Conservation of Momentum in Fluid Flow: The Navier-Stokes Equations

MEC516/BME516 Fluid Mechanics, Chapter 4 Differential Relations for Fluid Flow, Part 4: A brief discussion of the derivation of the Navier-Stokes equations, the differential equations for conservation of momentum in fluid flow. Euler's equation for inviscid flow is also briefly discussed. This video includes a sample problem that involves showing that a given velocity vector field satisfies the Navier-Stokes equations.
All of the videos in this course, sample exams (with solutions), and a copy (pdf) of this presentation can be downloaded at:
www.drdavidnaylor.net
Course Textbook: F.M. White and H. Xue, Fluid Mechanics, 9th Edition, McGraw-Hill, New York, 2021.
#fluidmatters #fluidmechanics #fluiddynamics

[@FluidMatters]

All the videos for this introductory Fluid Mechanics course are now available at: www.drdavidnaylor.net/

[@Ukraine-is-Corrupt]

Excellent lecture. It's obvious you've delivered this countless times

[@luistrejo7249]

Great explanation. It helps me understand easily how are derived Navier Stokes equations

[@alperenokur1272]

All semester fluid dynamic lessons is just 30 minutes in here. Thanks.

[@FluidMatters]

Glad to help!

[@deryakarakaya827]

It was a great lesson!! Thank you very much for your efforts :)

[@brianwachira6886]

G.O.A.T 🔥🔥🔥. It was simple and brief. Thanks

[@MohamedMohamed-mn7rk]

A complicated couse made simple, thanks doc!

[@FluidMatters]

Thanks for the kind words. Glad to hear it was helpful.

[@bruzanHD]

I once heard that any mathematical discovery one makes may as well be credited to themself and Euler since in all likelihood, Euler already mentioned it somewhere.

[@FluidMatters]

Ha. Ha. If only I could think on that level, I would be honored to do so.

[@hrkalita159]

Great 🙏

[@TahsinEngin-Akademi]

Which application and/or tools are you using to create this lessons? I did not able to display on the screen while I am talking about something

[@FluidMatters]

I use CamTasia (by TechSmith) which I can recommend. Comes with an add-in for PowerPoint. It's quick to learn. Reasonably good editing capabilities.

[@TahsinEngin-Akademi]

@@FluidMattersThank you

[@scientium8770]

At 15:05 , during calculation, how did you remove '2' from the expression which will come in the surface force section.
Also, how to calculate del(del v/del x)/del y as it comes in the tau(xy) term...
Please explain these...

[@FluidMatters]

Sorry, this is not intended to be a complete derivation of the Navier-Stokes, as this is a 1st course in undergraduate level fluid mechanics. You can find a more complete derivation in graduate-level textbooks (e.g. Viscous Fluid Flow by White)

[@scientium8770]

@@FluidMatters Thanks for the reply. I got it later; we split up the term with 2 infront of it in two, and take del/delx common, which gives the expression for continuity equation del (dot) u = 0. Putting it there gives the Navier-Stokes Equation.

[@SMD1999]

No kidding what I saw at the end was beautiful.
Are your slides available for non Ryerson students

[@FluidMatters]

The slides are available at the moment at my website www.drdavidnaylor.net

[@9wyn]

Thank you for making the pdf available.🙏🏽

[@hrkalita159]

@@FluidMatters great 😀🙏

[@jaypanchal1997]

​@@FluidMattershow did you write shear stress in x direction =2*myu*(dau u/ dau y)

[@FluidMatters]

@@jaypanchal1997 A full discussion of the viscous stresses is beyond a first course in fluid mechanics. You'd have to consult a more advanced-level book. That's why I say "It can be shown..."

[@comment8767]

Point of Confusion for me - The cube as 6 faces with viscous stresses on each face. Therefore, there are 6 x 3 = 18 stresses on the differential element. Are we somehow saying that the stresses on opposite faces are the same, so that it is really only one? Then, all stresses are referred to a central point? What happens to the extra nine stresses?

[@comment8767]

OK, so you are not claiming that the nine stresses cover the entire differential volume. You are simply positing the stresses on three faces, and then using taylor series to get the stresses on the opposite face. The posited stress plus the Taylor-derived stress provide the stress on the entire cube.

[@badelbadel3361]

I have a exercise can you help me please

[@FluidMatters]

Sorry. I can't help with specific problems.

[@nathans5773]

comment for the algorithm

[@syauqyahmadnails1tm113]

so, full course of Navier Stokes Equation is actually for post graduate?

[@FluidMatters]

It usually is a grad course in mechanical engineering (Advanced Fluid Mechanics).