The continuity equation

A fluid, that is, a liquid or a gas, can flow steadily along a pipe and that pipe can be made up of lengths of pipe of different diameters. It would be no surprise to learn that the fluid does not flow with uniform velocity at any point and nor is it surprising that finding out how the velocity actually varies with, say, radius is not easy. For engineering purposes we would look for some way to avoid this complication. The fact is that the only constant for all the different pipes is the mass flow . Then, for any given point in any one of the pipes:-

                                                             where  is the density of the fluid at the given point,  is the area of cross-section of the pipe and  is the mean velocity of the flow.

 

For a gas the mean velocity of flow and the density change along the pipe but, for most liquids the density does not change.

 

There is a consensus view that the best way to proceed for most purposes is to treat the flow as having physical properties that vary along the pipe but not across it. Then it is called one-dimensional flow and, having stated that the flow is to be treated as one-dimensional we change the average velocity  to  and write:-

                                                          for any fluid and:-

                                                          for a liquid where  is the volume flow.

 

This is called the continuity equation.