If everything works, you should see the food coloring begin moving around, leaving colored streaks and swirls on the milk surface. You may also see bursts of color suddenly appear from an otherwise plain white surface, just like colorful volcanoes or fireworks erupting. If nothing happens after several seconds, try touching another drop of soap to the milk surface, perhaps in a different spot. You can also try gently rubbing a little soap on the bottom of the bowl, or even carefully adding a drop of soap directly to the surface, but don't add too much soap. A drop or two is all you should need. Be patient- sometimes nothing may seem to be happening for a minute or more, but colors are moving around below the surface and will suddenly burst into view. Remember not to stir the milk or shake the bowl- we want the colors to move on their own, not because you move them!
Mother's milk is an amazing substance, which really isn't surprising considering that it contains all of the nutrients a baby needs to survive and grow. The plain white liquid you see is actually a uniform (homogenous) mixture consisting of fats or oils (composed of molecules called fatty acids), sugars, proteins, vitamins and some other chemicals, but mostly water molecules. This is called an emulsion or suspension- a mixture of liquids and solids, most of which are not truly dissolved. Normally these various components would separate (and in raw or non-homogenized milk some, notably the fats, still do), but the fatty acid and protein molecules have both hydrophilic (water-loving) and hydrophobic (water-hating) regions, which causes these long molecules to twist around and group together into more stable arrangements, making it possible for them to stay suspended in the water. The fatty acids form globules, sort of like little water balloons (or in this case, fat balloons), with their hydrophobic ends on the inside and their hydrophilic ends on the outside facing the water molecules. The proteins form similar balloon-like structures called micelles. Thus, the milk is in a complex state of equilibrium (i.e. not changing), with all the molecules not exactly "happy" to be where they are, but at least willing to stay put as long as they aren't "upset" too much. When you gently add the food color (which are dissolved in water) they are also happy to stay put in their own tiny puddles.
Molecules of dish detergent, like fats and protein, also have both hydrophilic and hydrophobic regions, so adding them to the surface of the milk completely disrupts the existing state of equilibrium as the soap molecules grab and pull on the fat globules and protein micelles. Scientists say that on a molecular level, this soapy region of the milk is in a state of chaotic disarray and all of the various molecules begin to move about, dragging the food colors with them as they try to return to some equilibrium. In other words, before the soap was added the molecules in the milk were at least where they liked to be and didn't want to be disturbed. Then come along these annoying soap molecules like bullies that want to disturb everything (on a molecular level, of course). Because milk is all white, we wouldn’t normally be able to see what's happening, but as the food coloring is sort of caught in the middle we can observe all the action taking place.
The fats in milk are largely responsible for the action in this experiment, so milks with more or less fat may behave differently, including milks from other mammals (see the link below). Temperature also affects many physical and chemical reactions because more energy is available at higher temperature.
References and links to more information:
Others versions of this activity:
Milk from other mammals:
What is homogenized milk?:
Learn (LOTS) more about milk chemistry: