Problem 2.14[Difficulty: 2]Given:Velocity fieldFind:Proof that the parametric equations for particle motion are xpc1eAt⋅⋅=and ypc2eA−t⋅⋅=; pathline that was at(2,2) at t = 0; compare to streamline through same point, and explain why they are similar or not.Solution:Governing equations:For pathlinesupdxdt=vpdydt=For streamlinesvudydx=Assumption:2D flowHence for pathlinesupdxdt=Ax⋅=vpdydt=A−y⋅=So, separating variablesdxxAdt⋅=dyyA−dt⋅=Integratingln x() At⋅C1+=ln y()A−t⋅C2+=xe⋅C1+=eC1e⋅⋅=c1e⋅⋅=yeA−t⋅C2+=eC2eA−t⋅⋅=c2eA−t⋅⋅=The pathlines arexc1e⋅⋅=yc2eA−t⋅⋅=Eliminating tt1Alnxc1⎛⎜⎝⎞⎠⋅=1A−lnyc2⎛
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This note was uploaded on 10/19/2011 for the course EGN 3353C taught by Professor Lear during the Fall '07 term at University of Florida.