Equation of Continuity

Equation of Continuity:

A bundle of streamlines constitutes a tube of flow. As the fluid velocity is parallel to the sidewalls, no fluid can flow through the sidewalls. A tube of flow is like a pipe in that the fluid that enters at one end must leave at the other.

The rate at which a fluid flows through a pipe can be easily obtained. First, consider the pipe to have a uniform area of cross-section. Let us find the volume of fluid passing per second through section X. We find that all the fluid contained in a cylinder of length ν, where ν is the velocity of the fluid, will pass through this section. Therefore, the volume, V of fluid passing per second through the section will be the volume of that cylinder whose length is XX’ where XX’ = ν and the area of cross-section is A. Hence the rate of flow, i.e., volume of fluid flowing per second through a cross-section A is given by-

If the pipe size varies, then flow velocity must also vary so that the fluid that enters at X must leave through Y. If the velocity of flow is ν₁ at X where the cross-section is A₁ and ν₂ at Y where the cross-section is A₂, then-

This equation is known as the equation of continuity. It is infact a statement of the law of the conservation of mass. It shows that where the tube area is large, speed is low, and vice-versa.