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 uxyzt vxyzt wxyzt dd u d v d w dt dt dt dt du x y z t u u dx u dy u dz dt t x dt y dt z dt uuu d u uvw tx y d z u u t d u dt t =++ ==++ ∂∂ =+ + + ∂∂∂ + + == + Vr i j k V ai j k V VV V a () Local Convective d vw xy d z t  ++ = +   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: iii CV ii CV dA V A V t d dxdydz tt x AV AV u u dx dydz udydz u dxdydz xx dxdydz t ρ ρρ ρρρ +− = ∂∂  −= + =   ∑∑ V V ( ) 0 ( ) 0 0 u dxdydz v dxdydz w dxdydz xyz uvw tx y z t ∂∂∂ +++ = = +∇⋅ = 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 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 11 Cylindrical: ( ) ( ) ( ) 0 Continuity equation for Incompressible Flow (Ma 0.3): Cartis rz uvw xyz rv v v rr r z θ ρρ ρ ∂∂∂ ++ = ∂∂ = ian: 0 or 0 Cylindrical: ( ) ( ) ( ) 0 xy z rv v v r z = = = V Å Linear differential equations