# Chapter 5

# First Applications of the Conservation Equations

## 5.1. Theorem of the dynalpy

In what follows, we restrict ourselves to flows where body forces are neglected (neither a heavy fluid nor an electrical conductor). If the motion is steady and (*S*) is a closed surface enclosing a finite volume (*ϑ*) that *does not contain a body*, then we write the equation of motion of section 4.4 as:

where is the tension vector representing the contact forces exerted by the fluid contained in (*S*) on the external medium (hence the minus sign to the right-hand side). This relation can also be written as:

The *dynalpy vector* is given by:

The relation above shows the following result:

which is read as:

In a steady flow, and in the absence of body forces, the flux of the dynalpy vector through a closed surface not containing any body is zero.

Let us consider an impermeable body (no mass exchange between the body and the external medium) placed in a flow and apply the above result to the fluid contained ...