the isotropic pressure force density in ideal HD with force densities arising from the viscous stress tensor for viscid HD. As tensor analysis is not a prerequisite for this course, the stress tensor is developed purely from a vector analysis of all stresses applied at a single point in a viscid fluid. This leads to the introduction of bulk and kinetic viscosity in a Newtonian fluid and the identification of ordinary thermal pressure with the trace of the stress tensor. Various flavours of the Navier–Stokes equation are developed including compressible and incompressible forms. The Reynold’s number is introduced as a result of scaling the Navier–Stokes equation which leads to a qualitative discussion on turbulent and laminar flow. Numerous examples are given in which a simplified form of the Navier–Stokes equation can be solved analytically, including plane-parallel flow, open channel flow, Hagen–Poiseuille flow, and Couette flow.