# Part I: Fluid Concepts

Before delving into the study of fluid mechanics⁠—or the study of anything, really⁠—it’s imperative to look at the underlying assumptions we make when we try to analyze the phenomena we wish to learn about. Namely, what is a fluid, in a conceptual/philosophical sense? And how does that description of a fluid create other related concepts that we can observe and measure?

For starters, most of the matter we interact with on a daily basis tends to come in incomprehensibly large clumps of smaller buildings blocks called atoms or molecules. These clumps can be either mostly stiff and with a specific shape, in which case we would usually describe the clump as solid, or flowing and with uncertain, changing forms, which we would describe as fluid.

In either case, trying to understand the behavior of a single clump really entails understanding the collective behavior of a hundred sextillion of these building blocks, each performing their own complicated dance through time and space. And if you think that trying to understand what each of these molecules is doing is effectively impossible, I’d agree with you! Even though it would be possible to describe the physics of each of these constituents fairly straightforwardly using models of molecular physics, not even the most powerful supercomputer would be able to easily and faithfully obtain the motion of these molecules from them due to their sheer numbers.

However, this incomprehensible complexity runs contrary to most of our daily experiences. I don’t expect my morning coffee to spontaneously crawl up the side of my cup and spill itself—and if I were to tilt my cup of coffee, I would reliably see its liquid surface stay parallel to the ground. Water from the faucet usually comes out in a steady stream, ketchup bottles doesn’t spontaneously explode or dissolve, and the ocean mostly stays put where it’s usually been. In short, the minuscule collective randomness we would expect to see from these massive assemblies of molecules averages out into behavior that is fairly uniform and easy to understand at the scales that we can see and feel.

As a result, the theory of fluids (and solids) is a theory that only cares about those scales that we can directly experience, and considers those tiny building blocks only when it really needs to. This means that the way we mathematically describe large clumps of fluid matter is as precisely that—continuous clumps. Formally, we refer to the theory of fluid mechanics as a continuum theory.

In a continuum theory, we usually focus on some abstract blob of “something” (in this case, matter) distributed over space, which has some properties that also vary over the space the blob occupies. For example, the atmosphere can be thought of as a moving continuum “blob” of air (and other chemicals) whose speed and density changes as you move around and above the Earth. But this brings up another question; what properties do we generally care about in a fluid?

Since the distinguishing characteristic of fluids is motion, surely you’ll agree that velocity is a property we care about. In the theory of fluid mechanics, we associate each point $(x,y,z)$ in a fluid at a time $t$ with a velocity $\vec{v}(x,y,z,t)$. This fluid velocity has three distinct components, $v_x(x,y,z,t)$, $v_y(x,y,z,t)$, and $v_z(x,y,z,t)$, each representing the speed of the fluid point in the $x$ / $y$ / $z$ direction. Its interpretation is straightforward; we expect a “point” of fluid located at $(x,y,z)$ to move with velocity $\vec{v}(x,y,z,t)$ at time $t$. But what does a “point” of fluid even mean?

Recall that, at the scales we experience, what we think is a “point” of fluid is actually a humongous mess of molecules randomly bouncing around at a molecular scale. Therefore, in order to make any sense of this at the continuum scale, we need to somehow “smooth” out all the molecular unpredictability into something predictable and measurable at the continuum scale. Luckily, we can do this without losing too much accuracy by selecting some specific volume size that is very small at the continuum scale, and treating that volume like a continuum “point”. We can then define the properties of that fluid “point” as an average of the properties of the molecules inside it. This is commonly referred to as the continuum approximation, and the properties we obtain from these averages are called continuum or fluid properties.

To be more precise, the properties of fluids we obtain using the continuum approximation are limits of ratios; ratios of volumes and the sum of the properties of molecules inside them. To demonstrate, consider a spherical molecular “net” in a fluid through which molecules pass in and out. At every instant, the net contains some well-defined amount of fluid molecules, and therefore some well-defined total molecular mass. If this net is small at the molecular scale, random perturbations to the total fluid mass within the net as a result of molecules moving in and out of it screw up our ability to define a consistent value of total fluid mass in the net over time. However, as you make the net bigger and bigger at the molecular scale—while keeping it point-like in the continuum scale—the randomness smooths itself out due to the sheer number of molecules, and one observes a total fluid mass within the net that is only a function of the net volume. The ratio of that total fluid mass to the net volume is called the density $\rho(x,y,z,t)$ of the fluid.

Likewise, we could do the same thought experiment but now counting up the energy of the fluid molecules in the net, specifically considering the energy coming from the unpredictable back-and-forth motions of molecules at the nanoscale. As the net becomes bigger, the ratio of the total energy divided by the net volume approximates a kind of molecular energy density usually referred to as pressure* $p(x,y,z,t)$. Note that this is unrelated to the energy from motion of fluid “points” at the continuum scale; pressure is entirely molecular.

*As I’ll talk about in Part II, most people get this definition wrong; even Wikipedia! It doesn’t affect much, but the Wikipedia definition obfuscates the molecular nature of pressure as an energy density, which is an important concept.

Finally, as you might expect, the velocity of a fluid point is simply the average velocity of the molecules within a molecular net of the appropriate size. Interestingly, although each individual molecule possesses a decent instantaneous velocity from random molecular motion, it turns out that the average velocity (i.e. the fluid velocity) of a group of molecules is very often quite low in comparison. As a result, almost all molecular motion is irrelevant to fluid motion. This doesn’t mean that what happens at the molecular scale is completely disconnected from what happens at the continuum scale, though; it just means that we should be able to describe most of what we see in fluid mechanics using only the averaged, continuum properties we described above.

This trinity of properties associated with points of fluid—fluid velocity, density, and pressure—are usually all we need in elementary fluid mechanics to be able to describe the motion of fluids. As a result, we expect the chief enterprise of any fluid mechanician to be describing how each of these properties is related to each other, and how things external to a fluid (forces, etc.) change them.