Can you perform a theoretical experiment in which we mix hot water with cold water and some material that changes the water color, and calculate under what circumstances the material will get evenly distributed in the water before the water temperature will become the same in all the places?

WattsAqualockBrass Fittings

After mixing the water, I can make it that the hot water is in A and the cold water is in B. Don't get me wrong it's not as hot and as cold as it was at time 0. But still there is a meaningful amount of hot water in A, that is too hot for my baby and, there is a meaningful amount of water in B that is too cold for my baby.

Sometimes the bath for my baby is too hot, so I mix into it some cold water. I open a cold water stream and mix the water with my hand waiting for the temperature to drop.

WattsAqualockBall Valve

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WattsAquaLock

But here is something that I don't understand. After doing a really good mix with my hand, I notice that the hot water is kept together. It's hard to explain so check out the image.

The reason for this unexpected behavior is the density of the hot water is lower than the cold water and so the hot water floats over the cold water. Even after the color already been mixed and, even after I mix the water a hot water concentration will be created on the top layer.

And this is not the only strange thing. I also like adding a color-bath-drops(It changes the water color) to the bath to make it more fun for the baby. The color sink/mix with the water much faster then it takes for the cold water and the hot water to reach the same temperature.