Module 9: Differential Analysis of Fluid Flow: Conservation of Momentum

In module 6, we covered that the conservation of momentum can be derived from Newton's second law that states that the time rate of change of the linear momentum of the system = Sum of external forces acting on the system. In module 9, we express this equation in the differential form, called the Euler's equations. We also introduce the alternative to Euler's equation for irrotational flow, Bernoulli's equation, which can be applied between any two points in the flow.

Student Learning Outcomes:

After completing this module, you should be able to:

* Apply the Euler's and Bernoulli's (where applicable) equations to obtain pressure gradient and pressure differences between various points in the flow

Lecture Videos:

Link to Module 9 Playlist     Link to Module 9 Lecture Notes

Links to Individual Module 9 Videos:

Lecture 1 - Derivation and Discussion of Conservation of Momentum in Differential Form: In this segment, we show step by step instructions on how to derive the conservation of momentum formula in the differential form. This equation is very important in that it is the basis of commonly used Euler and Naiver-Stokes Equations.

Lecture 2 - Euler's Equations of Motion: In this segment, we demonstrate how to obtain the Euler’s equations of motions from Conservation of Momentum equations we derived in the previous session

Lecture 3 - Alternative to Euler's Equations: Bernoulli's Equation: In this segment, we discuss Bernoulli’s equation as an alternative to Euler’s equations. We discuss the conditions that need to be satisfied as well as layout a three-option path for solving questions

Lecture 4 - Euler's and Bernoulli's Equation - An Example: In this segment, we go over a comprehensive example demonstrating when to use Bernoulli’s equation and when to use Euler’s equation

Lecture 5 - Comprehensive Example of Continuity, Velocity Potential, and Euler's equations: In this segment, we solve a comprehensive example of continuity, streamfunction, velocity potential, and Euler's equations. Errata at timestamp: 5:40, -4x should be multiple by the density. This error does not change the final answer.

Lecture 6 - Comprehensive Example of Velocity Potential, Euler's and Bernoulli's Equations: In this segment, we solve a comprehensive question, where the velocity potential is known. We check whether conservation of mass is satisfied or not, and find pressure difference between two points in the flow field

Lecture 7 - Module 9 Recap

               Congratulations, you just finished module 9! Please proceed to module 10