is there a problem in using the same water bath to cook different foods on succesive days? the food never touches the water, and the water is filtered. there is no residue in the water bath.
Nope, no problem at all. You can use it for several days.
If you are going to leave it unused for a week or more, I’d dump it though…you don’t want algae growing in your vessel. 
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Not at all. The only reason that I drain my water is to add really cold water and ice for the next ice bath cook.