What is it called to play a music with flowing tempo in piano ?
What I mean by "flowing tempo" is that when you, for example, start the music by playing it strong and little fast, but then you play soft and rather slow etc. like moving on a wave.
Rubato. This specifically means compensating for each slow-down with a speed-up. If there was a click track, you'd come out on the right beat at the end!
'Rubato' doesn't instruct you to play the slow bits soft, the fast bits loud though.
If you want the more common effect of slowing down at the 'expressive' bits but never really making up the time, just use one of the many words that mean 'Freely', 'with expression' etc.
It's what it sounds like... flowing. It is a feel thing. When you play enough music you start to get an innate sense of these things that are extremely nuanced. They cannot be described because they are a complex of interactive elements.
Tempo is not just a metronome. Tempo is a feeling of both time, rhythm, melody, harmony, dynamics, etc(and every one of those elements are elements of every other one).
For example, suppose a piece of music goes from point A to point B in whatever way it does. Now, suppose all you do is change one element such as the dynamics. For the music to flow, the tempo might have to be subtly changed.
Why? Because that is the way it is. It can be because of simple performance reasons such as when you play at ff it changes the physical way you play which alters the timing. If you try to play it the same way as pp it will sound/feel wrong(or not quite as good or right).
It is such a complex thing that when you try to overthink it you end up making it worse. Simply play, you will understand everything in due time. If it doesn't feel right, it isn't... and it's simple as that.
Now, you might say that what if we just change the expressive indication from flowing to calando or from allegro to adagio? Well, the composer tries to chose the closest tempo indicator that he can that works with all the other elements. Have you ever noticed that with some pieces that if you alter the tempo even a little that it destroys the feel while with others it doesn't seem to matter? Well, that is precisely because of all the other elements that are in play.
Again, we can try to parse everything build a tomb of what is going on, but your brain can figure this stuff out instantly if you just allow it to do it's job. It's called experience and it's something that you learn over a long period of time.
Don't get bogged down in the details. Most of the great musicians don't "know" shit as far as "theory" is concerned. They just played and learned to play well then people think it's great and special and try to figure out what makes it special and codify rules which only confuses people like you.
Also, each person is different and will have subtle differences in how they perform something due to their physical differences(hand size, finger length, muscle strength, etc) along with their experiences, etc.
Flowing means to flow, like water flows... it can flow smoothly or like rapids. It's just a word, it means nothing because water actually doesn't flow but it lets you know something is going on. Physics would tell you it's gravity at work, Zen Buddhists will tell you it's all in your head, it doesn't matter. It is what it is. If you really want to get somewhere, get nowhere! (which you probably don't get because you are too busy trying to get somewhere... so I will interpret, it means stop overthinking it and just do!)