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Electric current is a flow of a scalar quantity, "charge".
Fluid flow is a flow of a scalar quantity, "mass".
Heat flow is a flow of a scalar quanity, "heat".

What would "magnetic flow" be a flow of? A scalar quantity "north"? You would need magnetic monopoles. The question is why imagining a magnetic field to be constructed of magnetic monopoles doesn't work well.

Flows have conservation laws. If the flow into a region doesn't match the flow out, then there is a build-up of something in the region (charge, mass, heat). It would be hard to build up "north"-ness somewhere without "south"-ness building up somewhere else. How do you capture that with mathematics?

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