Introduction: Matter cannot be created or destroyed - (it is simply changed in to a different form of matter). This principle is known as the conservation of mass and we use it in the analysis of flowing fluids. The principle is applied to fixed volumes, known as control volumes (or surfaces), like that in the figure below:
For any control volume the principle of conservation of mass says.
For steady flow there is no increase in the mass within the control volume, so
This can be applied to a stream tube such as that shown below. No fluid flows across the boundary made by the streamlines so mass only enters and leaves through the two ends of this stream tube section
We can then write
Mass entering per unit time at end 1 = mass leaving per unit time at end 2
The flow of fluid through a real pipe (or any other vessel) will vary due to the presence of a wall - in this case we can use the mean velocity and write
When the fluid can be considered incompressible, i.e. the density does not change, ρ1 = ρ2 = ρ so (dropping the m subscript)
This is the form of the continuity equation most often used
This equation is a very powerful tool in fluid mechanics and will be used repeatedly throughout the rest of this course.