We teach that the breathing should be natural and not forced into any particular rhythm while doing form in order to allow the Qi to sink to the Dantian. Within that framework, you want the breathing to be deep and long, as if coming from deep in your abdomen/belly. The word "stomach" can mean the same thing as "abdomen" or "belly," especially when talking about the outside of the body; however, when talking about the inside, the word "stomach" tends to refer only to the cavity that is primarily responsible for digesting food.
From a scientific viewpoint, the diaphragm can contract to suck air into the lungs. There are multiple ways to do this, but Wikipedia
describes the two extremes well according to what I understand:
Cavity expansion happens in two extremes, along with intermediary forms. When the lower ribs are stabilized and the central tendon of the diaphragm is mobile, a contraction brings the insertion (central tendon) towards the origins and pushes the lower cavity towards the pelvis, allowing the thoracic cavity to expand downward. This is often called belly breathing. When the central tendon is stabilized and the lower ribs are mobile, a contraction lifts the origins (ribs) up towards the insertion (central tendon) which works in conjunction with other muscles to allow the ribs to slide and the thoracic cavity to expand laterally and upwards.
In other words, you can use the diaphragm to produce a feeling of sinking in the belly or a feeling of expansion in the chest to the sides. We want to use the first method, which many call belly breathing. In reality, the belly does little, but you will see movement and have a cascade of feeling there if you use the correct method.
From our viewpoint of Tai Chi theory, this type of belly breathing allows the Qi to sink to the Dantian to store the maximum amount of internal energy, like water resting at the bottom of a bucket. If the Qi sinks only partially, it is likely to be insufficient for your needs of the moment. When the internal is deficient, it will draw on the external and make it deficient. It is like water sloshing around in the bucket that requires special movement or a deeper bucket to prevent it from spilling.
One of our Ten Essential Principles is that internal and external should unite. That means that the body structure has to support the type of the belly breathing described above. For instance, you must "enfold the chest" and "loosen and open up the lumbar region" (i.e., "relax the waist"); otherwise the shape of the abdomen will not allow the Qi to sink properly.
Last week I had my push hands students experiment with this link so that they could feel it in their own bodies and how it affected their energy. You may want to try it yourself. Realize that abdominal muscles and the tissues around the lumbar spine are antagonistic with respect to controlling the pelvis. Try lifting the butt up and pointing the tailbone to the rear by forcefully contracting the tissues in the lumbar region. Then try to breathe out audibly, forcefully, and deeply. You will find the breathing labored, as if the lumbar region is squeezing the Qi out from the rear. Now try forcefully contracting the abdominal muscle to tilt the top of the pelvis backward and point the tailbone forward. Then try the breathing again. It will still be difficult, but this time because the Qi is squeezed out from the front. Now try opening up the lumbar region by imagining you are about to sit down and are letting your butt reach for the chair as you relax your abdominal muscles and are about to completely relax your legs. The tailbone will be "centered." If you try the breathing now, you should find that the sound comes from deep in your belly and is unconstricted.
I hope this is clear and is helpful.