Part II: Pressure & Hydrostatics

If our objective as fluid mechanicians is to understand how pressure, density, and velocity are related, we should then begin to look at simple examples of fluid mechanical phenomena to get a sense of how these properties interact with each other and with external forces—and in particular, how pressure and external forces interact.

Phenomenologically, it’s important to note that the amount of molecular energy present in some sample of fluid molecules is humongous here on Earth—even in fluids which are completely at rest at the continuum scale. As a result, fluid molecules tend to be pretty good at distributing themselves locally in a way that minimizes that molecular energy. And thanks to all the pushing and pulling between molecules, they tend to rapidly space themselves out so consistently that we can usually always define the number of particles in a given fluid point—and quantities connected to it, like density—as an exclusive function of the molecular energy density/pressure of that point. In short, \rho = \rho(p).

This may lead you to believe that applying a force to a fluid will lead to a change in its density. However, the amount of energy you are giving to a fluid molecule by applying a continuum-scale force to it is positively meager in comparison to the molecular energy the molecule already possesses through pressure. As a result, external forces and other continuum-scale effects almost never alter the density of a fluid—they usually just trigger continuum-scale fluid motion precisely so that the fluid can preserve its density. This assumption or property of most fluids is usually referred to as incompressiblity, and it causes the pressure to be uniquely defined solely by the requirement that the density be the same everywhere.

An everyday example of this can be found in a glass of water. Every point of water in the cup is feeling an identical downwards force density \vec{f}, proportional to the density of the point and the gravitational constant (\vec{f} = \rho \vec{g}), which we assume is the same everywhere. For future reference, a force distributed through a fluid like this is commonly referred to as a body force. So if Newtonian mechanics holds, then how are those points of water not moving? Why doesn’t all the water simply accumulate into a highly dense thin film at the bottom of the cup?

Common sense indicates that another force, namely a molecular force, needs to be countering gravity in order for the water in the cup to retain its density: \rho \vec{g} + \vec{f}_{molecular} = 0. This molecular force is coming from pressure—the molecular energy density of the water is redistributing itself to ensure the density stays the same. This redistribution causes gradients in pressure that manifest as continuum force densities, leading us to obtain the force balance equation for fluids not in motion, or the hydrostatic equation:

\displaystyle \vec{f} + \nabla p = 0 (in general)

\displaystyle \rho \vec{g} + \nabla p = 0 (when the external force is gravity)

Heuristically, this means that pressure increases in the direction of forces for static fluids. Noting gravity points “downwards”, this is why deep-sea divers can’t go too far down into the ocean without a submersible (they’d get crushed by the increased pressure/molecular energy of the water) and why astronauts wear full-body suits (our bodies’ molecular energy would get dumped out into the very low-pressure environment of space).

But body forces aren’t the only way forces can influence a fluid; we should also consider the influence from external forces that, rather than being distributed through a fluid like gravity, are concentrated on solid surfaces in contact with the fluid (which is how a cup holds water). These surface forces are a little trickier to interpret, but easy to describe with the right conceptual machinery.

Consider a point of fluid located right at the bottom of a cup, in contact with a tiny patch of cup. The point itself isn’t moving, but the molecules within the point are—and something needs to compensate for the lack of fluid points below the one of interest to ensure the fluid point doesn’t move. That compensation is coming from an increase of molecular energy inside that patch of cup to “match” the point’s surroundings. That increase, and its effect in ensuring the fluid point remains stationary, can be mathematically represented as a local force per area pointing into the fluid with the same magnitude as the local pressure \left(|\vec{t}_s| = p\right).

In a fluid point at the bottom of the cup, molecules under the influence of gravity bounce around against their neighboring fluid points and a tiny patch of cup. The neighboring pressure keeps the particles from wandering up or sideways, while a surface force from the cup patch keeps the particles from moving downwards due to gravity.

We can call that local force per area a surface traction, contact pressure, or simply a traction; there isn’t really a standard clear name for the concept, and many sources incorrectly refer to it as pressure without making the distinction between the local force per unit area and its molecular source.

To humanity’s benefit, this principle of induced surface forces due to pressure works both ways: if we impose a surface force on a fluid, a static fluid will locally increase its pressure by the magnitude of that force per unit area to compensate. This bidirectional principle lets us manipulate forces acting on objects in useful ways, all of which are variations of the following sequence of phenomena:

  1. An object imposes a net force on a fluid, in the form of a force per unit area distributed over a surface.
  2. The surface force generates a pressure increase throughout the fluid.
  3. The extra pressure is transmitted as a force per unit area onto another object with a different surface area, leading to a different net force acting on that second object.

The field of engineering that utilizes this force-multiplying principle to solve problems (among many other fluid-mechanical tools) is called hydraulics, and has many applications throughout varying fields of science & technology. Notable examples include hydraulic jacks, hydraulic suspensions, hydraulic presses, etcetra.

A bug and a car are placed on two ends of a hydraulic jack. The bug’s weight induces a force per unit area on the fluid, increasing its pressure. This pressure increase multiplied by the surface area of the large piston is enough to balance the weight of the large car.

The arguments we’ve made here are quite general, but don’t really account for what happens when the traction acting on a surface isn’t the same everywhere. To understand what happens in that situation, we’ll need to look at one of the first great scientific discoveries of fluid mechanics; buoyancy.

Things to Think About

1. Think of some simple examples of force fields and calculate/infer what the pressure distribution they induce in a still fluid might look like. Can you think of any force field that leads to nonsense pressure results? What would that mean?

2. Can you think of a way to determine when the incompressibility assumption is correct? What properties would you need to know? Can you think of a “metric” to determine how correct the assumption is?

3. If you heat up a liquid in a kettle or frying pan, you would usually reduce its density quite noticeably. How does this not contradict the statement that external continuum forces don’t affect fluid densities?

4. Solids can store energy in molecular bonds when pushed or pulled, behaving in a way similar to a tiny bunch of connected springs. How is this behavior different from the behavior of a fluid? How is it similar?

5. Why isn’t the density of a fluid a function of the fluid velocity—especially if we stated well-defined densities occurs as a result of molecular motion?

6. Can you think of a way to mathematically derive that |\vec{t}_s| = p? What would you need to consider?

7. What are the units of pressure, and it is a scalar or a vector? What are the units of traction, and is it a scalar or a vector? Can you think of reasons why people might confuse the two?

8. Can you think of some applications for the concept of hydraulics described above? Are there any you can identify that have already been made?

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