Chapter05

Chapter05 - Conservation of Mass and Momentum Momentum...

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Conservation of Mass and Conservation of Mass and Momentum Momentum z Using the Reynolds Transport Theorem, it is a simple matter to derive mass- and momentum-conservation laws for a control volume, i.e., the integral forms Global view Indirect computation of forces z Focusing on a differential-sized control volume, we can deduce the differential form of the conservation laws, which hold at every point in the flow Detailed view Direct computation of forces and all flow properties z From the differential forms we can derive Bernoulli’s Equation – mechanical energy conservation
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Conservation Principles Conservation Principles z The basis of our conservation laws is as follows, where B is an extensive variable and β is the corresponding intensive variable z We will concentrate on mass and momentum first, and do energy later (Chapter 7)
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Mass Conservation Mass Conservation - - 1 1 z We consider a general control volume whose bounding surface has velocity u cv z Definition of a system Mass, M , is constant By definition, Hence, B = M and β = 1 z dM/dt = 0 …the Reynolds Transport Theorem yields z In words, the sum of the instantaneous rate of change of mass in the control volume and the net flux of mass out of the control volume is zero
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Mass Conservation Mass Conservation - - 2 2 z Recall that for a differential-sized control volume… z Again, since dB/dt = 0 and d β /dt = 0 , the Reynolds Transport Theorem tells us that z This is called the continuity equation…it holds at every point in the flow – CV asymptotes to a point z Using the chain rule… “Conservation” form “Primitive- variable” form Æ
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Mass Conservation Mass Conservation - - 3 3 z We can use our results to further simplify the Reynolds Transport Theorem…as noted above, for a “point” we derived the following z Therefore,the Reynolds Transport Theorem for a “point” simplifies to Zero according to mass conservation
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Momentum Conservation Momentum Conservation - - 1 1 z We consider a perfect fluid so that only normal surface forces act z Momentum of a system By definition, Hence, B = P and β = u z Newton’s 2 nd Law says z We consider two types of forces 1. Surface force, F s : transmitted across surface S 2. Body force, F b : acts at a distance
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Momentum Conservation Momentum Conservation - - 2 2 z The only surface force acting is pressure The magnitude of the force on a differential surface element is pdS Since n is an outer unit normal, the force exerted by the surroundings on the control volume is –p n dS Thus, z Typical body forces…gravity, electromagnetic Express in terms of the specific body-force vector, f f is body force per unit mass…for gravity, f = - g k Thus,
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Momentum Conservation Momentum Conservation - - 3 3 z Using the Reynolds Transport Theorem… z The terms in this equation, from left to right, are 1.
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This note was uploaded on 10/12/2009 for the course AME 309 at USC.

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Chapter05 - Conservation of Mass and Momentum Momentum...

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