16100lectre20_cg

# 16100lectre20_cg - Problem#1 Assume Incompressible 1 2-D...

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Problem #1 Assume: Incompressible 2-D flow 0, 0 z V z ⇒= = Steady 0 = t Parallel 0 = r V a) Conservation of mass for a 2-D flow is: 1 ( r rV rr N 0 1 )( ) 0 ( ) 0 does not depend on () V r VV r θ θθ = += b) θ -mometum equation is: V t N r steady r +⋅ ∇+ K N 2 2 0 12 ( r r V pV V ν ρθ = ∂∂ =− + ∇ + N 2 0 ) r V V r = In cylindrical coordinates: r ⋅∇ = K N 0 1 V = + Thus, 1 V V r ⋅∇ + K N 0 from continuity 0 = = Also, 2 2 22 11 Vr r r  ∇= +   0 = ±²³²´ r 0 r 1 ω 1 ω 0

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Problem #1 16.100 2002 2 Combining all of these results gives: 2 this side is independent of 11 p VV r rr r r r θθ θ ν ρθ ∂∂   =−     ±²²²³²²²´ Since the right-hand-side (RHS) is independent of , this requires that constant p = for fixed r . But as varies from π 2 0 , it must be equal at 2 & 0 , that is ) 2 ( ) 0 ( = = = p p . If not, the solution would be discontinuous.
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## This note was uploaded on 11/06/2011 for the course AERO 100 taught by Professor Willcox during the Fall '03 term at MIT.

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16100lectre20_cg - Problem#1 Assume Incompressible 1 2-D...

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