Course Notes

• Syllabus
• Course Notes
• Exams
• Homework
• References
• Schedule
• Learning Suite
• Competencies

These are supplementary notes, examples, derivations, links and handouts related to the course. To help you have an improved in-class learning experience, I do not provide the full lecture notes here. Also, for copyright reasons, I cannot publish all of the handouts from class on the web.

General Resources

• National Committee for Fluid Mechanics Films. These are awesome videos made in the 1960s for teaching fluid mechanics. We watched part of the one on “Turbulence” in class.
• Concept Map. This is an overview of all of the concepts in the class and how they are connected.

Part I: Phenomenology and Dimensional Analysis

Lectures 2–3: Math and Python Review

• Dr. Tree’s Ch En 263 (Computer Tools) Course Notes
• Dr. Lignell’s Ch En 263 (Computer Tools) Course Notes
• Dr. Hedengren’s Ch En 263 (Computer Tools) Course Notes
• Kahn Academy Lectures on Linear Algebra
• Kahn Academy Lectures on Multivariable Calculus
• Fantastic Animated Videos on Linear Algebra, Multivariable Calculus, and Differential Equations by 3Blue1Brown on YouTube
• Lecture 2 - Math and Python Review I. See the video below for a recorded lecture reviewing Newton’s method, scalars, vectors, tensors and various vector products.

• Lecture 3 - Math and Python Review II (supplement). See the video below for a supplement to the lecture including an example of how to perform a double integral and a reminder about some identities such as the Gauss Divergence Theorem.

Lectures 4–6: Fluid Properties and Dimensional Analysis

• Coursera Lecture on Surface Tension. This is a nice introduction to the concept of surface tension by Professor Philip Roberts of Georgia Tech. This is part of a broader set of Coursera lectures that review the material on the FE exam.
• Lecture 6 - Dimensional Analysis. See the video below for an example of making an equation dimensionless.

• Supplemental Notes on the Buckingham Pi Theorem. Here are my notes on (i) the integration of a 2nd order ODE for our in-class example for the lecture on dimensional analysis, and (ii) a proof of the Buckingham Pi theorem.

Lectures 7–9: Pipe Flow

• Lecture 7 – Pipe Flow I, Pressure Drop Example. This clip contains an example of how to calculate the pressure drop of a fluid flowing through a steady pipe using the Reynods number and the friction factor. Note that the link contains a pdf of these notes.

• Video on Laminar and Turbulent Pipe Flow. We watched this clip in class when we introduced the concepts of Laminar and Turbulent Flow.

• Lecture 8 – Pipe Flow II, Force Balance Derivation Details. These notes contain details of the derivation relating the pressure drop to the wall shear stress.
• Lecture 8 – Pipe Flow II, Laminar Flow Example (non-circular pipe).
• Lecture 8 – Pipe Flow II, Turbulent Flow Example (roughness).
• Lecture 9 – Pipe Flow III, Non-Newtonian Flow Example. For your reference, this paper by Dodge and Metzner and this paper by Hartnett and Kostic give the correlation for the friction factor of a power-law fluid in turbulent flow.

Lectures 10–11: External Flow, Drag

• Lecure 10 – Drag I, Example of Drag Force Calculation. This is an example of a drag force cacluation using the drag coefficient on a disk.

• Lecture 11 – Drag II, Example on Terminal Velocity.. Here is an example of a terminal velocity problem with a “smart” guess. This way of solving the problem uses the Archimedes number and mirrors the examples in the textbook.

• Lecture 11 – Drag II, Example Drag Calculation on a Flat Plate.. Here is an example problem where we calculate the drag force on a flat plate.

• Lecture 11 – Drag II, Question on the Approach to Terminal Velocity. A question came up in class in 2017 about whether a larger particle approaches terminal velocity slower because it has a larger mass or because it has a larger volume. Implicit in this question is whether it is gravity forces or drag forces causing the slower approach. The short answer is that it is a larger mass. A particle with a larger mass takes longer to approach terminal velocity and a particle with larger drag (all else equal) reaches terminal velocity faster. (See the link for a derivation and more detailed explanation.)

