p0416 - dx = aUydt and dy = Udt Integrating the equation...

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4.16. CHAPTER 4, PROBLEM 16 501 4.16 Chapter 4, Problem 16 Problem: Consider a flow for which the velocity vector is u = U ( ay i + j ) ,whe re U and a are characteristic velocity and inverse length scales, respectively. Determine the Lagrangian description of the fluid-particle position vector, r = x i + y j ,inte rm so f the constants U and a ,t ime t and the initial values of the coordinates, x o and y o . Solution: By definition, the velocity components in the Lagrangian description are u = X x t ~ x o ,y o = aUy and v = X y t ~ x o ,y o = U
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Unformatted text preview: dx = aUydt and dy = Udt Integrating the equation for y , we find y = y o + Ut where we assume y = y o at t = 0 . Substituting for y , the equation for x becomes dx = aU ( y o + Ut ) dt = x = x o + aUt w y o + 1 2 Ut W Thus, the particle location is given by x = x o + aUt w y o + 1 2 Ut W and y = y o + Ut Therefore, the Lagrangian description of this flow is r = } x o + aUt w y o + 1 2 Ut W] i + [ y o + Ut ] j...
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