# Part III: Buoyancy

With a decent understanding of hydrostatics and the way pressure and forces interact under our belt, we can then begin to ask some important questions about how objects react to the presence of fluids. For example, how do I know if a boat will sink or float? Is thinking about sinking and floating the only thing I need to thinking about when I’m designing a boat?

To be able to answer those questions, we need to recall the example we saw before of the fluid point at the bottom of the cup. There, we saw that a surface force $\vec{f}_s$ of the same magnitude as the pressure within the fluid point ensures that the fluid point remained stationary. This fact turns out to generalize for any interface between two continuum objects, be they solid or fluid; at every point of the interface between two continua, there is a traction with magnitude equal to the pressure at that point, pointing perpendicular to the interface in the direction of the continua of interest. This is what we saw in the cup example— in that scenario, our continuum of interest was actually the water, and the cup was exerting a traction on the fluid point proportional to its local pressure $|\vec{t}_s| = p$.

This concept allows us to understand & calculate the hydrostatic forces acting on an object embedded in a fluid through the following analysis:

1. Determine the pressure distribution of the fluid.
2. Determine the interface between the continuum of interest and the fluid.
3. Determine the pressure of the fluid at the interface.
4. Calculate the pressure-induced traction at each point on the interface.
5. “Add up” (integrate) those tractions to get the total hydrostatic force on the object.

We can express this last step of the process succinctly using mathematical language:

$\displaystyle \vec{F}_b = \int_{A} \vec{t}_s\ dA$

This procedure involves applying tools from vector calculus, which beginning practitioners of fluid mechanics might find daunting. Luckily, we can obtain the total buoyancy force on any object embedded in a static liquid on Earth without having to do so. In short, the buoyancy force on an object in a fluid is always equal to the weight of the fluid it displaces, and always points up. This is mathematically represented as Archimedes’ law:

$\vec{F}_{\text{b}} = -\rho V_{\text{displaced}} \vec{g}$

For the sake of ideological consistency, we can show how Archimedes’ simple law is derived from the more complicated process of adding up tractions described above. This relies chiefly on the fact that the pressure distribution in a given static fluid on Earth is essentially always the same, stemming from the solution to $\nabla p = \rho \vec{g}$. Since the gravitational force points strictly downwards, we can simply integrate in the “depth” direction $z$ to find that:

$p = p_{0} + \rho|\vec{g}|z$

where $p_0$ represents the pressure at the surface of the fluid and $z$ represents the depth of the fluid at which the fluid point is located. Clearly, the pressure of a fluid point here only depends on its depth within the fluid. As a result, horizontal pressure-induced tractions that act on the surface of an object in such a fluid must cancel out, since a pressure-induced tractions on the “left” side of the object will inevitably be canceled by tractions on the “right” side with the same net magnitude.

Now consider a cube of side length $a$ within the fluid. Its interface is determined by six distinct surfaces; the top, bottom, and four sides. Because the traction on the side surfaces must necessarily cancel by the argument above, the only contributions to the buoyancy force are going to come from the top and bottom, in which the tractions are uniform but distinct.

Therefore, the total traction on the cube is just the difference in tractions between the top and bottom of the cube, multiplied by the area over which they act:

$\vec{F}_{\text{b}} = \int_{A} \vec{t}_s\ dA = a^2 \vec{t}_{\text{top}} + a^2 \vec{t}_{\text{bottom}}$

$\vec{F}_{\text{b}} \cdot \hat{y} = a^2(p_{0} + \rho|\vec{g}|z_{\text{bottom}}) - a^2(p_{0} + \rho|\vec{g}|z_{\text{top}})$

$\vec{F}_{\text{b}} \cdot \hat{y} = a^2 \rho|\vec{g}|(z_{\text{bottom}} - z_{\text{top}})$

$\vec{F}_{\text{b}} \cdot \hat{y} = a^3 \rho|\vec{g}|=V_{\text{cube}} \rho|\vec{g}|$

Because this force is additive in the volume, and because every solid body can be approximated to arbitrary accuracy as a combination of sufficiently small cubes, we have by consequence derived Archimedes’ law for arbitrarily shaped solid objects. Voilà!

This begs the question; why bother thinking about this in a way that requires vector calculus if we don’t need it to calculate the buoyancy force? Luckily, the answer is simple—rotation!

Think of a cylinder wrapped in string. If you pull the string on both sides with equal and opposite force, the center of mass will surely remain stationary by virtue of Newton’s law, but the cylinder as a whole will spin about its axis. Clearly, just the total force on an object doesn’t paint the whole picture of how the cylinder moves; the location of those forces on the object also matters! And since we don’t want our boats to spontaneously capsize as we sail on the ocean, understanding this phenomenon is imperative for any practical application of buoyancy.

In essence, hydrostatic pressure-induced tractions don’t only induce a net buoyancy force $\vec{F}_b$ on an immersed object, but a buoyancy torque $\vec{\tau}_b$ as well. And unlike the buoyancy force, we don’t really have a neat torque version of Archimedes’ law that lets us calculate this buoyancy torque without using vector calculus. As a result, we find the process of calculating it nearly identical to the original process of determining the net buoyancy force:

1. Determine the pressure distribution of the fluid.
2. Determine the interface between the continuum of interest and the fluid.
3. Determine the pressure of the fluid at the interface.
4. Calculate the pressure-induced traction torques ($\vec{r}\times \vec{t}_s$) at each point on the interface.
5. “Add up” (integrate) those traction torques to get the total hydrostatic torque on the object.

We can again list out this last step succinctly using mathematical language:

$\displaystyle \vec{\tau}_b = \int_{A} \vec{r}_{cm} \times \vec{t}_s\ dA$

where $\vec{r}_{cm}$ represents the position of the point being analyzed relative to the object’s center of mass.

As it turns out, this process of summing up small contributions through integration is a critical tool in all areas of fluid mechanics, which we shall soon observe.