Reynolds Transport Theorem Bcv = Boundary Layer Equations ( ! (x) << x ) # "v "v & "2 v "p ! % vx x + v y x ( = ) + µ 2x $ "x "y ' "x "y "p 0=! "y Form A: " !vx !v y % $# !x + !y '& = 0 ( Free Surface Flows; p ! "# $2 h $x 2 Form B: ) h !h !Q" + = 0 where Q! = " vx! dy 0 !t !x 2 !h $ " gh sin # ' !h " g cos # ! * 3 !h +& ) = ,h / !t % µ 3µ !x + !x . ( !x 0! dt cv Incompressible Flow !"v = 0 { } "p + # gy "y Inviscid fluid Re ! ! (Euler Equation) Dv ! = "#p + !g Dt Steady Irrotational Flow ( ! = 0 ) Velocity potential v = !" (! " v) = ! # = 0 2 ! =0 ! =0 Steady Bernoulli Equation along a streamline for an inviscid flow of an incompressible fluid 1 1 2 1 2 ( p2 ! p1 ) + g ( z2 ! z1 ) = 0 2 v 2 ! 2 v1 + " ( ) Hydrostatics !p = "(g # a) Across streamlines (outward pointing normal n) !p " v 2 = R !n Unsteady Bernoulli Equation along a streamline 2 # !v " dr + % $ !t 1 ! = $ " # dA = ! $ v # ds L = !V" D=0 Young-Laplace Equation # 1 1 & !p = " % + ( $ R1 R2 ' Navier-Stokes Equation for isothermal flow of an incompressible Newtonian fluid Dv % "v ( ! = ! & + v #$ v ) = +$p + µ$ 2 v + !g Dt ' "t * Very viscous fluid (Stokes’ Flow; Re ! 0 ) 0 = !"p + µ" 2 v + #g Kutta-Zhukowski Theorem © Gareth H. McKinley December 2006 !" =# % dV + ( ( &b)v ' n dA CS $ !t Newtonian Constitutive Equation ! = µ "v + ("v)t = µ#! 2 #2 v #p + µ 2x + $ gx #x #y dBsys The Equations of Fluid Mechanics d " !dV + "CS (#b) ( v $ vc ) % n dA dt cv Continuity 1 D! = " (# $ v ) ! Dt h L << 1 , ReL ( h L ) << 1 , h (!" ) << 1 ) 0!" dt = Cauchy Momentum Equation Dv ! = "#p + # $ % + !g Dt Quasi-Fully Developed Flow (Lubrication approximation) 2 dBsys "cv !dV = "cv #b dV ( 2 # dp 1 2 1 2 + ( (2 & (1 ) = 0 2 v 2 & 2 v1 + % $ ' ) 1