Lecture_Note_Ch_4

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

Info icon This preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Motivation 2
Image of page 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
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 4
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
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 6
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.
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 8
This is the end of the preview. Sign up to access the rest of the document.