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

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Discussion Overview

The discussion revolves around performing a force analysis on a fluid element within a fully developed laminar flow in a horizontal circular pipe. Participants explore the vertical and horizontal force balances, particularly focusing on the implications of pressure gradients and viscous forces.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Homework-related

Main Points Raised

  • One participant expresses confusion about how to approach the vertical force balance after performing a horizontal force balance that suggested the partial derivative of pressure with respect to x is zero.
  • Another participant questions whether the pipe is assumed to be horizontal or vertical, clarifying that it is horizontal.
  • There is a discussion about the rationale behind the zero pressure gradient in the x-direction, with one participant asserting their belief in its correctness.
  • A later reply challenges the correctness of the initial claim regarding the pressure gradient and suggests considering either shell momentum balances or the Navier-Stokes equations for a more accurate analysis.
  • One participant acknowledges the omission of viscous forces and inquires about the appropriate representation, questioning whether to use Stokes' Law and which radius to consider.
  • Another participant asks about the familiarity with fluid mechanics concepts and textbooks, indicating that Stokes' Law may not be suitable for this problem.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correctness of the initial pressure gradient claim, and there are competing views on the appropriate methods for analysis, indicating that the discussion remains unresolved.

Contextual Notes

Participants reference various methods and concepts in fluid mechanics, but there is uncertainty regarding the application of these methods and the definitions involved, particularly concerning the use of Stokes' Law and the appropriate forces acting on the fluid element.

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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Are you assuming that the pipe is horizontal or vertical?
 
The pipe is horizontal in the diagram, sorry should have mentioned that.
 
cruckshank said:
The pipe is horizontal in the diagram, sorry should have mentioned that.
Is x the direction along the pipe axis?
 
Yes, along the pipe axis.
 
cruckshank said:
Yes, along the pipe axis.
What was your rationale for saying that the partial derivative of the pressure with respect to x is zero?
 
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
 
  • #10
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.
 

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