Anything you can suspend in water that will look white will be a particulate of some kind (like paint pigment).
Any particulate larger than colloidal size will fall out of suspension over time.
Dyes will stay dissolved (as food coloring will) but dyes don't have the property of reflection, so can't look white.
The best I can suggest is colloidal chalk, which is produced by a reaction between lime water and carbon dioxide.
Children's chemistry sets used to include instructions on how to make lime water, and you can blow your breath through the solution to provide the carbon dioxide.
An alternative solution would be to find a way to recirculate the water, so that whatever you suspend (water color pigment, for instance) will stay suspended rather than settling out.
An aquarium pump and airstone is one option.
You will surely have trouble, over time, with algae growth that will turn your pretty white water green, brown, or reddish, but there are algicides you can add to the water to slow or prevent this (they're sold for fountains, to keep them from filling with pond scum, and for water beds).
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You could try adding something to milk to sterilize it.
Household bleach may work, but may also turn it clear
(bleach turns coloured things white, but I'm not cure if it turns white things clear).
Campden tablets (sodium metabisulfite) as used by homebrewers for sterililising might be a better choice.
You didn't say what you need this for, so I don't know if it has to be a liquid or just look like a liquid.
I have heard that food photographers will use PVA glue instead of real milk.
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Commenting on another answer gave me the idea to use an anis spirit (ouzo, sambuca, pastis, raki) mixed with water.
The ouzo-effect results in a milky solution which, according to the Wikipedia article, is very stable due to the small particle size in the range of microns and the chemical interplay.
I will try this (and suggestions in the other answers) out and update the answer (comment on answers).
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The stuff only has to look milky; not be milky to achieve your goal.
When I shoot liquids for advertising that I want to glow (or look like they do) such as a glass of frosty beer or a jar of marmalade jam, etc, I put a reflector card behind the subject and let the light show through the liquid.
Using this technique allows me to vary the lighting in a very controlled way that mixing unknown substances don't.
If your liquid stuff is too thick or too thin you won't achieve the effect you're striving for.
All you have to do is slightly angle the reflectors for each test tube or flask in your set-up.
Product photographers have been doing this for years.
If you choose the colours for the card reflectors, you can get any colours you want sitting right next to each other.
Green next to fluorescent red is easy to achieve this way without using coloured liquids at all.
For white, use a white card.
For blue, use a blue one, etc.
Trim the cards carefully
so they don't show.
LED lights are adjustable but you'll find the stray light hard to harness.
Different hues of stray light will mix giving you neutrals which will diminish your intended effect.
You'll see the problems in the final shot.
If you were shooting functional props for motion pictures, your way would be necessary due to handling by the talent, angles for shots would be uncontrolled, and the lighting problems for that situation are difficult.
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Inorganic soluble cutting oil used in industrial machines such as lathes and mills.
It's mixed with water, stays in solution as far as I know, and doesn't mold.
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Mica powder.
I made a hallow key chain shaped like a milk carton with “milk” inside.
I used baby oil and mica powder.
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I want to color water white/cloudy/milky, so that it disperses light nicely. I have tried to dissolve soap in it, which worked very well, but the dissolved particles sink down after half a day and the effect is gone.  Are there any better ideas than buying food color (since we're talking about a life hack)? The fluid should also not be prone to molding, which I guess will happen when milk is used.
How to color water milky/white, - non-molding, non-sinking?
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It is unclear whether there is a non-particulate solution to this problem and the particles will therefore sink over time. However, colloidal chalk might be an effective way to produce this effect. It's possible that to achieve this effect you could use milk, PVA glue, soluble cutting oil or mica powder. The options are to use colloidal chalk, circulate the water with an aquarium pump and use a steriliser to keep clean, use sterilised milk or PVA glue, mix water with an anis spirit, position card refelectors behind the container, use baby oil mixed with mica powder or use inorganic soluble cutting oil in water.
Use colloidal chalk or find a way to circulate the water for example with an aquarium pump. You could sterilise the milk or use diluted PVA glue. Mix some anis spirit (ouzo, sambuca, pastis, raki) into water to produce a stable cloudy solution. Use card reflectors of the right colour positioned behind the container. Use inorganic soluble cutting oil. Use baby oil mixed with mica powder.