Fundamentals of Fluid Mechanics

When we talk about fluid mechanics, we aren’t just talking about the second state of matter (liquid \(\subset\) fluid). Fluid mechanics is the study of flow, which applies to both liquids and gases.

Continuum Mechanics

One of the common assumptions we make when working with fluids is that it forms a continuum. The individual atoms of a fluid are assumed to be tightly packed enough that we can approximate the collective molecular interactions. What are these collective molecular interactions? Well, pretty standard stuff actually. We assume that applying a force causes an object to move, or a body to deform. (Technically speaking, that stress is related to strain). And honestly well, that’s kinda it.

The reason this assumption is so powerful is because it makes our model continuous. This may seem pretty obvious at a glance, but working with stuff that is continuous is very different from working with stuff that is discrete! The reason as you may already have guessed: derivatives.

Viscous Flow

Unlike solids, fluids deform constantly under stress (in other words, a fluid flows). How it flows is very much dependent on it’s viscosity.

Viscosity is to fluids what the modulus of elasticity is to solids. It relates the stress (amount of force) to strain (amount of deformation). Qualitatively, viscosity is a measure of a fluid's resistence to flowing. Things like honey have a very high viscosity, while air and water have comparatively lower viscosity.

Viscous flow then, is the study of how a fluid’s viscosity affects its behavior. Viscous flow covers topics like boundary layers, low-speed flows, and small-scale flows. Coincidentally, the regimes in which viscous forces dominate is often incompressible, but this is not a hard and fast rule.

Compressible Flow

While viscous flow deals with flows that are very slow, compressible flow is for your rockets and airplanes. In this Reynold’s number regime, the density of a fluid cannot assumed to be constant, but luckily we can ignore viscous effects.

Turbulent Flow

In general, turbulent flow is modeled statistically. Averages values are used instead of exact ones. Before laminar flow becomes fully turbulent, it experiences transition. Transition begins with flow instability. The point at which a flow trips into instability (critical point) is distinct from the point at which flow transitions fully to turbulence (transition point).

Non-Dimensional Numbers

Non-dimensional simply means unitless. Why unitless? Because we’re taking the ratio of two similar properties and comparing their relative magnitude.

There’s a whole bunch of these, but here are the main ones that you might come across.