Nondimensionalizing the Navier-Stokes Equation

In this video, I convert the Navier-Stokes equation for an incompressible, Newtonian fluid to a dimensionless form. The Reynold's number comes up as one of the parameter groups that results from the nondimensionalization.
Any questions/comments? Leave them below and I'll get back to you!
Prereqs: Previous nondimensionalization video. Even though Navier-Stokes is mentioned, you don't need to know much Fluid Mechanics. It helps to know about Reynold's number, but the lesson is basically a giant exercise in algebra so not knowing some of the dimensionless parameters won't screw you over. Some really basic knowledge of partial derivatives also helps.
Lecture Notes: drive.google.c...
Patreon Link: www.patreon.co...

[@justinwilliamswi]

Absolutely great explanations. I have background in fluids and the method they taught nondimensoinalization with was very non-intuitive (White's book). This makes complete sense and I have now applied it to some solid mechanics functionals with ease.

[@alfazero1423]

Thx for the feedback. I have a request. Can you please derive the Navier stokes eqn. entirely from beginning till end, explaining each step. for example I have seen many videos of derivation of NS eqn. But when it comes to the shear stresses and substantial derivation of acceleration they just put this eqns without explaining what is it why we should write in that way. what is the Stokes hypothesis. why we should use them. neglecting the explanation of some parts of the eqn. regarding second viscosity, strain rate tensor. and so on.

[@FacultyofKhan]

Sure! I'll be glad to get to it eventually. That's going to be part of my Fluid Mechanics series that I'm planning to start in the winter break or so. Right now I have a couple of requests for my ODE videos already and some other videos that I'm planning to do myself, but I will definitely get to it!

[@DanielSerranowD]

Hey i don't know if you're still checking your comment section, i really want to congratulate you. You really know how to explain in such a simple way something that might not be as obvious at first glance; Anyway i've been in the hunt for books that can further explain the scaling process for constitutive equations, could you recomend any? cheers!

[@foxmccloud9548]

scrub? how dare ye!

[@FacultyofKhan]

My sincerest apologies, Mr. McCloud. If it's any consolation, I'll have you know that I always chose you over your buddy Falco when playing Smash Bros (yes, it took me 2 months to come up with that response).

[@archiebrew8184]

@Prince Douglas It's pretty obvious that no one will give a shit, please go to somewhere like Reddit where people of your intellectual capacity can exist in an ignorant bliss.

[@adampax]

Finally an explanation I can follow! Thank-you :D

[@mohamedibr752]

Thanks for the video. In Frank White's Viscous Fluid Flow, the dimensionless equation does not contain gravity term, why it was neglected ?

[@FacultyofKhan]

Hi Ibrahim, Thank you for the feedback! As for Frank White's book, I'm not entirely familiar with it and I don't know if you're referring to what I read just now (2nd edition page 85-86), but here's what I believe is going on: 1) When he starts nondimensionalizing Navier-Stokes (the momentum equation which I've done here), he considers 'high-speed gas flow', in which gravity is negligible, so he neglects the gravity term. Fast-moving gases don't have their flow profiles influenced much by gravity, which is why it's neglected. 2) For 'low-speed flow', gravity isn't neglected, but is made part of another dimensionless number called the Grashof number. That's just another dimensionless number that contains a thermal expansivity coefficient as well. Hope that helps! If you have any more questions, feel free to ask!

[@jaychou4579]

Hello, thanks for your video. I have a question. If I use dimensionless ns-equation for CFD, I need to simulate air or water, I only need to change the viscosity to make the fluid water or air? Because the density disappeared in the Pressure term when the equation is dimensionless.

[@nashimchowdhury4419]

Thanks for this, just one question - after you divide by Re, what happens to the constants at the first term of the LHS (rho*Ls^2) / (Mu*ts) ? shouldnt there be Ls/us*ts left over ?

[@nashimchowdhury4419]

ok im stupid, i just realised u set ts=Ls/us, thanks anyhow, best explanation ive come across so far

[@FacultyofKhan]

Glad I could help!

[@carbon273]

in my textbook the Froude number is U^2/gl. So it should be 1/Fr towards the end. Or did i miss something?

[@yarooborkowski5999]

Great. Could You also derive from zero the Navier-Sotkes equation in Your beautiful way, please? Best regards

[@faezeg1579]

Thanks for the good explanation Can you help me change the Navier-Stokes equation if it is a non-Newtonian fluid?

[@rakesh_velpur]

This is always a headache for me when dealing with Microfluid dynamics

[@captainkielbasa5471]

Karniadakis Microfluidics textbook is the way to go

[@b.hemalathapremkumar3663]

Excellent explaination....

[@mehdibadami5942]

I know with considering the N-S for compressible Fluid in non-dimensional form , Mach number and gamma=Cp/CT appears but I dont know how ,can someone help me please?

[@FacultyofKhan]

I'm not sure which compressible N-S equation you're referring to. Can you post a link?

[@charithjeewantha]

Thank you soooo much!!!

[@alfazero1423]

great video, thx. Q: what's meaning (point) of scaling parameters (ts, us, etc.) ?

[@FacultyofKhan]

Hi Ilyas, Thank you for the kind feedback! The scaling parameters are just there to get rid of the dimensions and make the quantities dimensionless. For instance, 'ts' is the scaling parameter for time, and so it divides out the time dimension from the dimensional time to make it a dimensionLESS time. Professor Khan

[@nishanthsp1517]

Hi Prof, what is the use though? You can still solve a problem using the normal Stokes' equations no?

[@FacultyofKhan]

You can, but as I mention in this video: kzbin.info/www/bejne/fJKVaZycdpx7b9E Nondimensionalization helps simplify your equations and reduce the number of parameters. This makes the system easier to analyze and it also makes your writing/notation less complicated.

[@mauriciomani9284]

Great video! Thanks!!!

[@readyforeveryone5654]

Sir please solve navier stokes equation for square cylinder in non dimension form please sir

[@erickrobles6365]

why is density not non-dimensionalized???

[@FacultyofKhan]

Density is just a parameter; not a variable that's being differentiated (e.g. v) or being differentiated with respect to (x,y,z,t). So there's no point in non-dimensionalizing it, since it's supposed to be part of the dimensionless constants (e.g. Reynold's number).

[@erickrobles6365]

Wow that was fast! My test is in 3 hours. Thank you so much.

[@FacultyofKhan]

No problem, glad I could help!

[@abud3775]

why isn't Tilda there in Laplace form (∇2) but do have Tilda in the single derivative (∇) form?

[@LosVideosDeLouis]

Why yes.....yes I am a scrub 😥

[@tulgatbolderdene7493]

Scrub :')