- #1
FrogPad
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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
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
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