Derivation of the Continuity Equation for Fluid Flow

MEC516/BME516 Chapter 4 Differential Relations for Fluid Flow, Part 2: Derivation of the general continuity equation for three dimensional unsteady incompressible flow. Vector notation and cylindrical coordinates are also discussed. A numerical example is also presented. (Rev: Minor correction, June 2021)

All of the videos in this Introductory Fluid Mechanics course, sample exams (with solutions), and a copy (pdf) of this presentation can be downloaded at:

Course Textbook: F.M. White and H. Xue, Fluid Mechanics, 9th Edition, McGraw-Hill, New York, 2021.

#fluidmechanics #fluiddynamics

[nature_boy]

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[Appleandcalculator]

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!! :)

[surprisemoleele]

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[MechanicalEngineerFromMETU]

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.

[PrasanjayaEkanayake]

Most clear and well explained lecture slides about Fluid Mechanics, Much appreciate professor

[nadeemjan]

Nice Presentation. Thanks Prof @FluidMatters

[fatemehamini]

thank you so much professor. it helped me a lot.

[wyn]

How do we get from the continuity equation in the divergence form to the convective form please Dρ/Dt + ρ ∇ . V = 0. ?

[alibalouch]

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

[jamboreejackson]

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.

[jameyatesmauriat]

Hi, thank you for the lecture. Is there any video lecture on partial differential equation derivation of tidal wave in incompressible fluid?

[ericjung]

hi professor, why are there no videos for content past chapter 5?

[tw]

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.

[tonymunene]

Why are we taking the Taylor expansion??

[surry]

You used Cartesian coordinates not cylindrical coordinates.That is a mistake on the first slide.