Mathematics for Engineers: The Capstone Course

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The Hong Kong University of Science and Technology

Mathematics for Engineers: The Capstone Course

This course is part of Mathematics for Engineers Specialization

Instructor: Jeffrey R. Chasnov

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3 modules

Gain insight into a topic and learn the fundamentals.

42 reviews

Intermediate level

Recommended experience

1 week to complete

at 10 hours a week

Flexible schedule

Learn at your own pace

3 modules

Gain insight into a topic and learn the fundamentals.

42 reviews

Intermediate level

Recommended experience

1 week to complete

at 10 hours a week

Flexible schedule

Learn at your own pace

What you'll learn

Fundamentals of computational fluid dynamics
How to construct and solve a numerical method for computing the flow field around an infinite cylinder

Skills you'll gain

Tools you'll learn

Details to know

Shareable certificate

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Assessments

1 assignment

Taught in English

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This course is part of the Mathematics for Engineers Specialization

When you enroll in this course, you'll also be enrolled in this Specialization.

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There are 3 modules in this course

Mathematics for Engineers: The Capstone Course provides a capstone project for students who are completing the Mathematics for Engineers specialization. Students will first learn some basic concepts in computational fluid dynamics, and then apply these concepts to compute the fluid flow around a cylinder. Access to MATLAB online and the MATLAB grader is given to all students who enroll.

Before enrolling, students should have already taken courses in matrix algebra, differential equations, vector calculus and numerical methods, and be able to program in MATLAB. The course contains 22 short video lectures and a full set of lecture notes. After each lecture, there are problems to solve, and at the end of the second and third weeks, there is a substantial MATLAB programming assignment. Download the lecture notes from the link https://www.math.hkust.edu.hk/~machas/flow-around-a-cylinder.pdf Watch the promotional video from the link https://youtu.be/FlM1de9Sxh0

We learn the governing equations for the flow around a cylinder. We discuss the Navier-Stokes equations and the continuity equation, and derive a pair of coupled equations for the stream function and scalar vorticity. We nondimensionalize these equations so that they contain only a single dimensionless parameter called the Reynolds number. We then simplify the nondimensional governing equations using log-polar coordinates.

What's included

10 videos16 readings1 assignment1 plugin

10 videos Total 67 minutes

Course Overview 2 minutes
Week One Introduction 1 minute
Navier-Stokes Equations | Lecture 1 9 minutes
Vorticity Equation | Lecture 2 8 minutes
Geometry of the Flow | Lecture 3 8 minutes
Two-dimensional Flow | Lecture 4 8 minutes
Stream Function | Lecture 5 7 minutes
Streamlines | Lecture 6 7 minutes
Reynolds Number | Lecture 7 10 minutes
Log-polar Coordinates | Lecture 8 7 minutes

16 readings Total 147 minutes

Welcome and Course Information 2 minutes
MATLAB Online 5 minutes
Plane Couette Flow 10 minutes
Channel Flow 10 minutes
Pipe Flow 10 minutes
Vector Identities 10 minutes
Flow Boundaries 10 minutes
Curl of the Vorticity-Velocity Cross Product 10 minutes
Scalar Vorticity in Cartesian Coordinates 10 minutes
Scalar Vorticity Equation in Cartesian Coordinates 10 minutes
Stream Function in Cartesian Coordinates 10 minutes
Scalar Vorticity in terms of the Stream Function in Cartesian Coordinates 10 minutes
Stream Function as a Vector Potential 10 minutes
Streamlines in Cartesian Coordinates 10 minutes
Stream Function Equation when Re = 0 10 minutes
Steady Flow Equations 10 minutes

1 assignment Total 30 minutes

Week One Assessment 30 minutes

1 plugin Total 17 minutes

Deep Dive into the Flow Around a Cylinder 17 minutes

We formulate the computational fluid dynamics problem of the steady flow around a cylinder. We introduce the finite difference method and derive iteration equations. We derive boundary conditions and discuss the outline of a MATLAB program. Students will write a MATLAB code to compute the stream function at a Reynolds number of ten.

What's included

8 videos6 readings1 app item

8 videos Total 63 minutes

Week Two Introduction 1 minute
Finite Difference Method | Lecture 9 10 minutes
Iteration Equations | Lecture 10 11 minutes
Free-stream Boundary Conditions | Lecture 11 7 minutes
Cylinder Boundary Conditions | Lecture 12 15 minutes
Summary of the Boundary Conditions | Lecture 13 5 minutes
MATLAB Program (Steady) (Part A) | Lecture 14 7 minutes
MATLAB Program (Steady) (Part B) | Lecture 15 8 minutes

6 readings Total 37 minutes

Even and Odd Functions 5 minutes
SOR Equations 10 minutes
Vorticity Free-Stream Boundary Condition 10 minutes
Test the Cylinder Boundary Condition 10 minutes
Starting SOR with zero vorticity 1 minute
Reference solution to "Steady Flow at Re = 10" 1 minute

1 app item Total 60 minutes

Steady Flow at Re = 10 60 minutes

We formulate the computational fluid dynamics problem of the unsteady flow around a cylinder. We introduce periodic boundary conditions in the polar angle, and show how to solve for the stream function using matrix methods. We show how to use a MATLAB ODE integrator to solve for the scalar vorticity. Students will write a MATLAB code to compute the time-dependent scalar vorticity at a Reynolds number of sixty.

What's included

9 videos8 readings1 app item

9 videos Total 50 minutes

Week Three Introduction 1 minute
Periodic Boundary Conditions | Lecture 16 6 minutes
Finite Difference Equations | Lecture 17 6 minutes
Stream Function Computation | Lecture 18 7 minutes
Stream Function Boundary Conditions | Lecture 19 10 minutes
Vorticity Computation | Lecture 20 5 minutes
MATLAB Program (Unsteady) (Part A) | Lecture 21 9 minutes
MATLAB Program (Unsteady) (Part B) | Lecture 22 5 minutes
Concluding Remarks 1 minute

8 readings Total 57 minutes

Ghost Points 10 minutes
Vorticity Finite Difference Equation 15 minutes
MATLAB Coding of Right-Hand Side 10 minutes
Three-by-Three Grid 10 minutes
Computational Time 10 minutes
Reference Solution to "Unsteady Flow at Re = 60" 1 minute
Please Rate this Course 1 minute
Acknowledgements 0 minutes

1 app item Total 60 minutes

Unsteady Flow at Re = 60 60 minutes

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Instructor

Instructor ratings

(17 ratings)

Top Instructor

Jeffrey R. Chasnov

The Hong Kong University of Science and Technology

17 Courses 249,700 learners

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The Hong Kong University of Science and Technology

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Specialization

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Reviewed on Jun 10, 2023

This is by far the most comprehensive specialization or course related to mathematics out there. Loved every video, reading, assignment, quizzes, and projects

Reviewed on Feb 13, 2026

Great Course! Thank you Professor Jeffrey R. Chasnov!

Reviewed on May 26, 2025

A great course for people who want to get started writing their own CFD solvers!

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