Stream Function: 2D Notation Explained

In summary, the stream function in two dimensions is a scalar function that can be used to satisfy Laplace's equation for an incompressible fluid. In this context, it represents the curl of the velocity field and can be treated as a vector field in the plane. However, the curl operation is only defined in three dimensions, so it is necessary to consider the 2D space as a plane in a larger 3D space. The stream function can be used to represent the velocity field in this way, and its curl is used to satisfy Laplace's equation.
  • #1
FrogPad
810
0
I'm having some trouble trying to decihper the notation used for the stream function in two dimensions.

Say we have a velocity field:
[tex] \vec V(x,y) [/tex]

The fluid is incompressible, thus Laplaces equation must be satisfied.
[tex] \nabla^2 u = 0 [/tex]
Where: [tex] u = \Nabla \vec V [/tex]
Thus: [tex] u_x = V_1 [/tex]
[tex] u_y = V_2 [/tex]

Where [itex] u_x [/tex] is short hand for the partial derivative of [itex] u(x,y) [/tex]

So now here comes the stream function.
Is it a vector function? It has to be right?

The definition I have is that the stream function satisfies:
[tex] u = -\nabla \times s(x,y)\hat k [/tex] (1)

Now the curl is supposed to return a vector right?
So how is this satisfied with (1). I'm guessing that it must deal with the [itex] \hat k [/tex]

But if someone could help me clear this up that would be cool. Also please, note that we are ONLY dealing with 2 dimensions for right now.

Thanks
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
I assume you mean for the fluid to also be irrotational, and for u to be the fluid potential. Is that right? Then you should have [itex]\vec V = \nabla u[/itex], although from the next two lines it seems like you know this, and the above was a typo.

Technically the curl can only be taken of a vector field, and it is another vector field, and this operation is only defined in 3D. You can use the curl in 2D if you think of your 2D space as a plane in a larger 3D space. Then the curl of any vector field in the plane is perpendicular to the plane (ie, something times [itex]\hat k[/itex]), and so may be treated as a scalar (this happens with the vorticity [itex]\vec \omega[/itex]). Conversely, if you have a vector field that is everywhere normal to the plane, it may be treated as a scalar, and its curl is a vector field in the plane. This latter case is what happens with the stream function. Also, I think you want to set the velocity, not the potential, equal to the curl of the stream function.
 
Last edited:

1. What is stream function in 2D notation?

Stream function is a mathematical concept used in fluid mechanics to describe the flow of a two-dimensional, incompressible fluid. It is a function that maps the position of a particle in the fluid to its velocity components in the x and y directions.

2. How is stream function represented in 2D notation?

In 2D notation, stream function is represented by the Greek letter psi (Ψ) and is a function of the x and y coordinates. It can also be represented as a function of streamlines, which are curves that are tangential to the velocity vector at every point.

3. What is the significance of stream function in fluid mechanics?

Stream function is an important tool in the study of fluid mechanics because it allows for the simplification of the Navier-Stokes equations, which describe the motion of a fluid. It also helps to visualize and understand the flow patterns in a fluid.

4. How is stream function related to vorticity in 2D flow?

In 2D flow, vorticity is directly related to stream function. The vorticity is equal to the negative of the second derivative of the stream function with respect to the x-coordinate. This relationship is known as the vorticity-stream function equation.

5. Can stream function be used to model 3D flow?

No, stream function can only be used to model 2D flow. In 3D flow, the fluid particles can move in any direction, so stream function is not a suitable representation. Instead, other mathematical concepts such as velocity potential and vorticity are used to model 3D flow.

Similar threads

  • Calculus and Beyond Homework Help
Replies
9
Views
699
  • Calculus and Beyond Homework Help
Replies
6
Views
971
  • Calculus and Beyond Homework Help
Replies
20
Views
386
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
966
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
2K
Back
Top