Lecture_Note_Ch_4

# Lecture_Note_Ch_4 - ME 342 Fluid Mechanics Lecture Note on...

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Lecture Note on Ch. 4: Differential Relations for Fluid Flow Prof. Chang-Hwan Choi Stevens Institute of Technology Department of Mechanical Engineering ME 342 Fluid Mechanics Spring 2008

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Motivation 2
The Acceleration Field of a Fluid ( , ) ( , , , ) ( , , , ) ( , , , ) ( , , , ) ( ) t u x y z t v x y z t w x y z t d du dv dw dt dt dt dt du x y z t u u dx u dy u dz dt t x dt y dt z dt u u u du u v w t x y dz u u t d u dt t = + + = = + + = + + + = + + + = + ⋅∇ = = + V r i j k V a i j k V V V V a ( ) Local Convective d v w x y dz t + + = + ⋅∇ V V V V V Å Chain rule

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out in out in ( ) ( ) 0 For an infinitesimal fixed control volume: For a face of : ( ) ( ) ( ) ( ) For all faces: i i i i i i CV i i CV d AV AV t d dxdydz t t x AV AV u u dx dydz udydz u dxdydz x x dxdydz t ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ + = = + = V V ( ) ( ) ( ) 0 ( ) ( ) ( ) 0 ( ) 0 u dxdydz v dxdydz w dxdydz x y z u v w t x y z t ρ ρ ρ ρ ρ ρ ρ ρ ρ + + + = + + + = + ∇⋅ = V The Differential Equation of Mass Conservation Elemental cartesian fixed control volume showing the inlet and outlet mass flows on the x faces. Å Continuity equation in cartesian coordinates
The Differential Equation of Mass Conservation (cont.) z z x y y x r = = + = 1 2 / 1 2 2 tan ) ( θ 0 ) ( ) ( 1 ) ( 1 ) ( ) ( 1 ) ( 1 = + + + + + = z r z r v z v r v r r r t A z A r rA r r ρ ρ θ ρ ρ θ θ θ A Definition sketch for the cylindrical coordinate system. Å Continuity equation in cylindrical polar coordinates Å Divergence in cylindrical polar coordinates

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The Differential Equation of Mass Conservation (cont.) Continuity equations for Steady Compressible Flow: Cartisian: ( ) ( ) ( ) 0 1 1 Cylindrical: ( ) ( ) ( ) 0 Continuity equation for Incompressible Flow (Ma 0.3): Cartis r z u v w x y z r v v v r r r z θ ρ ρ ρ ρ ρ ρ θ + + = + + = ian: 0 or 0 1 1 Cylindrical: ( ) ( ) ( ) 0 r z u v w x y z rv v v r r r z θ θ + + = ∇⋅ = + + = V Å Linear differential equations
u dydz ρ V ( ) u u dx dydz x ρ ρ + V V ( ) ( ) out in out in ( ) ( ) ( ) For a face of : ( ) ( ) ( ) ( ) Similarly for faces of and , i i i i CV CV d m m t d dxdydz t t x m m u u dx dydz u dydz x u dxdydz x y z ρ ρ ρ ρ ρ ρ ρ = + = + = & & & & F V V V V V V V V V V V V V then ( ) ( ) ( ) ( ) ( ) u v w dxdydz t x y z u v w dxdydz t t x y z d dxdydz dt ρ ρ ρ ρ ρ ρ ρ ρ = + + + = + ∇⋅ + + + + = F V V V V V V V V V V V The Differential Equation of Linear Momentum Elemental cartesian fixed control volume showing the linear momentum in the x direction only.

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