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enter image description here Let me explain the picture above: We have a uniform flow and a source sheet cover the the surface of an arbitrary closed body. Let s be the distance measured along the source sheet, λ=λ(s) is the source strength per unit length along s. Assume that we already have the λ=λ(s) along the source sheet that when we superimpose the uniform flow with this source sheet, we obtain the streamline over surface of the body. Could we have this formula: $$\oint_{body} \lambda(s).ds = 0$$ ?? If yes, is there any way of thinking to pop out this formula right away before using math to prove it? An idea is that: "the body itself couldn't be adding or absorbing mass from the flow" but I can't relate this state with the formula above.

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  • $\begingroup$ ..a source sheet covering a body sounds like a murder took place. $\endgroup$ – Koyovis Aug 23 '17 at 23:34
  • $\begingroup$ @Koyovis Haha, my mistake, had to put some other words $\endgroup$ – Dat Aug 24 '17 at 3:16
  • $\begingroup$ Is lambda here the flow, ie, a scalar magnitude? If yes I am unsure how that circular integral can be anything but non-zero for any realistically possible values of lambda(s) $\endgroup$ – AEhere Aug 24 '17 at 12:58
  • $\begingroup$ @AEhere $\lambda$ is scalar, it is a parameter of the source sheet and the sheet produces flow. More clearly, the sheet lies on the surface of an imaginary body, but when we combine the unifrom flow and the flow produced by the sheet, we obtain a streamline exactly like the surface of the body. i think you should know how the sheet produces flow and what a source sheet really is . $\endgroup$ – Dat Aug 24 '17 at 16:18
  • $\begingroup$ @AEhere: Values of $\lambda$ have to make sure that when the flow produced by the sheet (this flow is governed by $\lambda$) combines with the uniform flow, we obtain streamline along the surface of the body. $\endgroup$ – Dat Aug 24 '17 at 16:32
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If a streamline is closed, then there's no mass flow that can go throught it. That means that inside the streamline the net mass flow produced is null. You can have for example sinks and sources inside a closed streamline, but they must compensate each other to make the streamline remain closed.

Another important aspect about source sheets is that they actually don't match with the border of the body, i.e, with the closed streamline. If they did, then some flow produced by the sources would go out of the source sheet, and some other flow would be in. The source sheet needs to be inside the border of the body. Anyway you could find a source distribution that makes the source sheet to almost match the border of the body.

What I wanted to say, is that if the streamline is closed, then the sum of sources and sinks inside the body is null. If the sinks and sources are the source sheet itself, then this proposition is equivalent to:

$$\oint_{Source\hspace{1pt}sheet} \lambda(s)ds=0\hspace{40pt}(1)$$

Note that each infinitesimal element of sheet, $ds$, produces an infinitesimal source strength of $\lambda(s)ds$ in the $s$ position. The relation $(1)$ means that the sum of all infinitesimal source strengths along the source sheet is null, or, in other words, that the net source strength is null.

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