Buoyancy and Stability

What is the vertical force acting on a body which is completely submerged in a fluid? Answer to such a question can be very well found in the theory developed in the previous section. Archimedes seems to have discovered the laws concerning submerged bodies as well as floating bodies. What is well known as Archimedes principle states -

1. The vertical buoyant force experienced by a body immersed in a fluid is equal to the weight of the fluid displaced.
2. A floating body displaces its own weight of the fluid.

Figure 2.16 : Forces about a body immersed in a fluid-1

Figure 2.17 : Forces about a body immersed in a fluid-2

 

Proof is straight forward. Consider an elemental volume within the immersed body as shown in Fig.2.16 . Now the buoyant force is given by,

$$F_B~=~\int_{body}~~(p_2~-~p_1) dA_H$$ (2.47)

where $dA_H$ is the area of cross section of the elemental volume chosen. We have from Eq. 2.48

$$F_B~ =~-\gamma~\int_{body}~~(z_2~-~z_1) dA_H$$

$$=~\gamma~~(body volume)$$

$$=\texttt{weight of the fluid displaced}$$ (2.48)

It can be shown that the buoyant force, $F_B$ passes through the centroid of the displaced volume as shown in Fig.2.17. The point where this force acts is called "Center of Buoyancy", denoted as $CB$.

The above result holds good even in the case of a partially submerged body i.e., a floating body. It is assumed that part of the body above the liquid level is in air. The weight of air displaced as a consequence is ignored. (Fig. 2.18). For this case as well,

$$F_B~=~\gamma~~(diplaced~ fluid~ volume) =\texttt{weight of the body}$$ (2.49)

Figure 2.18 : Partially Submerged Body

 

The theory developed so far does hold good in case of a fluid for which specific gravity $\gamma$ is not a constant, a layered fluid for example. However now the buoyant force may not act at the centroid of the displaced volume.

The theory developed is also applicable where the fluid involved is a gas, say air. Convection currents established in atmosphere depend upon the buoyant forces generated.

(c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney