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# pset_4 - 2.29 Numerical Fluid Mechanics Problem Set 4...

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2.29: Numerical Fluid Mechanics Problem Set 4 MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139 2.29 NUMERICAL FLUID MECHANICS— SPRING 2007 Problem Set 4 Totally 120 points Posted 04/21/07, due Thursday 4 p.m. 05/08/07, Focused on Lecture 18 to 22 Problem 4.1 (120 points): Steady state unidirectional fully developed incompressible laminar flow in a rectangular pipe a) Write down the Navier Stokes equation for the fully developed incompressible laminar flow in a pipe with arbitrary constant cross section and simplify the equations 1 . Furthermore simplify the equation for the case when the pressure gradient is fixed. Explain when your assumption holds. b) What is the name of this equation? Categorize that and give at least five examples of other cases (possibly from other physical domains) where we encounter the same equation. 2b 2a x y Now consider the upper rectangular pipe where . c) Find an analytical solution by separation of variables. d) (EXTRA CREDIT 10 Points) Explain how your solution (method) changes if we had a pulsating flow. You do not need to solve it thoroughly.

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