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 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 in a fluid at a time with a velocity . This fluid velocity has three distinct components, , , and , each representing the speed of the fluid point in the // direction. Its interpretation is straightforward; we expect a “point” of fluid located at to move with velocity at time . 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** 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** . Note that this is unrelated to the energy from motion of fluid “points” at the continuum scale; *pressure is entirely molecular*.

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 all that random motion *usually* averages out into nothing. As a result, **random molecular motion is almost always independent from fluid motion**. This, for example, means that uniformly increasing the local molecular energy density (i.e. pressure) of a fluid doesn’t result in the fluid flowing any faster, since all that extra molecular kinetic energy doesn’t contribute to the fluid point’s kinetic energy. The only exception to this occurs when there are gradients of density within a fluid, in which case molecular motion causes fluid molecules to shuffle around in a way that appears like the fluid is trying to smooth out those density gradients. This type of motion is called **diffusion**, and is often ignored by fluid mechanicists although it is a perfectly valid (albeit quite small) type of flow.

This trinity of properties associated with points of fluid—fluid velocity, density, and pressure—are almost always 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.

## Things to Think About

1. How can you preemptively check if the continuum approximation is appropriate? Can you think of some “measure” of how good it is?

2. Can you think of other useful fluid properties you could obtain using the porous sphere technique detailed above?

3. Why does it make sense to assume that the properties of many similar, randomly-moving molecules average out to something non-random? Does this arise from a physical phenomenon, or a mathematical one?

4. How would you try to estimate the molecular energy of a group of molecules? Are there other concepts in physics & engineering that could help you do that?

5. Can you make any guesses as to how some continuum properties affect each other based on their molecular equivalents?

6. From an experimental perspective, can you think of any reason why this definition of fluid properties is convenient? How would you as an experimentalist measure them?

7. Can you think of a reason why diffusion relies on density gradients? Think of some physical process by which molecules in a density gradient like to randomly move in a way that erases those gradients.