What an absolutely fantastic lecture series! Im taking my first course in advanced fluid mechanics next semester and this really helps with my preperations, thank you!! :)
Dear professor, thank you very much for making such hard concepts so easy to understand in chunks and bits. This service of you make the world a more equal place, it is so valuable. By the way this is the part 2 of this chapter but in the playlist the order is mixed, just saying for your information.
Thanks for the kind words, and for letting me know the playlist issue. For anyone reading this, my website is the best place to get the most up-to-date version in the correct order: www.drdavidnaylor.net/
Hi professor, many thanks for the excellent lectures. I tried the derivation of the equation in cylindrical co-ordinates and I couldn't quite get there so ended up looking at another reference. Should the dA in the outlet term you show at 14:30 actually be (r+dr)dΘdz (rather than rdΘdz) to account for the slightly larger area at the outer radius? This is what the derivation I saw suggested.
Sorry for the slow reply. I just tried this derivation. I think you are getting hung up by second order effects, which always go to zero. Remember that this equation applies at a point, so r+dr --> r as you shrink to the differential volume to be infinitesimally small.
We seek to calculate the inlet mass flow minus the outlet mass flow. Just look at the differential cube in the top right side of the slide: We have rho*u*dx*dz flowing in and rho*u*dx*dz flowing out. So, they cancel out when calculating inlet mass flow minus outlet mass flow. I hope that helps.
I wouldn't say that the rest of the analysis is easy (at 14:34). Firstly, because the term (r+dr)[dθdz] is missing in the formulation of the mass outflow on the infinitesimal area; if you follow this video only you can't arrive at the correct answer. Secondly, when you isolate the inflows and outflows across this face, you have to use the curious mathematical fact that dr^2 = 0 to get rid of one of the terms. Then, you have to see the odd step of multiplying some terms by r/r so you can factor a 1/r term out. Then, after more simplification, you're left with a term like (1/r)[ρVr+∂(ρVr)/∂r], and you have to recognize that the term in brackets is the same as ∂(rρVr)/∂r by virtue of the product rule of derivatives.
Sorry for the slow reply. I just tried this derivation. I think you are getting hung up by second order effects, which always go to zero. Remember that this equation applies at a point, so r+dr --> r as you shrink to the differential volume to be infinitesimally small. (So, it's not "curious" that second order terms like dr^2 equal zero.)
The change across the faces of the differential volume (at 2:59) only considers the first order term. I am pointing out that the true variation is not linear --- so there will be higher order terms (as described by the infinite Taylor expansion). So you might wonder: Is a first order approximation sufficiently accurate? The point I am trying to make is that, as Delta_x --> 0 and becomes dx, these higher order terms vanish more quickly than the linear term, and hence, are not needed. This is a standard feature of these kinds derivations in fluid mechanics, heat transfer, solid mechanics, etc.
@@FluidMatters thankyou for your reply and the video is a great explanation. Final question ... Since this is an isolated system what is causing the change in the flow rate?
@@tonymunene1592 It's not an "isolated system" in the thermodynamic sense of that term. It is just a control volume, with mass flow in and out. So, consider this to be a arbitrary control volume taken anywhere in any real flow, such as delivered by a fan or pump. A real flow will have have local velocity variations for various reasons, e.g. due to changes in the cross sectional area of a pipe.
All the videos (and pdf downloads) for this introductory Fluid Mechanics course are available at: www.drdavidnaylor.net/
This is the best explanation over the internet which i was searching for weeks.
Thanks. Glad to hear it was helpful.
What an absolutely fantastic lecture series! Im taking my first course in advanced fluid mechanics next semester and this really helps with my preperations, thank you!! :)
Thanks for the kind words. Best of luck with your studies!
I hope your course went well! I am in the same position for this January, and I totally agree these videos are a huge help!!!
This channel is helpful for my research project.
Glad it helps. What's the project?
Dear professor, thank you very much for making such hard concepts so easy to understand in chunks and bits. This service of you make the world a more equal place, it is so valuable.
By the way this is the part 2 of this chapter but in the playlist the order is mixed, just saying for your information.
Thanks for the kind words, and for letting me know the playlist issue. For anyone reading this, my website is the best place to get the most up-to-date version in the correct order: www.drdavidnaylor.net/
Most clear and well explained lecture slides about Fluid Mechanics , Much appreciate professor
Glad to hear you are finding the videos helpful!
