Mathematics for Engineers: The Capstone 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

• 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.