The Continuity Equation is used to study the flow of many different quantities, ranging from actual physical quantities to highly abstract ideas. However, it is most commonly studied in fluid mechanics due to its wonderful ability to represent fluid flow very succinctly.
Hey everyone, I’m back with another Equation Explained video! Today, we’re looking at an equation used throughout physics, based on common sense and classical physics. I like to think of the equation as preserving the continuity of the flow of any substance we happen to consider, i.e. avoiding continuity errors like you seen in TV shows – hence its name! This equation is most commonly studied in fluid mechanics, but can be applied to any quantity that cannot randomly teleport from one place to another, and must move in a smooth and continuous way.
The first thing to understand is the meaning of the terms used in the equation. If we are studying the behaviour of a particular substance, then q is the amount of that substance in a particular region of space. The first term in the equation deals with the net rate of change of the amount of substance in that region. For example, if we are looking at a group of particles, then q represents the number of particles in a given region of space, and the first term of the equation represents the net rate of change of this number in the region. We can also apply this to the flow of energy, so the first term represents the net rate of change of energy in our region. The most common use of this equation is in the field of fluid mechanics – studying the flow of either viscous or non-viscous fluids is really elegant! Or we could apply the equation to something as abstract as the “probability density” of a wave function in quantum mechanics. Even though this equation cannot directly be applied to quantum objects, we can treat the probability density of finding a quantum object like a fluid that can flow smoothly and continuously. Therefore, the continuity equation can be applied here too!
The second term deals with the flux of our chosen quantity through the closed surface that surrounds our region of space. Flux basically is a measure of how much stuff (chosen quantity) is passing through any given surface. And so in this case we are looking at the total amount of stuff either entering or leaving our chosen region of space, through the closed surface that naturally surrounds it, per unit time. We discuss a few important restrictions on the surface that we are allowed to use in this equation too.
The final term in the equation represents the amount of stuff either created or destroyed in the region we happen to be considering, per unit time. This term is zero if we happen to be studying conserved quantities. For example, if we are looking at the energy in a region, then this final term must be zero as energy cannot be created or destroyed anywhere. It only make sense for energy to move around from one region to another.
So overall, this equation basically reads “net rate of change of amount of stuff in a given volume” = “amount of stuff created in that region per unit time” – “amount of stuff leaving the region per unit time”. It’s essentially saying that the stuff we are considering flows smoothly and cannot teleport from one place to another, and accounts for both the creation/destruction of this substance, and the movement of this substance in our chosen region of space.
Obviously, this equation cannot be applied to objects that can teleport. It is firmly a classical equation, and even more so, one seemingly based on common sense. But it is extremely useful and is used in many different areas of physics as we can see in this video.
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