Why Do Velocity Components Depend on All Spatial Variables in Fluid Mechanics?

[R Power]

HI
I was a under a little confusion about vector field.
Consider velocity field of fluid flow:

V = u i + v j + w k

here V is vector and consider a cap over i, j, k (since they represent x,y,z directions)

now we know that u,v,w are functions of x,y,z,t. This is where i am confused.

u is velocity component in x direction then it should be function of only x and t. Why y and z also??

In a velocity field we see arrows representing magnitude and direction of velocity at any position, are these arrows over fluid particles? I mean can i assume each arrow as a particle at that position having certain velocity represented by the arrow?

I think i lack understanding of fluid velocity field. Can anyone expalin a bit.

 

[tiny-tim]

Hi R Power!

(use i j and k )

… u is velocity component in x direction then it should be function of only x and t. Why y and z also??

Imagine a river of width 60m flowing steadily in the x-direction.

The water is fastest in the middle, say 3 m/s, and zero near the bank.

So in the middle, the velocity vector is 3i, and at distance y from the middle, it is (|30 - y|/10)i …

u is a function of y, but is independent of x.

In a velocity field we see arrows representing magnitude and direction of velocity at any position, are these arrows over fluid particles? I mean can i assume each arrow as a particle at that position having certain velocity represented by the arrow?

Yes, that's exactly correct, the value of the velocity field at each point is the velocity of the actual particle that is (instantaneously) at that point.

 

[R Power]

Yes, that's exactly correct, the value of the velocity field at each point is the velocity of the actual particle that is (instantaneously) at that point.

If this is so streamlines should be same as pathlines whether flow is steady or unsteady. Am I correct?
Tiny Tim, please answer my another post relating to streamlines,streaklines,pathlines in mechanical engineering section.