Part II: Fundamentals and Differential Theory

Lectures 12–13: Fluid Statics

• Lecture 12 – Fluid Statics I, Derivation of Fluid Statics Equation. This video goes through the detailed algebra necessary to derive the Fluid Statics Equation. See also section 4.2 in Deen’s book on pages 85-87. A pdf copy of the notes from the video can be found here.

• Lecture 12 – Fluid Statics I, Manometer Example. This is an example manometer problem.

• Lecture 12 – Fluid Statics I, Clip on Pascal’s Paradox. This clip is a great demo by Katerina Visnjic at Princeton on “Pascal’s Paradox.” This should help with your homework as well.

Lectures 14–15: Fluid Kinematics

• Lecture 14 – Video about Eulerian and Lagrangian Descriptions. We watched the first part of this clip in class when we were discussing velocity fields.

• Lecture 14 – Python Plots of Velocity Fields. This is a Python Jupyter notebook showing the velocity field in planar pressure-driven flow and a slice of the velocity field of flow around a sphere. To download the notebook right-click here and select “Save link as”.

• Lecture 14 – Kinematics of Simple Shear Flow. These are some examples calculating some of the kinematic quantities of simple shear flow, including the vorticity and the streamfunction.

• Lecture 15 – Example Calculation of Local and Total Acceleration. These notes and video show an example of how to calculate a local acceleration (partial derivative) and total acceleration (material derivative) for a given velocity field.

• Lecture 15 – Example Differential Balance in Cylindrical Coordinates. These notes and video shows some of the steps in the derivation of the continuity equation in cylindrical coordinates.

Lectures 16–17: Fluid Dynamics (The Navier-Stokes Equation)

• Lecture 16 – Example Force Caclulation. These notes and video show an example force calculation for a flowing fluid. Given a velocity field and a pressure field, one can calculate the rate-of-strain tensor, the viscous stress tensor and finally the total stress tensor. Using the latter quanitity, the force on a surface can be calculated.

• Lecture 17 – Derivation of the Navier Stokes Equation. These notes give a detailed and complete derivation of Cauchy’s momentum equation and the Navier-Stokes equations for Cartesian coordinates.

Lectures 18–21: Unidirectional Flow, Creeping Flow, Inviscid Flow

• Lecture 18 – Calculation of the Flow Profile for Couette and Poiseuille Flow (and the laminar friction factor). These notes and video show an example of a derivation of the flow profile for Couette flow in a parallel plate geometry and for Poiseuille flow in a cylinder. I also drive the laminar friction factor for pipe flow.

• Lecture 19 – Velocity Profile for a Power-Law fluid in a Cylindrical Pipe. These notes and video show an example of a derivation of the flow profile for Poiseuille flow in a cylinder of a power law fluid. Also, here are some bonus notes showing how to calculate the laminar friction factor derivation for power-law fluids..

• Lecture 20 – (Supplement) Potential Flow around a Sphere. This is a derivation of potential flow around a sphere. Your book (“Introduction to Chemical Engineering Fluid Mechanics”, William Deen, Cambridge Univ. Press, 2016) has a derivation for potential flow around a cylinder, but not a sphere. We talk about this in the lecture on creeping flow and inviscid flow.
• Lecture 21 – Drag Force Calculations. See the video below for a recorded lecture on drag force calculations that goes with these notes.

Lectures 22–24: Advanced Flow Concepts

• Lecture 22 - Boundary Layer Theory. See the video below for a recorded lecture on boundary layer theory that goes with these notes.

• Lecture 22 (Supplement) - Bernoulli’s Equation. A derivation of Bernoulli’s equation from Euler’s Equation.
• Lecture 22 (Supplement) Derivation of Boundary Layer Equations. These notes contain a derivation of the boundary layer equations from the Navier-Stokes equation via non-dimensionalization. They accompany the above lecture on boundary layer theory.
• Lecture 23 (Supplement) - Turbulent Flow Profile Derivations. These are some supplemental lecture notes containing the algebraic details of the derivation of the average velocity profile for turbulent flow.
• Lecture 23 (Supplement) - Turbulent Friction Factor. These notes accompany the video below, which show how to get the friction factor (i.e. the Prandtl-Karman equation) from the turbulent velocity profile.

