Some people call this "glycerin rivers", but that is a bit of a misnomer, since a true glycerine river would be soft and goopy. Some call it stearic streaking, but there's more than one fatty acid involved, not just stearic. Probably the most accurate name is just "streaking" or "mottling".
Soap is made of many different kinds of soap molecules. As the soap cools in the mold, some parts of the soap may harden before other areas do. That affects where a colorant such as titanium dioxide ends up in the soap. Also some soaps are clear and others are opaque, and the clear soap surrounded by colored or opaque soap can also create a streaked or mottled effect.
Any soap can streak, but slow cooling, allowing the soap to gel, adding titanium dioxide, and using palm oil seem to increase the chance of streaking/mottling. Here are some things to try if you want to reduce the chances of streaking:
Soap cool
Avoid gel
Put your powdered colorants in glycerin, oil, or water and let them stand for several hours before use so they get thoroughly wetted
Mix palm oil well before measuring it
Use oil-soluble titanium dioxide rather than a water-soluble or a water-and-oil soluble product
I also suspect the way the soap is put into and manipulated in the mold is also a factor, since the streaks are often parallel to the path the soap took when it was poured into the mold or parallel to the path created when the soap was swirled. My guess is that a thin batter is more prone to streaking, but some decorative effects can't be done with a thicker batter. It's a bit of a tradeoff.
