Lecture 19 Boundary Layers and Drag

# Lecture 19 Boundary Layers and Drag - Thermal and Fluids...

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Thermal and Fluids Engineering I Prof. Deborah A. Kaminski 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 Lecture 19 Page 1

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Thermal and Fluids Engineering I Prof. Deborah A. Kaminski Consider a differential control volume of size xyz ∆∆ and apply conservation of mass xy x y xx x yy mm m m +∆ y += + ±± ± ± xyx y x y y zx z y ρρ ρ ∆∆+ ∆∆ = VV V z y x y y x y x ∆+ ∆= 0 y x xy =+ Taking the limit as and 0 x ∆→ 0 y 0 y x ∂∂ 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 Lecture 19 Page 2
Thermal and Fluids Engineering I Prof. Deborah A. Kaminski from the control volume form of conservation of momentum. Without derivation, the results are: Conservation of Momentum in x -direction 22 1 x xx xyx P g xy y µ ρρ ⎛⎞ ∂∂ += + + ⎜⎟ ⎝⎠ x V VV Conservation of Momentum in y -direction 1 yy y xyy P g x y x y

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Lecture 19 Boundary Layers and Drag - Thermal and Fluids...

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