Nice Presentation. Thanks Prof @FluidMatters
Thanks. Best of luck with your studies.
Hi professor, many thanks for the excellent lectures. I tried the derivation of the equation in cylindrical co-ordinates and I couldn't quite get there so ended up looking at another reference. Should the dA in the outlet term you show at 14:30 actually be (r+dr)dΘdz (rather than rdΘdz) to account for the slightly larger area at the outer radius? This is what the derivation I saw suggested.
Sorry for the slow reply. I just tried this derivation. I think you are getting hung up by second order effects, which always go to zero. Remember that this equation applies at a point, so r+dr --> r as you shrink to the differential volume to be infinitesimally small.
im so glad this comment exists! thank you both
thank you so much professor. it helped me a lot.
Glad it helped!
hello Professor how did you cancel the inlet and outlet (Rho u dy dz ) with the outlet Rho u
time of the video @7.36
We seek to calculate the inlet mass flow minus the outlet mass flow. Just look at the differential cube in the top right side of the slide: We have rho*u*dx*dz flowing in and rho*u*dx*dz flowing out. So, they cancel out when calculating inlet mass flow minus outlet mass flow. I hope that helps.
How do we get from the continuity equation in the divergence form to the convective form please Dρ/Dt + ρ ∇ . V = 0. ?
Sorry, that's not really something I can answer here in the comments.
I wouldn't say that the rest of the analysis is easy (at 14:34). Firstly, because the term (r+dr)[dθdz] is missing in the formulation of the mass outflow on the infinitesimal area; if you follow this video only you can't arrive at the correct answer. Secondly, when you isolate the inflows and outflows across this face, you have to use the curious mathematical fact that dr^2 = 0 to get rid of one of the terms. Then, you have to see the odd step of multiplying some terms by r/r so you can factor a 1/r term out. Then, after more simplification, you're left with a term like (1/r)[ρVr+∂(ρVr)/∂r], and you have to recognize that the term in brackets is the same as ∂(rρVr)/∂r by virtue of the product rule of derivatives.
Sorry for the slow reply. I just tried this derivation. I think you are getting hung up by second order effects, which always go to zero. Remember that this equation applies at a point, so r+dr --> r as you shrink to the differential volume to be infinitesimally small. (So, it's not "curious" that second order terms like dr^2 equal zero.)
Hi, thank you for the lecture. Is there any video lecture on partial differential equation derivation of tidal wave in incompressible fluid?
Sorry. That is not a topic in engineering fluid mechanics, which is the focus of this channel.
hi professor, why are there no videos for content past chapter 5?
That's where the intro fluid mechanics course ends at my university. I don't teach "Fluids II", at least for now.
Excellent!
Thanks. Glad it was helpful.
Why are we taking the Taylor expansion??
The change across the faces of the differential volume (at 2:59) only considers the first order term. I am pointing out that the true variation is not linear --- so there will be higher order terms (as described by the infinite Taylor expansion). So you might wonder: Is a first order approximation sufficiently accurate? The point I am trying to make is that, as Delta_x --> 0 and becomes dx, these higher order terms vanish more quickly than the linear term, and hence, are not needed. This is a standard feature of these kinds derivations in fluid mechanics, heat transfer, solid mechanics, etc.
@@FluidMatters thankyou for your reply and the video is a great explanation.
Final question ...
Since this is an isolated system what is causing the change in the flow rate?
@@tonymunene1592 It's not an "isolated system" in the thermodynamic sense of that term. It is just a control volume, with mass flow in and out. So, consider this to be a arbitrary control volume taken anywhere in any real flow, such as delivered by a fan or pump. A real flow will have have local velocity variations for various reasons, e.g. due to changes in the cross sectional area of a pipe.
@@FluidMatters thankyou for taking your time to answer, much appreciated!
I like this series
great lecture
thank you so much sir.
Glad to hear it was helpful. Good luck with your studies.
You used Cartesian coordinates not cylindrical coordinates.That is a mistake on the first slide.
This is definitely NOT a mistake. This is an outline of the presentation. I cover the derivation in cylindrical coordinates at about 12:34
@@FluidMatters Ok thanks
amazing lecture