When the box is upside down and filled with water, is it possible to rotate the white wheel freely without water being leakage by any method or any existing technology?
White is wheel and green is axle.
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This sort of sliding seal is very difficult to achieve, especially when you have a square corner.
Rotary engines have this problem and tend to accept a certain amount of leakage as inevitable and design around it.
Changing the wheel profile to give it a more rounded shoulder would certainly help.
One possible solution is an expanding seal, either an inflatable tube or a material which expands when wet, this tends to be the solution used for inboard propeller shafts on boats, even then you tend to accept a certain amount of leakage. Indeed the new Royal Navy aircraft carrier had exactly this problem is sea trials which goes to show how difficult a problem it is.
Addling additional layers of seals ie a labyrinth seal may also help.
another potential issue is that if the wheel if open to the environment contamination by grit etc may be an issue, either disrupting the interface or damaging the seals themselves. Similarly the wheel would need good concentricity and surface finish.
I'm not entirely sure what this is intend to achieve either, if you don't want water to escape why not just have the wheel fully enclosed on one side or the other ? The only similar application I can think of is something like the self loading rollers used for line painting on sports fields etc or indeed a ball point pen.
It's hard to be more specific without more information about the intended application.