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Whenever I add milk to my morning coffee I often enjoy watching the patterns which are created. These patterns have a striking resemblance to certain fractals and my question is, "Why?" Oh dear, that is never a good question, so let's try "Why shouldn't they?"

The obvious observations are that the water is very hot when the milk is added, so we expect the "coffee particles" (non-milk part) to have a higher velocity relative to the "milk particles". Secondly, the coffee part, having been recently stirred, is often still rotating (clockwise in my case). Finally, the image we observe is a projection of some distinct portion of a "top layer" of the milk and coffee mixture.

Add diffusion into the mix and it really seems like some crazy stuff should go down! But I'm afraid I'm not satisfied. Why physics.SE... why?

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To some extent a full answer might require more data.

I think it likely that most of what you see is stretching in a chaotic fluid flow.

If you consider a small ball of milk within the flow, the flow evolves in a complex non-linear way. This means that nearby elements within the ball have trajectories in the flow that rapidly diverge. If the ball is to remain constant in volume (conservation of mass), then what we see is that it stretches in one to two dimensions and contracts in the other one or two.

The net effect is for long streamers to be created which show as the complex patterns you observe.

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    $\begingroup$ The Navier Stokes equations that describe the fluid flow have a tendancy to produce chaotic behaviour. For a summary of why this happens see Terence Tao's blog. $\endgroup$ Jan 22, 2014 at 9:39

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