Mathematics for Engineers: The Capstone Course

The Hong Kong University of Science and Technology via Coursera

Go to Course: https://www.coursera.org/learn/mathematics-engineers-capstone

Introduction

# Course Review: Mathematics for Engineers: The Capstone Course on Coursera ### Overview Mathematics for Engineers: The Capstone Course is an advanced course designed for students who have successfully completed the Mathematics for Engineers specialization on Coursera. This capstone project not only consolidates your knowledge but also introduces exciting applications of mathematics in engineering, specifically in the realm of computational fluid dynamics (CFD). The course presents a hands-on opportunity to engage with real-world problems, such as fluid flow around a cylinder, using powerful tools like MATLAB. ### Course Structure The course is divided into several key sections, each building upon essential concepts and progressively leading students to more complex applications in CFD. 1. **Governing Equations**: In this module, students delve into the fundamental governing equations crucial for understanding fluid dynamics. The Navier-Stokes equations and the continuity equation take center stage as students derive coupled equations for the stream function and scalar vorticity. The process of nondimensionalizing these equations to incorporate the Reynolds number is an essential skill for aspiring engineers. 2. **Steady Flows**: This section formulates the CFD problem related to steady flow around a cylinder. Students are introduced to the finite difference method and develop iteration equations alongside boundary conditions. A significant highlight here is the practical application, where students will write MATLAB code to compute the stream function at a Reynolds number of ten, grounding theoretical knowledge in practical execution. 3. **Unsteady Flows**: Expanding on the previous section, this module focuses on unsteady flow around a cylinder. It incorporates periodic boundary conditions and matrix methods to provide insights into time-dependent phenomena. Students will employ MATLAB ODE integrators to compute scalar vorticity at a Reynolds number of sixty, enhancing their computational skills and understanding of dynamic flow behavior. ### Practical Aspects One of the biggest advantages of this course is the provision of access to MATLAB online and the MATLAB grader. This ensures that learners can code, test, and refine their understanding in a supportive virtual environment. ### Target Audience Before enrolling, students are expected to have prior knowledge in matrix algebra and differential equations. This prerequisite ensures that all participants have a solid foundation upon which to build more complex concepts in fluid dynamics. ### Review The Mathematics for Engineers: The Capstone Course stands out for several reasons: - **Relevance**: The course ties mathematical theory directly to engineering applications, making it incredibly relevant for those pursuing careers in various engineering fields. - **Hands-On Learning**: The opportunity to work with MATLAB allows students to transition theoretical concepts into practical skills, a major plus in today’s job market. - **Robust Curriculum**: The course covers both steady and unsteady flow concepts, ensuring that students appreciate the broader spectrum of fluid dynamics. While the course is ambitious, it is recommended for dedicated learners keen on pushing their understanding of mathematics in engineering contexts. The use of MATLAB adds a layer of complexity, which might be challenging for some, but it is precisely this challenge that provides an invaluable learning experience. ### Recommendation I highly recommend the Mathematics for Engineers: The Capstone Course to any engineering students or professionals looking to solidify their mathematical skills in a practical, applicable context. The course not only enhances your understanding of critical fluid dynamics concepts but also equips you with the coding skills necessary for modern engineering tasks. If you’re motivated to merge mathematics with engineering principles, this course will be a significant step in your academic and professional journey.

Syllabus

Governing Equations

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.

Steady Flows

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.

Unsteady Flows

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.