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# f10_mud - Lecture F10 Mud Momentum Theorem Applications(37...

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vector Lecture F10 Mud: Momentum Theorem Applications (37 respondents) 1. What’s the difference between D () /Dt and d () /dt ? (1 student) What we’re after is the time rate of change of some ﬂowfield quantity like p ( x, y, z, t ), as seen by a sensor moving as a ﬂuid 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 Dρ/Dt differs from ∂ρ/∂t . (2 students) The partial derivative ∂ρ/∂t merely gives the rate of change of density seen by fixed sensor sitting at one particular x, y, z location. But we’re after the rate seen by a sensor which moves with the ﬂow velocity, so the sensor’s x, y, z are continually changing. Using the chain rule then brings in the three spatial partial derivatives ∂ρ/∂x, y, z whose contributions are added to the ∂ρ/∂t term. See point 1 above.

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f10_mud - Lecture F10 Mud Momentum Theorem Applications(37...

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