When I am taking a shower, the shower curtain slowly moves towards my legs.
Also, it seems that the hotter the water, the faster it gets to my skin.
Why is that?
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It is due to the convection in the shower, as mentioned in Steeven's comment.
The water is significantly above room temperature. Obviously it is going to heat the air (and you can feel the warm air/vapour yourself).
The air in the shower is both warming up, and becoming more humid. Both of those factors lower the density of the air, making it raise up.
When the air flows up, it creates a low pressure zone in the shower. Once the pressure gets low enough compared to the room outside, the pressure difference pushes the curtain into the shower until enough air can get in to account for the air lost to convection.
Warmer water will cause greater convective flows, so the effect should be more noticeable.
Disclaimer: Note that the heat and humidity are not the only cause of convective flows; and convective airflow is likely only part of the situation. As some comments (and the wikipedia page on this subject) mentioned, the flow of the water downwards may actually generate an air vortex in the shower. This vortex creates a low pressure zone, which drives the convective process regardless of water temperature.
There are also other potential reasons, and without experiments and potentially (multi-physics) simulations, it's very unclear what the primary cause of this phenomenon is.
I still feel this answer properly addresses the second question about why warmer water makes the process more noticeable.
EDIT on reflection, my answer, as noted in the comments below, is not correct (to be specific, it's dead in the water), it simply would not produce the mass of less dense air required. END EDIT
The Bernoulli Principle is probably the main cause. This states that an increase in velocity, of the air associated with the falling water, produces a reduction in air pressure around your body, and then ambient air pressure in the bathroom pushes the curtain towards you.
Another possible reason is the Coanda Effect, which you can observe when you hold a spoon near a stream of water from a tap and as the water moves over the curved back of the spoon, the spoon will move inwards toward jet center.
I have to say I include this Coanda Effect for interest only, I believe the first reason to be far more significant.
There are numerous theories and this has been a subject of eternal debate. However, given that we now have mountains of evidence, I believe at this point we can state conclusively that the mechanism is as follows:
The water falls in the front of the shower. The falling water carries air downward with it. The air that moves downward at the very front of the shower moves upward further back. This pattern of downward moving air in the front and upward moving air in the back closes to form a stable, horizontal vortex. The vortex has a low-pressure zone in the middle, which produces a constant, low-pressure against part of the curtain. The higher air pressure on the other side of the air curtain therefore pushes it inward.
I have personally never been convinced of any explanation that involves a difference in air-pressure causing this effect.
When I first used a shower that suffered from this problem - about 25 years ago - I tried various things to understand it. One of those things was to make sure I left a large gap at one end of the curtain to allow the air to freely flow around the end of the curtain. This should allow cold air to enter the shower and warm air to leave with little obstruction. It made virtually no difference.
I fail to see how the pressure difference between the outside and inside can be so great that an opening of perhaps half a square metre cannot allow equalisation to take place.
The temperature of the room compared to the water also seemed to make little difference.
Unfortunately, I have never reached a conclusion, but my pet theory is that it may be electrostatic in nature. Indeed, the very small difference you do observe when creating an opening in the curtain is far closer amount you would expect from decreasing the surface area of the curtain than the very much larger change in the pressure difference that would result from the opening.
EDIT: Have just remembered the rest of my reasons for leaning toward electrostatic forces and away from pressure differences.
I noticed that the effect was much smaller before I got into the shower - long after the water got to full temperature. It worsened after I entered. If the effect is caused by some interaction between the water and air, surely it would DECREASE as enter the shower, not INCREASE...? @DavidSchwarz says the effect is caused by a "vortex" but when I am in the shower, the falling water column is hitting my HEAD and is therefore about 5 or 6 times shorter than when it is unoccupied.
Also, I noticed that my wife (with a significantly smaller body than mine) never complained about the problem at all. Observation showed is was much reduced for her than for me.
If the cause was electrostatic, then my larger body would increase the "charge" - AND I would be closer to the curtain to make the effect much larger. As observed.