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

[cruckshank]

"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!

 

[Chestermiller]

Are you assuming that the pipe is horizontal or vertical?

 

[cruckshank]

The pipe is horizontal in the diagram, sorry should have mentioned that.

 

[Chestermiller]

The pipe is horizontal in the diagram, sorry should have mentioned that.

Is x the direction along the pipe axis?

 

[cruckshank]

Yes, along the pipe axis.

 

[Chestermiller]

Yes, along the pipe axis.

What was your rationale for saying that the partial derivative of the pressure with respect to x is zero?

 

[cruckshank]

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.

 

[Chestermiller]

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.

 

[cruckshank]

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

 

[Chestermiller]

Are you currently taking a course in Fluid Mechanics? What textbook are you using? Have you ever heard of the book Transport Phenomena by Bird, Stewart, and Lightfoot? Are you familiar with the following concepts: stress tensor, Newton's law of viscosity?

Stokes Law would not be appropriate for this problem.