**Chapter 4 Videos: Differential Equations (Continuity and the Navier-Stokes Equations)**

**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 |

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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.

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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.

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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.

YouTube Video

Chapter 4 Part 6 (pdf) | |

File Size: | 513 kb |

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**An Additional Solved Navier-Stokes Problem **

**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 |

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**Toronto Metropolitan University**