ONLINE FLUID MECHANICS, Professor D. Naylor
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Chapter 4 Videos: Differential Equations (Continuity and the Navier-Stokes Equations)


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Part 1: Differential Relations for Fluid Flow (20:23)
Introduction to Chapter 4. The Continuity and Navier-Stokes equations are presented and briefly discussed. Exact solutions and the CFD applications of these equations are briefly discussed. 
YouTube Video

Chapter 4 Part 1 (pdf)
File Size: 754 kb
File Type: pdf
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Part 2: Differential Relations for Fluid Flow (18:35) 
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.
YouTube Video

Chapter 4 Part 2 (pdf)
File Size: 1295 kb
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Part 3: Differential Relations for Fluid Flow (27:41)
Derivation of the fluid acceleration field. The difference between local acceleration and convective acceleration is discussed in some detail. Two mathematical example problems are also presented.
YouTube Video

Chapter 4 Part 3 (pdf)
File Size: 1244 kb
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Part 4: Differential Relations for Fluid Flow (31:49)
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. A sample problem that involves showing that a given velocity vector field satisfies the Navier-Stokes equations. 
YouTube Video

Chapter 4 Part 4 (pdf)
File Size: 2125 kb
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Part 5: Differential Relations for Fluid Flow (21:12)
Two exact solutions to the incompressible continuity and Navier-Stokes equations. One solution is for laminar steady flow between fixed parallel plates, where the flow is driven by a pressure gradient (Poiseuille Flow). The other solution is for laminar steady flow between parallel plates where the flow is driven only by the motion of the upper plate (Couette Flow).
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Chapter 4 Part 5 (pdf)
File Size: 1271 kb
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Part 6: Differential Relations for Fluid Flow (21:28) 
The exact solution of the Navier-Stokes and Continuity equations for fully developed laminar flow in a round pipe (Hagen-Poiseuille Flow). The video concludes with a numerical example.
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Chapter 4 Part 6 (pdf)
File Size: 513 kb
File Type: pdf
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Intro to Boundary Layer Flow (pdf)
File Size: 3416 kb
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Part 7:  Introduction to Boundary Layer Flow (12:44) 
A brief discussion of the viscous boundary layer flow. The presentation starts with a discussion of the historical aspects of  Prandtl's idea. Transition from laminar to turbulent flow and the critical Reynolds number are described. Experimental and CFD flow visualization of a boundary layer on a flat plate are also presented.
YouTube Video


An Additional Solved Navier-Stokes Problem 

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Solved Exam Problem: Navier-Stokes Solution (12:39)
An additional solved problem for study purposes. A final exam question (Fall 2022) on solving the Navier-Stokes equations. The velocity and pressure fields are calculated for a gravity-driven liquid film on an inclined plate. The no-shear stress boundary condition at the top of the liquid film is also discussed.  
YouTube Video

Navier-Stokes Solution. Final Exam Question (pdf)
File Size: 1224 kb
File Type: pdf
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