p0506 - x = 0 For the given density ρ x,t = Cxe − t we...

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5.6. CHAPTER 5, PROBLEM 6 585 5.6 Chapter 5, Problem 6 Problem: The density of a one-dimensional unsteady flow is ρ ( x, t )= Cxe λ t ,where C and λ are constants of dimensions M/L 4 and 1 /T , respectively. Derive a differential equation for the x component of the velocity, u . Solution: Since the flow is one dimensional and unsteady, the continuity equation is ∂ρ t + u ∂ρ x + ρ u
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Unformatted text preview: x = 0 For the given density, ρ ( x,t ) = Cxe − λ t , we have ∂ρ ∂ t = − λ Cxe − λ t , ∂ρ ∂ x = Ce − λ t Hence, the continuity equation becomes − λ Cxe − λ t + Cue − λ t + Cxe − λ t ∂ u ∂ x = 0 Dividing through by Ce − λ t and rearranging terms yields x ∂ u ∂ x + u − λ x = 0...
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This note was uploaded on 02/09/2012 for the course AME 309 taught by Professor Phares during the Spring '06 term at USC.

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