







Advanced Engineering Mathematics






Organization:  U.S. Naval Academy 
Department:  Department of Mathematics 






College






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, AddisonWesleyLongman, 1998. The text is augmented with notes that I have placed on my homepage at http://web.usna.navy.mil/~rmm/






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 NavierStokes equation in a rotating frame with special emphasis on the Coriolis force and its impact on the socalled 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) RayleighBenard flow, d) Lorenz and Veronis models of convection, and e) Serrin's tornado model. Topics:  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 NavierStokes Equations
 NavierStokes Equations in a Rotating Frame
 Ekman Layer












http://web.usna.navy.mil/~rmm/ http://web.usna.navy.mil/~rmm/COURSES/sm311o.html