• Lecture 24 - CFD. These are notes from our in-class discussion of computational fluid dynamics. They include a discussion of the pressure-Poisson equation and finite differences.
• Jupyter Notebook for a CFD solution of Drag on a Square Cylinder. To download the notebook right-click here and select “Save link as”.

Part III: Macroscopic Systems and Pipe Network Design

Lectures 25–28: Integral Engineering Balances

• Lecture 25 - Integral and Engineering Mass Balance. See the video below for a recorded lecture on the Integral and Engineering Mass Balance (recorded in 2017).

• Lecture 25 - Examples for the Integral and Engineering Mass Balance. Here are some examples from the end of class for solving some problems using integral mass balances.
• Lecture 26 - Integral and Engineering Momentum Balance. See the video below for a recorded lecture on the Integral and Engineering Momentum Balance (recorded in 2019).

• Lecture 26 (Supplement) - Derivation of the Engineering Mometum Balance from the Integral Momentum Balance. These notes contain a derivation of the engineering momentum balance from the integral momentum balance. This derivation explains why the assumptions we list in class (e.g. discrete inlets and outlets, constant density at inlets and outlets, etc.) are necessary for the engineering momentum balance.
• Lecture 27 (Supplement) - Total Energy Equation. These notes highlight the difference between the total energy equation, the mechanical energy equation and the thermal energy equation. (For yet more details, see “Incompressible Flow” 3rd Ed., R. L. Panton, Wiley, 2005.) The total energy equation is an expression of the first law of thermodynamics for a continuum. In this class, we are mainly concerned with the mechanical energy equation. You will encounter the thermal energy equation again in heat transfer.
• Lecture 28 - Combined Engineering Balances. These are my lecture notes on engineering design and combined engineering balances.
• Additional Examples of Integral Engineering Balances. These are some extra examples of worked out mass, momentum and mechanical energy balances.
• Lecture 28 (Supplement) - Summary Handout of Integral Engineering Balances. This handout summarizes the integral balances for mass, momentum, and mechanical energy as well as the assumptions that lead to the more convenient engineering form.

Lectures 29–31: Pipe Network Design

• Lecture 29 - Minor Losses Example.
• Lecture 30 - Jupyter Notebook Example of a System Demand Curve. To download the notebook right-click here and select “Save link as”.
• Lecture 31 - Jupyter Notebook Example of Serial and Parallel Networks. To download the notebook right-click here and select “Save link as”.

Lectures 32–34: Valves and Turbomachinery

• Lecture 32 - Example of Flow through Globe Valves, Obstruction Flow Meters and Pitot Tubes.
• Lecture 32 - Slides on Valves Pressure and Flow Rate Measurements. These are a few slides about (i) pipes and tube and (ii) techniques for measuring pressure and flow rate. See also the excellent Visual Encylopedia of Chemical Engineering Equipment hosted by the University of Michigan.
• Lecture 33 - Slides on Pumps and Turbines. These are some slides containing the pictures of pumps and turbines that were shown in class.
• Lecture 33 Supplemental - Ideal Centrifugal Pump Curve. This is a derivation of the ideal centrifugal pump curve using Euler’s turbine equation.
• Lecture 33 Example - Finding the Operating Point with a Pump (Jupyter Notebook). This example shows how to use a pump performance curve and a system demand curve to find the operating point of a pipeline system. The video below goes through the notes (video notes here recorded in 2019).

• Lecture 34 - Example with NPSH. To download the notebook right-click here and select “Save link as”. The video below shows the example with NPSH and also shows how to calculate the operating point for a system with pumps in parallel (notes here recorded in 2019).

Lecture 35: Compressible Flow

• Lecture 35 Supplemental - Compressible Flow Derivations. This has two derivations: (i) A proof showing the converging nozzles have a maximum velocity at a Mach number equal to one; (ii) A proof of Equations 12.5-28 in Deen relating temperature and pressure to their respective stagnation value.
• Lecture 35 - Choked Flow Example . To download the notebook right-click here and select “Save link as”.