(adsbygoogle = window.adsbygoogle || []).push({}); Fluid Mechanics: Point like "sink"

I am badly badly confused by the following problem and its solution. I hope that someone would be kind enough to help me out! Thank you for your help and your time!

1)The plunger of a hypodermic needle (a point-like "sink") is being pulled up with speed v_p as shown in the first link/picture. The fluid at the needle is v_n given by the equation of continuity. What is v(r), the speed field, in the surrounding fluid? (Hint: draw a sphere of radius r centered at the tip of the needle)

http://www.geocities.com/asdfasdf23135/phy002.JPG

The model solution is:

http://www.geocities.com/asdfasdf23135/phy003.JPG

NOTE: Q is the volume flow rate.

And I am completely lost after raeding this solution. I don't understand a single step from the beginning to the end.

(i) What is the REASON of drawing a sphere around the tip of the needle?

(ii) Why is it true that dQ=v(r)dA? What I have learnt is that Q=vA, not dQ=v(r)da.

(iii) What is dA?

(iv) Why did they take the indefinite integral to both sides?

(v) How come v(r) is brought OUT of the integral sign? v(r) should be varying and is NOT (vi) a constant, right?

(vii) How actually did they get the 4pi(r^2) out of nowhere?

(viii) Is the origin r=0 set to be at the centre of the sphere (i.e. the tip of the needle)? What does v(1) represent, for example?

(ix) v_p and v_n are not used in any part of the model solution. Does it mean that the speed fluid given by v(r)=Q/(4*pi*r^2) would exist even if the plunger is stationary (i.e. not pulled)? How is this possible? Shouldn't the surrounding fluid be STATIC?

(x) One more question: when it says that the plunger is being pulled up at speed v_p, but this is not the speed of the FLUID. Then how can it be related to the equation of continuity? Is it true that the fluid below the plunger but above the top of the needle is ALSO moving up at speed v_p? If so, why?

I apologize for the amount of questions being asked, but any help to any part of the problem is greatly appreciated!

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# Fluid Mechanics: Point like sink

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