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	Volume 5, Issue 2 Pg 87 - 171 (November 2010)
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Far East Journal of Mathematical Education

Far East Journal of Mathematical Education
Volume 5, Issue 1, Pages 53 - 85 (August 2010)

TEACHING FLUID-DYNAMIC CASCADES FOR UNDERGRADUATES: ANALYSIS AND SIMULATION
Victor A. Miroshnikov, Nour Aqeel, Robert Bararwandika, Stephanie Chavez, Meghan Conroy, Ryan Foti, Hussain Gardezi, Yanell Innabi, Karen Laurent, George V. Miroshnikov and Elizabeth M. Toribio

Abstract:
Fundamental solutions of the Navier-Stokes equation for cascade flows in one spatial dimension constitute a golden fund of undergraduate mathematics, since teaching their constituents in various mathematics courses across the undergraduate curriculum, ranging from algebra and calculus to computational mathematics, scientific computing, and partial differential equations, synthesizes simplicity of analysis with generality of results and through interactive animations makes connections between contemporary research and standard topics in mathematics courses. We study spatiotemporal cascades as a family of fundamental solutions of biological fluid dynamics for the Couette flow, the Stokes flow, and the Poiseuille flow with moving boundaries, which model contractions of biological channels, and summarize results in the existence theorem of a general solution in the considered class of flows with moving boundaries. The revealed connections between classical topics, like the fundamental theorem of algebra, mathematical induction, and convergence of power series, and contemporary topics, like boundary layers, coherent structures, dissipative waves, multiscale problems, and the Kolmogorov-Batchelor cascades, have been presented by the faculty and the students at undergraduate mathematics conferences. The paper meets the needs and intellectual interests of a broad community of college mathematics teachers and students, since it presents a novel approach in teaching parabolic partial differential equations of fluid dynamics, diffusion, and heat transfer.

Keywords and phrases: undergraduate mathematics, parabolic partial differential equations, fundamental and general solutions, the existence theorem, the Couette flow, the Poiseuille flow, the Stokes flow, the Bernoulli flow, biological flows, coherent structures, multiscale problems, the Kolmogorov-Batchelor cascades.

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