Welcome back. I hope that all of my readers have had an excellent summer and a great start to the fall semester. Today, I wanted to gently continue our discussion of aerospace engineering by reviewing a core concept that underpins nearly every macroscopic fluid flow analysis. This concept is the continuum hypothesis, which states that a fluid can be treated as one smooth, differentiable continuum upon which we can do calculus. In reality, fluids are not continuums. Indeed, they are composed of billions of tiny molecules that come in discrete chunks. However, this continuum approximation is highly accurate when we are analyzing flows on “large enough” scales. The specific goal of this blog is to explore the continuum hypothesis in a little more depth and quantify phrases such as “large enough.”
To understand the continuum hypothesis, it is first necessary to become familiar with the concept of mean free path. In a liquid or gas, the mean free path is usually denoted by the symbol l, and is defined to be the average distance that one molecule travels before it collides with another molecule. For some gasses, the mean free path is approximated by
l = A µ ρ-1 (RT)-½,
where A = 1.26 is a constant, µ is dynamic viscosity, ρ is density, R is the ideal gas constant, and T is temperature. For air near sea-level, this yields l ≈ 70 nm, which is indeed very small.
Let’s also denote by L the length scale on which we are interested in measuring a fluid. If we are analyzing a Boeing 777, then L might be 1 m, and if we are analyzing a hot air balloon, then L might be closer to 10 cm. Now define the ratio of the mean free path to L to be the Knudsen number, Kn = l/L. Suppose there is now a box with side lengths equal to L, and suppose we look at a small subsection of that box that we want to treat as an infinitesimal sub-volume. If Kn = l/L << 1, then there will be lots of molecules inside the small sub-volume. This large quantity of molecules will allow us to take averages of fluid properties over the sub-volumes, and then subsequently perform calculus procedures on the sub-volumes. However, if Kn >> 1, then there will not be enough particles to take averages of fluid quantities inside small sub-volumes. This will prevent us from treating the fluid inside the large box as a continuum.
Therefore, we can roughly say that the continuum hypothesis holds whenever Kn << 1, and that the continuum hypothesis breaks down whenever Kn ≥ 1. Luckily, because the mean free path of air is so small, the Knudsen number in most aerospace applications is well below Kn = 10-3. This allows engineers to use the powerful Navier-Stokes equations to analyze fluids. For incompressible fluids, these equations are
∇ • u = 0
ρ ∂u/∂t + ρ (u • ∇) u = – ∇p + µ ∆u + ρ g,
which are formulated in the language of calculus.
While this discussion was brief, I hope it provided some details and clarity to the very important continuum hypothesis. Next week, we will continue our exploration of fluid dynamics by studying the Finite-Time Lyapunov Exponent, which can help us identify coherent patterns in messy flows. Until then, please take care.