Use the complex method for obtaining the quasi steady

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Use the complex method for obtaining the quasi-steady solution.
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Fluid Mechanics (Spring 2020) – Chapter 3 - U. Lei (李雨)Quasi-steady internal flow (3),cosh(1)2iKAiaωων=+Apply B.C.’s0=BThen++=aiyiKiywνωνωω2)1(cosh2)1(cosh1)(()1cosh2(, )Re()Re11cosh2ititiyNiKauy tw y eeiNωωω+==+andνω/2aNwitha dimensionless frequency
Fluid Mechanics (Spring 2020) – Chapter 3 - U. Lei (李雨)Quasi-steady internal flow (4)For smallN,......211......2121121cosh222++++++ayiNayNiayNi()2221cos21),(aytKatyuωνThusthe quasi velocity has a parabolic profile.same as the steady solution if the steady pressure gradientis replaced by the periodic pressure gradient under this lowfrequency limit (i.e., forN<< 1).
Fluid Mechanics (Spring 2020) – Chapter 3 - U. Lei (李雨)Quasi-steady internal flow (5)For largeN,ayNiayNiayNieeeayNi212121212121cosh++++=+112(, )Re1iyNitaiKuy teeωω+Thussin(, )ReitiKKtu y teωωωω=For region not close to the wall (called thecore region),it further reduces to (note thaty/a< 1)Naswhich is also the solution with the viscous term set to zero.Thus the flow in the core region is essentially inviscid !
Fluid Mechanics (Spring 2020) – Chapter 3 - U. Lei (李雨)Quasi-steady internal flow (6)For region next to the wall (called theboundary layer region),we may re-scale the problem by setting,ayNaη=122sin1(, )Re1sin2iitiKKtKu y teeetηηωωωηωωω+=≈ −+)1(O=ηsuch thatwithin the boundary layer.We then havesame as the core flowviscous effect
Fluid Mechanics (Spring 2020) – Chapter 3 - U. Lei (李雨)Other examples :the pulsating flow in a circular straight pipe drivenby an imposed periodic pressure gradient,the oscillating plane Couette flow driven by theharmonic oscillation of the upper and/or lowerplate(s),the pulsating flow in a circular straight pipe drivenby the axial oscillation of the pipe wall.Quasi-steady internal flow (7)
Fluid Mechanics (Spring 2020) – Chapter 3 - U. Lei (李雨)Flow with circular streamlines (1)We have found that the nonlinear convection termin the Navier-Stokes equation is identically zero forcases with parallel streamlines.

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