GPT-4 Conversation #1095071702276390962

[tranquil pumice]

let's say i have a circle named circle 1 with radius r1 and it has a uniformly distributed vorticity w (which is greater than zero). flow is incompressible. there is no vorticity outside of circle 1. let's say i draw a larger circle around circle 1 with radius r2. what would be the circulation around the boundary of the larger circle?

[mortal dune]

To calculate the circulation around the boundary of the larger circle (circle 2), you need to use the following equation:

Circulation (Γ) = ∫ V·dl

where V is the velocity vector of the fluid, and dl is the differential length around the boundary.

Since the vorticity (ω) is uniformly distributed inside circle 1 (radius r1), and no vorticity is present outside the circle, the circulation will only be due to the flow within circle 1. This means that you don't need to consider the flow in the annular region between circle 1 and circle 2.

Using the relation Γ = ω * A (where A is the area of the region with vorticity), we can calculate the circulation as follows:

Γ = ω * π * r1^2

This gives the circulation around the boundary of the larger circle, circle 2, since no extra vorticity is added beyond circle 1.

[tranquil pumice]

what would be the speed along the boundary of the larger circle

[mortal dune]

To find the speed along the boundary of the larger circle (circle 2), we can use the circulation we calculated earlier.

Considering the flow to be irrotational (except for within circle 1), we can assume a potential flow outside circle 1. In potential flow theory, one of the important relations is the Biot-Savart law, which states the relation between the speed V, circulation Γ, and distance from the vortex center.

V = (Γ / 2πr)

Here, V is the speed along the boundary, Γ is the circulation around the larger circle (which we found out earlier), and r is the radial distance (in this case, r = r2, the radius of the larger circle).

Let's substitute our earlier result for circulation:

Γ = ω(>0) * π * r1^2

Now, calculate the speed V at the boundary of the larger circle:

V = (ω * π * r1^2) / (2π * r2)

The π terms cancel out, leaving the final expression for V:

V = (ω * r1^2) / (2 * r2)

This equation gives you the speed along the boundary of the larger circle (circle 2).

[tranquil pumice]

what would the direction of the velocity vector be at a point along the larger circle