**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.

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).

YouTube Video

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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**Optional Chapter 4 Videos**

**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. This unique element of this problem is the no-shear stress boundary condition at the top of the liquid film.

YouTube Video

Navier-Stokes Solution. Final Exam Question (pdf) | |

File Size: | 1224 kb |

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**Toronto Metropolitan University**