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Advanced Engineering Mathematics

Reza Malek-Madani
Organization: U.S. Naval Academy
Department: Department of Mathematics
Education level


To introduce the basic mathematical tools for generating and solving the governing equations of fluid dynamics. Vector calculus and partial differential equations are the primary topics covered.

The main text is the second volume of the book I wrote with the title Advanced Engineering Mathematics, Addison-Wesley-Longman, 1998.

The course, which is primarily taught to junior level students majoring in oceanography, mechanical engineering and mathematics, covers the basic materials of vector calculus and partial differential equations in the context of fluid flows. After a thorough review of the vector operations (grad, div and curl), conservation laws of mass and linear momentum are introduced. Numerous examples of flows that one typically encounters in a basic fluid dynamics and geophysical fluid dynamics setting are introduced and visualized using Mathematica's symbolic and numerical capabilities. The course ends with the derivation of the Navier-Stokes equation in a rotating frame with special emphasis on the Coriolis force and its impact on the so-called Ekman tranport solution.

The primary goal of the course is to demonstrate the natural relationship between several topics in mathematics and fluid dynamics and oceanography. Mathematica is the primary tool used throughout the course as a symbolic manipulator as well as a numerical workhorse. Visualizing flows, especially through animations, is one of the main strengths of this course. Another strength of the course is the set of computer projects that the students carry out as part of the grade requirement. Examples of past projects include a) Flow past cylinder, b) Oseen vortex, c) Rayleigh-Benard flow, d) Lorenz and Veronis models of convection, and e) Serrin's tornado model.

  • Conservation of Mass and Incompressibility
  • Curl and Vorticity
  • Line Integrals and Circulation
  • Separation of Variables
  • Normal modes
  • Fourier Series
  • Balance of Linear Momentum
  • Special solutions of the Navier-Stokes Equations
  • Navier-Stokes Equations in a Rotating Frame
  • Ekman Layer

*Applied Mathematics
*Mathematics > Calculus and Analysis > Differential Equations
*Science > Geology and Geophysics
*Science > Physics > Fluid Mechanics