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

In summary, Tiny Tim said that the streamlines of a fluid flow are always the same as the pathlines, regardless of the flow's state.
  • #1
R Power
271
0
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.
 
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  • #2
Hi R Power! :smile:

(use i j and k :wink:)
R Power said:
… 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. :wink:
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. :smile:
 
  • #3
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.
 

1. What is a vector field in fluid mechanics?

A vector field in fluid mechanics is a mathematical representation of a fluid's velocity at different points in space. It is characterized by a vector at each point, with the direction of the vector showing the direction of fluid flow and the magnitude of the vector indicating the speed of the fluid at that point.

2. How is a vector field related to fluid flow?

A vector field is directly related to fluid flow as it describes the velocity at each point in space. This information can then be used to calculate other important properties of fluid flow such as pressure, acceleration, and vorticity.

3. What are some applications of vector fields in fluid mechanics?

Vector fields are used in a wide range of applications in fluid mechanics, including aerodynamics, hydrodynamics, and weather forecasting. They are also important in studying the flow of liquids and gases in pipes, channels, and other structures.

4. How are vector fields represented and visualized?

Vector fields can be represented by arrows on a graph, with the length and direction of each arrow indicating the velocity at that point. They can also be visualized using computer simulations, which can show the changing patterns and behaviors of fluid flow over time.

5. What are some challenges in working with vector fields in fluid mechanics?

One of the main challenges in working with vector fields in fluid mechanics is accurately measuring and predicting the complex behavior of fluids. This requires advanced mathematical models and computational tools. Additionally, the accuracy of vector fields can be affected by factors such as turbulence, viscosity, and external forces, which can be difficult to account for in real-world scenarios.

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