Applying the Navier-Stokes Equations, part 2 - Lecture 4.7 - Chemical Engineering Fluid Mechanics

Simplifying conservation of mass and momentum for analysis of flow through a pipe.
[NOTE: Closed captioning is not yet available for this video. Check back soon for updates.]
This video is part of a series of screencast lectures presenting content from an undergraduate-level fluid mechanics course in the Artie McFerrin Department of Chemical Engineering at Texas A&M University (College Station, TX, USA).
The screencasts have been prepared in 720p HD with accompanying closed captioning for enhanced accessibility.
My inspiration for producing this series of videos has been my lifelong personal journey to understand fluid mechanics and explain its beauty to others in a straightforward way. I have received no external support for this project...the effort is purely a labor of love.
I would like to acknowledge Aashish Priye and Jamison Chang for assistance in developing the materials and preparing the captioning.
Please feel free to share any comments or suggestions.
Best wishes,
Victor Ugaz

[@theneonpython]

>an 11 minute long video on youtube is more valuable than 4 hours of lecture at my uni
really makes you think...

[@giseleroscoe3401]

Thank you! You explain it like you are genuinely amazed that all this stuff works out nicely and you want to let others in on the secret. Appreciated.

[@altairfilipinijunior6073]

I'm an engineering student from Brazil. I'm grateful for the explanations.

[@estefaniarnico]

Thank you a lot I really needed some slowly explained example. Cheers!

[@silvachifla]

u dumb?

[@estefaniarnico]

Yes ;)

[@silvachifla]

me too :'(

[@myzas2992]

Thanks! Much appreciated for explaining this. Keep up the good work

[@MrDrumz98]

You are my savior.

[@ultraslashism]

amazing video, explained everything. I was confused by what terms need to be cancelled out for a while

[@BeleteAssefa]

it is nice everyone should look it &understand the real world

[@JamesVestal-dz5qm]

Motion towards taking another hit

[@dsbupadrasta2385]

Thank you. This video was really helpful

[@STWNoman]

Awesome sir

[@alias40anon]

Very complicated problem for me but the uploader tamed it i must admit

[@JK-sy4ym]

wonderfully explained, thank you.

[@Wisconsin_Cheese]

Why in a pressure driven flow are we able to neglect gravity? Wouldn't gravity slightly change the magnitudes of the vectors in z (smaller for the bottom half and larger for the top half) due to gravity pulling particles down onto a surface we assumed to have 0 velocity? Or are we assuming that this problem is in a 0 gravity environment so the orientation of the pipe makes no difference. It seems like gravity would almost create a shift of the Vz(r) that was calculated

[@click.shot.record8661]

If the pipe is held horizontal ,then gravity can be neglected . And if its vertical or diagonal , gravity would contribute in it. As you see there is a "-" before pressure gradient,if the pipe is horizontal, the gravitational force will be only in y direction which would be zero ,by equating the pressure gradient with gravitational force which are opposite to each other. Just the terms should be converted x,y,z from r,Q(theta),z to get this

[@dannychoo1985]

Hello Victor- Wouldn't it be more appropriate to say that d(vz)/dz = 0 is an outcome of continuity equation (at 3.4 mins), rather than saying that we know in advance v does not change with z?
Then this outcome of the continuity equation i.e. d(vz)/dz =0, is adopted later in momentum equation at 8.43 mins and 9.33 mins for d2(vz)/dz2.

[@JamesVestal-dz5qm]

All things are destiny!

[@sophiamento3790]

stay at home mom stumbled upon this & fascinated. stuff just jumps out at me & it seems obvious. I never took calc & please, please don't think i'm here to talk trash. i have a formula but don't know how to write it out. lol Just have to say , thrilled i found this & curious to see if my entire "formula" continues to be correct. now i know why teachers want you to show your work, lol. Also I cant help but notice & seems obvious ...a lot of Nothing-from Nothing Equals Nothing. I assume all the incorrect formulas are for the process of elimination.

[@ximenaortiz9723]

You are my hero😭😭😭 gracias :D

[@sardorerjigitov8479]

Thanks for explaining. It is very interesting and challenging for me.
I have question. In applying the conservation laws how we got 1/r before the partial derivatives? Sorry if it is too easy

[@thefish1711]

This is actually quite a difficult question, and it relates to the derivation from first principles around circular or cylindrical coordinates. A fundamental level physics/engineering book (maybe Maxwells) will contain the solution from the initial volume equations in the cylinder. An easier way to think of it is in terms of units; as the theta term is dimensionless, the r variable adds the required dimensional element to the term. Check the units on every term to see how it balances out.

[@sardorerjigitov8479]

@@thefish1711 Thank you very much for the precise explanation!

[@lukepapaspyrou1191]

What are Vr and Vtheta? Is Vtheta angular velocity? What direction are they going in?

[@Abbas_alfayli]

Thanks alot ❤️

[@bernardkrasnisky747]

In the Navier-Stokes how do you derive the cylindrical coordinate form for the laplacian * the velocity vector?

[@Matanoify]

awesome

[@deyvisonsa2729]

How is it possible for p to vary with z, if v_z doesn't. I mean, you eliminated ∂v_z/∂z, but kept ∂p/∂z. It doesn't make sense to me (I'm starting to believe that this pressure is not the same pressure that we use in other equations, such as Bernoulli's equation) D;

[@ericg6772]

Thank you!

[@keturahbethel6966]

Great video. You actually make this exciting. I have a question though; how would this change if this was a gravity driven flow, and not pressure? I simplified it to be rho*g= 0. However, wouldn't that make my entire equation go to zero?

[@JamesVestal-dz5qm]

Remember in high school when I said weed never again?

[@adam12many]

Gravity in the r direction shouldn't be neglected according to my textbook.

[@benjaminsmus8553]

At this point, one would hope that the Navier-Stokes equations would be open source

[@NadeemAhmed-9950]

When velocity varies in r-direction then how we have v subscript r=0????

[@deyvisonsa2729]

What varies in the r-direction is the z-component of the velocity (v subscript r, in turn, remains at 0)

[@KuldeepSingh-kr4fn]

I am on last stage I solved navier Stokes equation

[@zee9000_]

i dont get vz(r) ? how r is another direction and z is another direction?

[@tirannnus]

Imagine having a pipe of water where the velocity on the direction Z varies depending on the position in the R axis.

[@DadotheGK]

Mark Yeri sent me this

[@dr.satyanarayanareddys6564]

any reference book ?

[@JamesVestal-dz5qm]

Some day later in life sam revolinski and I will smoke weed together again.

[@JamesVestal-dz5qm]

My Indian neighbor in kansas supported donald trumps manufacture in America first and impose high tariffs and taxes on wine imports policy.

[@JamesVestal-dz5qm]

Donald trump was a modern day tea party member.