**To**:**nevensf@xxxxxxxxxxxxxx, ansley@xxxxxxxxxxxxx****Subject**:**Re: Calculating stream function from known velocity field.****From**:**kessler@xxxxxxxxxxxxx (Billy Kessler)**- Date: Mon, 14 May 2001 10:38:04 -0700 (PDT)
- Cc: ferret_users@xxxxxxxxxxxxxxxxxxx
- Sender: owner-ferret_users@xxxxxxxxxxxxxxxxxxx

A streamfunction is only defined if the velocity field is non-divergent. ("Streamlines", which are lines always parallel to the flow, but with arbitrary spacing, can be defined for any flow, and I think that is what Ansley was describing in her first e-mail). I have some old scripts to do streamfunctions that I would have to dig out and check before posting them to the group, but the math goes like this: Define a streamfunction, Psi(x,y), such that u = d(Psi)/dy and v = -d(Psi)/dx. (1) By definition: du/dx+dv/dy = 0 (2) so the field is non-divergent. Given (1), u=d(Psi)/dy -> Psi = Int{y0:y}u dy + a(x) (3) and therefore: d(Psi)/dx = Int{y0:y}du/dx dy + da/dx Given (2), d(Psi)/dx = Int{y0:y}(-dv/dy) dy + da/dx = -v(x,y) + v(x,y_0) + da/dx (4) But also, by definition: d(Psi)/dx = -v(x,y) Therefore: da/dx = -v(x,y_0) a(x) = -Int{x_0:x}v(x,y_0) dx (5) >From (3) and (5): Psi = Int{y0:y}u(x,y) dy - Int{x_0:x}v(x,y_0) dx (6) Algorithm: Choose an (x_0,y_0). Integrate the second (indefinite) integral of (6) along y_0. The result is a function of x. Then at each x, indefinitely integrate the first integral of (6) along y. The streamfunction is the sum of those two. Billy K

**Follow-Ups**:**Re: Calculating stream function from known velocity field.***From:*E.D. (Ned) Cokelet

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