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Let's say, I have half bucket of boiling water and I have added equal amount of freeze cold water to it. I want to have equilibrium temperature reached as quick as possible. Can I speed up the convection manually? Does manually mixing different levels of water work?

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Unless I'm missing something here, you can speed up reaching the equilibrium temperature simply by stirring the water mixture.

Better still, pour in the cold water from a height so it produces turbulent currents as the hot and cold water mix.

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