lecture 19 - Thermal and Fluids Engineering I Lecture 19...

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Thermal and Fluids Engineering I Lecture 19 Page 1 Lecture 19 – Boundary Layers and Drag Boundary Layer on a Flat Plate Consider the flow of a viscous fluid parallel to a flat plate: Within the boundary layer, viscous effects are important. Outside the boundary layer, viscous effects are negligible. We would like to determine the drag force that the flow exerts on the plate. The shear stress at the wall depends on the velocity profile. x x d dy τµ = V To find the velocity profile, we use the differential versions of conservation of mass and conservation of momentum. Assume the flow is - incompressible - steady - two-dimensional
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Thermal and Fluids Engineering I Lecture 19 Page 2 Consider a differential control volume of size xyz ∆∆∆ and apply conservation of mass xy x y xx x yy y mm m m +∆ += + && & & xyx y x y yz xz ρρ ρ ∆∆ + ∆∆ = ∆∆ VV V V xy x y x y yx y x ∆+ ∆= V V 0 y x =+ Taking the limit as 0 x ∆→ and 0 y 0 y x ∂∂ V V This is called the continuity equation or conservation of mass. By a similar but much more elaborate process, one can derive the differential conservation of momentum equation
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Thermal and Fluids Engineering I Lecture 19 Page 3 from the control volume form of conservation of momentum. Without derivation, the results are: Conservation of Momentum in x -direction 22 1 x xx x xyx P g xy y µ ρρ ⎛⎞ ∂∂ += + + ⎜⎟ ⎝⎠ VV V V Conservation of Momentum in y -direction 1 yy y y xyy P g yx y + + V V
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This note was uploaded on 04/08/2008 for the course ENGR 2250 taught by Professor Borca-tasciuc during the Spring '08 term at Rensselaer Polytechnic Institute.

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lecture 19 - Thermal and Fluids Engineering I Lecture 19...

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