Problem 6.113[Difficulty: 4]Given:Complex functionFind:Show it leads to velocity potential and stream function of irrotational incompressible flow; Show that df/dzleads to u and vSolution:Basic equations:uyψ∂∂=vxψ∂∂−=uxφ∂∂−=vyφ∂∂−=First considerxf∂∂xz∂∂zfdd⋅=1zfdd⋅=zfdd=(1)and alsoyf∂∂yz∂∂zfdd⋅=izfdd⋅=izfdd⋅=(2)Hence2xf∂∂2xxf∂∂⎛⎜⎝⎞⎠∂∂=zzfdd⎛⎝⎞⎠dd=2zfdd2=and2yf∂∂2yyf∂∂⎛⎜⎝⎞⎠∂∂=izizfdd⋅⎛⎝⎞⎠dd⋅=2zfdd2−=Combining2xf∂∂22yf∂∂2+2zfdd22zfdd2−=0=Any differentiable function f(z) automatically satisfies the Laplace
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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.