f10_mud - vector Lecture F10 Mud: Momentum Theorem...

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Unformatted text preview: vector Lecture F10 Mud: Momentum Theorem Applications (37 respondents) 1. Whats the difference between D () /Dt and d () /dt ? (1 student) What were after is the time rate of change of some owfield quantity like p ( x, y, z, t ), as seen by a sensor moving as a uid element. Just writing dp/dt is very sloppy math, since p depends on four variables x, y, z, t , not just on t . So we must work with the four partial derivatives of p . The correct expression for the time rate of change seen by the sensor is p p p p + u + v + w t x y z but this is long and tedious to write. So we use the shorthand Dp/Dt for this whole expression. 2. Not clear how D/Dt differs from /t . (2 students) The partial derivative /t merely gives the rate of change of density seen by fixed sensor sitting at one particular x, y, z location. But were after the rate seen by a sensor which moves with the ow velocity, so the sensors x, y, z are continually changing. Using the chain rule then brings in the three spatial partial derivatives /x, y, z whose contributions are added to the /t term. See point 1 above....
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This note was uploaded on 01/28/2012 for the course AERO 16.01 taught by Professor Markdrela during the Fall '05 term at MIT.

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f10_mud - vector Lecture F10 Mud: Momentum Theorem...

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