Applicable domain of Stokes Flow

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Discussion Overview

The discussion revolves around the applicability of Stokes flow in fluid mechanics, particularly regarding the Reynolds number (Re) and its implications for flow through a rectangular duct. Participants explore the limits of Stokes flow as an approximation and share references for further reading. Additionally, there are tangential inquiries about solving boundary layer flow and heat transfer using numerical methods.

Discussion Character

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

Main Points Raised

  • Michiel questions the upper limit of the Reynolds number for which Stokes flow remains a valid approximation and seeks references for this topic.
  • Matt notes the challenge in determining when one is outside the domain of Stokes flow and suggests that experimental validation would be beneficial.
  • Another participant mentions that Stokes' law begins to fail around Re ~0.5, although they do not provide a reference.
  • Michiel later finds a journal article indicating that for porous structures, a Reynolds number of Repore < 0.8 yields 99% accuracy, which aligns with the previously mentioned value for flow around spheres.
  • Several posts diverge into a separate discussion about solving boundary layer flow and heat transfer using the finite difference method, with participants expressing frustration about the relevance of these posts to the original topic.
  • One participant suggests that a similarity transform method could provide a good approximation for the boundary layer problem, while another questions the necessity of coding the solver by hand versus using existing CFD software.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of Stokes flow, particularly regarding the Reynolds number thresholds. There is no consensus on the exact limits or the accuracy of the approximation. Additionally, the discussion about boundary layer flow and numerical methods remains unresolved, with participants offering various suggestions without reaching an agreement.

Contextual Notes

Some participants reference specific values for the Reynolds number related to Stokes flow, but these values are not universally accepted or verified within the discussion. The transition between topics regarding boundary layer flow and numerical methods introduces additional complexity and may distract from the main focus on Stokes flow.

MichielM
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Hi,
I'm working on a calculation of flow through a rectangular duct and I'm assuming I'm in the Stokes' flow regime (Re<<1), but I also want to experiment on this system and I was wondering if anyone knows until what Re-number Stokes' flow is still a good approximation (and how good, i.e. in percentages error): Re<0.1, Re<0.01 ?!

Does someone maybe have/know a scientific paper or book on this?

Thanks in advance,

Michiel
 
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... what Re-number Stokes' flow is still a good approximation...

That is one of the problems of fluid mechanics, knowing when you are outside of the equations known domain.

Are you able to test your flow field in an experiment to back up the calculations? I am assuming that you are not, since then you probably wouldn't have asked the question.

EDIT: In my copy of "Transport Phenomena" 2nd Edition by Bird, Stewart, and Lightfoot that this book "Microhydrodynamics: Principles and Selected Applications" that there is a thorough discussion of "creeping flow" problems.

The book is fairly inexpensive. See link below.

https://www.amazon.com/dp/0486442195/?tag=pfamazon01-20

END EDIT:

Thanks
Matt
 
Last edited by a moderator:
For flow around a sphere, Stokes' law begins to fail around Re ~0.5 I don't have a decent (inexpensive) reference for this,
 
Hi
I want to solve boundary layer flow and heat transfer using Finite difference method.
Can anyone help me.
 
Thanks for the answers Andy (gives me some clue about order of magnitude) and CFD (potentially interesting book!)

This morning I found a journal article completely devoted to this topic in which criterion is proposed which matches well with a large number of experiments. Turns out that (working with porous structures like I do) Repore<0.8 for an accuracy of 99% or better, which is similar to the 0.5 Andy gave for flow around spheres (to which some porous media are very similar).

For those who are interested:
Comiti et al. / Chem. Eng. Sci. 55 (2000) 3057-3061
 
Hi
I want to solve boundary layer flow and heat transfer using Finite difference method.
Can anyone help me.

Why is that in this thread? This thread has nothing to do with finite difference methods.

Please create a new thread with your question.

Also, provide enough information so that someone can actually help you. Such as, the specifics of your problem, any graphs/drawings of the system. Your question is so vague no one could possibly help you out.

Matt
 
Hi
i want to solve convection boundary layer flow past a vertical plate problem using Finite difference method, that is, i have to solve momentum (NavierStokes) and energy equations. Most of the boundary layer problem is solved by similarity transform method. That is they convert PDE to ODE then they solve ODE by Runge Kutta Shooting method. I would like to solve the PDE directly.
Momentum Eqn
U_t+U U_x + V U_y = U_yy + Gr T
Energy Eqn
T_t+U T_x + V T_y = (1/Pr) T_yy
Boundary Condns
U=0; T=0; at X=0
U -> 0; T -> 0; as Y -> infinity

When i am solving the problem i could not get the correct velocity and temperature profiles as given in many papers. So i need help in this regard, particularly how to handle the infinite boundary condition in the computer code.
Thanks in advance
 
Two things:
1. You're still posting in the wrong place, this topic is on the Applicable domain of Stokes Flow. This can severely limit the amount of response you get because some people just don't want to answer anymore if they see a question in a topic where it doesn't belong.

2. That aside, I have two questions for you:
a) By the looks of your equations you want to solve the boundary layer problem in 2D, how accurate do you want to have your solution? If accuracy is not of vital importance: a very good approximation (less than 1% error) is obtained when you do the similarity transform that you already mentioned and then solve the resulting ODE via an integral method (i.e. analytically but with an approximation).

b) Are you coding the solver for the equations by hand? Because a standard CFD package such as Comsol or Fluent should be able to solve your set of equations easily and in that case the boundary conditions shouldn't be an issue at all.

Final thing: maybe there is a moderator who can shift this thing to a new topic?!
 

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