Lecture_Note_Ch_4

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

Info iconThis preview shows pages 1–9. 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
Background image of page 1

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

View Full DocumentRight Arrow Icon
Motivation 2
Background image of page 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
Background image of page 3

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

View Full DocumentRight Arrow Icon
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
Background 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 A Definition sketch for the cylindrical coordinate system. Å Continuity equation in cylindrical polar coordinates Å Divergence in cylindrical polar coordinates
Background image of page 5

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

View Full DocumentRight Arrow Icon
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
Background image of page 6