I am wondering about the flow of current through the short circuit in the attached diagram. To my mind it is undefined for the following reason:
Voltage accross the short circuit = IR. As resistance is zero then V is also 0.
Current is V/R - 0/0. Undefined.
Is this the case? In reality I know a short circuit does indeed have current running through it.

 

In analyzing the lumped element model of the circuit the nodes not separated by elements are considered to be the same node for the purpose of analysis. i.e. for all intents and purposes, the left side of the 1Ω resistor is physically the same point as the positive end of the 12V source (as well as the top and bottom of the 4Ω and 12Ω.)

In a real circuit there may be an actual wire, in which case the wire always has a non-zero resistance.

 

Look, I like your circuit.
Basically you don't have a short circuit in there,
You just have a battery of 12 volts connected across a resistor of 1 ohm.
The "short circuit" is just a wire connected to the positive side of the battery.
Haha. (sorry)
And the other resistors are shut down, those parallel ones (4 and 12 ohm) they have no voltage or current runnign through them at all. It's like they're not even there.

The current through the "short circuit" is 12 amps!

But it's not a short circuit.

 

There is current in the loop consisting of the 12V battery, the 1 ohm resistor and the short. I =V/r=12/1=12 amps flowing through the short.

 

But it's not a short circuit.

Of course it's a short circuit.

This looks like a simple model of a 12V battery with a 1 ohm source resistance driving 4 Ohm and 12 Ohm loads in parallel. The battery (realistic model of ideal source with source resistance) is shorted.

 

Just a matter of one's perspective.