Force Analysis in Circular Pipe: How to Approach Vertical Force Balance?

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cruckshank
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"Consider a fully developed laminar flow in a circular pipe, perform force analysis on an element of the real fluid."

I've just started this type of question, and I'm a bit confused about where to go from here:

I've managed to do a force balance horizontally to show that the partial derivative of pressure with respect to x should be zero (which is obvious anyway?), but I have no idea what to do with the vertical force balance and what the question is really asking me to do.

Thanks in advance!
 
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The pipe is horizontal in the diagram, sorry should have mentioned that.
 
Yes, along the pipe axis.
 
I did a force balance on the horizontal and it came out with that, which I believe is correct?

I have no idea about the vertical force balance though.
 
cruckshank said:
I did a force balance on the horizontal and it came out with that, which I believe is correct?
It is not correct. How are you supposed to be doing this: (a) using shell momentum balances or (b) using the Navier Stokes equations?

What forces are acting horizontally on the fluid, besides the pressures.
 
I haven't heard of either of these methods, and upon looking them up they don't look familiar to me either.

I forgot about the viscous force acting on the fluid element I think. Would I represent this using Stoke's Law or some other way? If using Stoke's law would the radius be the pipe radius or the fluid element's radius?

Thanks