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Mathematics for Engineers: The Capstone Course

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

Top Instructor

6,380 already enrolled

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3 modules

Gain insight into a topic and learn the fundamentals.

45 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.

45 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

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Assessments

1 assignment

Taught in English

91%

of learners achieved a positive career outcome

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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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Gain a foundational understanding of a subject or tool
Develop job-relevant skills with hands-on projects
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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

Module details

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 videosTotal 67 minutes

Course Overview2 minutes
Week One Introduction1 minute
Navier-Stokes Equations | Lecture 19 minutes
Vorticity Equation | Lecture 28 minutes
Geometry of the Flow | Lecture 38 minutes
Two-dimensional Flow | Lecture 48 minutes
Stream Function | Lecture 57 minutes
Streamlines | Lecture 67 minutes
Reynolds Number | Lecture 710 minutes
Log-polar Coordinates | Lecture 87 minutes

16 readingsTotal 147 minutes

Welcome and Course Information2 minutes
MATLAB Online5 minutes
Plane Couette Flow10 minutes
Channel Flow10 minutes
Pipe Flow10 minutes
Vector Identities10 minutes
Flow Boundaries10 minutes
Curl of the Vorticity-Velocity Cross Product10 minutes
Scalar Vorticity in Cartesian Coordinates10 minutes
Scalar Vorticity Equation in Cartesian Coordinates10 minutes
Stream Function in Cartesian Coordinates10 minutes
Scalar Vorticity in terms of the Stream Function in Cartesian Coordinates10 minutes
Stream Function as a Vector Potential10 minutes
Streamlines in Cartesian Coordinates10 minutes
Stream Function Equation when Re = 010 minutes
Steady Flow Equations10 minutes

1 assignmentTotal 30 minutes

Week One Assessment30 minutes

1 pluginTotal 17 minutes

Deep Dive into the Flow Around a Cylinder17 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 videosTotal 63 minutes

Week Two Introduction 1 minute
Finite Difference Method | Lecture 910 minutes
Iteration Equations | Lecture 1011 minutes
Free-stream Boundary Conditions | Lecture 117 minutes
Cylinder Boundary Conditions | Lecture 1215 minutes
Summary of the Boundary Conditions | Lecture 135 minutes
MATLAB Program (Steady) (Part A) | Lecture 147 minutes
MATLAB Program (Steady) (Part B) | Lecture 158 minutes

6 readingsTotal 37 minutes

Even and Odd Functions5 minutes
SOR Equations10 minutes
Vorticity Free-Stream Boundary Condition10 minutes
Test the Cylinder Boundary Condition10 minutes
Starting SOR with zero vorticity1 minute
Reference solution to "Steady Flow at Re = 10"1 minute

1 app itemTotal 60 minutes

Steady Flow at Re = 1060 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 videosTotal 50 minutes

Week Three Introduction 1 minute
Periodic Boundary Conditions | Lecture 166 minutes
Finite Difference Equations | Lecture 176 minutes
Stream Function Computation | Lecture 187 minutes
Stream Function Boundary Conditions | Lecture 1910 minutes
Vorticity Computation | Lecture 205 minutes
MATLAB Program (Unsteady) (Part A) | Lecture 219 minutes
MATLAB Program (Unsteady) (Part B) | Lecture 225 minutes
Concluding Remarks1 minute

8 readingsTotal 57 minutes

Ghost Points10 minutes
Vorticity Finite Difference Equation15 minutes
MATLAB Coding of Right-Hand Side10 minutes
Three-by-Three Grid10 minutes
Computational Time10 minutes
Reference Solution to "Unsteady Flow at Re = 60"1 minute
Please Rate this Course1 minute
Acknowledgements0 minutes

1 app itemTotal 60 minutes

Unsteady Flow at Re = 6060 minutes

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Instructor

Instructor ratings

(19 ratings)

Top Instructor

Jeffrey R. Chasnov

The Hong Kong University of Science and Technology

17 Courses255,786 learners

Offered by

The Hong Kong University of Science and Technology

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5 stars
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Reviewed on Aug 12, 2025

Outstanding course and instructor, would highly recommend to anyone wanting to learn or refresh their matrix algebra, calculus, and Matlab programming fundamentals

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 May 26, 2025

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

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