Question

What is the basic, fundamental meaning of the Continuity equation? What are some examples and the importance of this rule?

Answer 1

I'm assuming that you are referring to the equation which forms the fundamental bedrock of continuum fluid mechanics.

Imagine section of pipe with a segment that is transparent so that you can view the fluid flowing within it. Now imagine that you have been presented by the Alien Qxahrl, of the planet Fluzoop, a sensory device that looks just like an ordinary pair of spectacles, but is in fact a pair of independently configurable view screens that can resolve objects upto 1 nano-meter in diameter and can simultaneously record temperature, pressure, velocity and the density of any flow.

You put on this device and call up the "Eulerian view" option while looking at the transparent section of the pipe. Voila, you see the transparent section being lightly outlined in red, with the words INLET and OUTLET placed at the appropriate locations where the flow enters and exits the transparent section. Now by choosing filters you can colour those particles according to temperature, density, pressure or velocity.

Obviously some very high end computing is going on here. But what are the equations that are being solved?

Well, several of them actually. One of them is our good friend the continuity equation.

Set out in words it is basically a statement of conservation of mass. It says, for non-reactive flows, which contain no sources within the domain, what goes in through the inlet, must either stay in the domain and increase the volume of the domain or it must flow out of the outlet.