Streamlines around a sphere (qualitative)

[member 428835]

Hi PF!

Attached is an figure from Bird Stuart and Lightfoot. I'm wondering if anyone can comment about the difference in spacing of these streamlines assuming each streamfunction is evenly spaced (or does this requirement make this picture invalid)?

My interpretation assuming each streamfunction sketched is evenly spaced, meaning $\psi = 1,2,3...$: the difference between any two streamfunctions is volumetric flow rate. Then streamlines closer together would imply faster flow, so that the flow is the fastest in the far field before and after the solid body in figures a, b, that flow is fastest before the solid body in figures c and d, and that flow is fastest close to the solid body in figure e. Assuming the streamfunctions are evenly spaced, is this interpretation true?

 

[Chestermiller]

Your interpretation is correct, but it seems to me that the figures are only schematic, and don't represent plots of the actual streamlines.

 

[member 428835]

Thanks, I was thinking the same!

 

[boneh3ad]

I'd also like to point out that the distance in physical space between steamlines is meaningless unless the inflow is constant. It's the distance in $\psi$ space that matters. Two pairs of streamlines can both originate $\Delta y$ apart from each other, but they still may not have the same $Q$ between them.

 

[member 428835]

Totally! This is an important point boneh3ad.

 

[Chestermiller]

I'd also like to point out that the distance in physical space between steamlines is meaningless unless the inflow is constant. It's the distance in $\psi$ space that matters. Two pairs of streamlines can both originate $\Delta y$ apart from each other, but they still may not have the same $Q$ between them.

In this system, far upstream and far downstream, equal delta y identify equal Qs.

 

[boneh3ad]

In this system, far upstream and far downstream, equal delta y identify equal Qs.

Sure, but that's not a general rule.