The continuity equation

By Marcus Wilson 25/10/2011

Yesterday, being a warm, sunny Labour Day holiday  (those words don’t usually go together) we decided we needed to get out of the house and go somewhere interesting, and chose the Waitomo area. Didn’t go into the show caves this time (done those a few times before) but chose to keep the bank account under a bit of restraint and do some of the amazing walks through the area.

The Ruakuri bush walk is a favourite – you see some fantastic limestone scenery, including a natural tunnel through which the river runs. And it’s free.  If you get the timing right, you can also watch the intrepid ‘blackwater’ rafters jump into their rubber tubes and start their journey down the river. Must do that one day.

With some fairly simple physics application, it was easy to tell that the jumping-in-point must be pretty deep. That’s because the water flows past very slowly, compared to the raging torrent it is further up the river. The width of the river hasn’t changed much, the volume of water going down it won’t have changed much, and so, the natural conclusion is that the river must have got deeper.

This argument is based on ‘continuity’. Specifically, in this case, the flow of water down the river is constant everywhere (since water doesn’t get created or destroyed – at least not in a vast amount over a short distance – and it doesn’t squash); the flow rate is given by the product of the cross-section area of the river and the river’s velocity (or, for the more mathematically inclined, the surface integral of the area and the velocity); therefore, if the velocity goes down the cross-sectional area must increase. If the width hasn’t, then the depth must have.

Sure enough, my conclusion was quickly verified by a blackwater tour guide clad in wetsuit running off the bank at speed and bombing into the river.

The continuity equation raises its head in many forms throughout physics. Fluid flow is the obvious example, but we also see it in electromagnetism. Here, lines of electric and magnetic field don’t just end in empty space – electric field lines have to end on charges, and magnetic field lines never end at all. Therefore, the strength of the field very much depends on the area to which it is confined. Squash it into a small region, and the field strength is going to be larger than if it is allowed to spread over a wide volume. 

The flow of heat through an object is another example. In this case, heat energy is the thing that doesn’t get created or destroyed, i.e. is ‘continuous’. The same mathematical equations apply to all these cases, meaning that solving one physics problem often means you’ve solved other physics problems.

You can probably think of other examples yourself – some are quite fun and not so obviously physicsy – such as the flow of people down a street.

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