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Dynamic Forces

Dynamic Forces. Fig. from Warrick et al. 2005. Nature. F. Solids vs. Fluids. S. . Solids resist shear deformation: they care how far they are deformed. Fluids care about how rapidly they are deformed: rate of shear. F/S = (/t). F/S = G . Dynamic viscosity. Shear modulus.

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Dynamic Forces

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  1. Dynamic Forces Fig. from Warrick et al. 2005. Nature

  2. F Solids vs. Fluids S  Solids resist shear deformation: they care how far they are deformed Fluids care about how rapidly they are deformed: rate of shear F/S = (/t) F/S = G Dynamic viscosity Shear modulus Rate of shear Shear strain

  3. No-slip condition & Boundary layer U F S (F/S) = (U/z) z Lingulodinium polyedrum

  4. Viscosity U F S z F/S = (/t) Viscosity is a measure of the resistance to rate of shear

  5. Physical coupling of temperature and viscosity Viscosity of water has a sever dependence on temperature (Viscosity doubles from 30 to 5ºC)

  6. Temperature effects on viscosity * More viscous blood should flow more slowly * Slower mass flow can minimize heat loss Q’ = CJΔT

  7. Temperature effects on viscosity Leopard Seal Crabeater Seal Weddell Seal Blood viscosity triples as temperature decreases (38 - 3ºC)

  8. Principle of Continuity A1V1 = A2V2

  9. Principle of Continuity: example Total Area Flow speed Aorta 200 mm2 200 mm s-1 Capillaries 1 mm s-1 5.7 x 104 mm2

  10. Principle of Continuity: example

  11. Continuity in flow fields Narrow streamlines A1V1 = A2V2 The principle of continuity must apply between a pair of streamlines

  12. Pressure exerted in all directions by a fluid at rest Static Pressure Dynamic Pressure Δp U Δp = ½ρU2 • Measurable pressure when fluid is brought to a halt

  13. Bernoulli’s Principle Pressure Drop = ρgh P1+ ½ ρV1 = P2 +½ρV2 Fluid has a local static and dynamic pressure: the sum of these is constant

  14. Bowhead Whale: Continuous filter feeder (Werth 2004, J. Exp. Biol. ) Anterior Opening Posterior Opening

  15. A1 A2 A2 A1 Venturi manometer Showing pressure drop A1 A2 Bernoulli: ΔP = ρu2/(1-A12/A22) Pressure drop (Werth 2004, J. Exp. Biol. )

  16. Difference in dynamic pressure creates drag Momentum = mU Dynamic pressure = ½ ρU2 1.0 Cp = Measured pressure 0.5 ΔP → Drag Cp Dynamic pressure 0 -0.5 • Flow is redirected around the body, flow is robbed of its momentum • Momentum lost in wake (eventually in the form of heat)

  17. Difference in dynamic pressure can also create lift No lift, little drag Lift, Little Drag Stall: Lift & Drag FL = ½ CLSρU2 Fluid separates from surface: vortices & eddies are generated

  18. Vortex shedding caused by flow past a flat plate

  19. Vortex Street st = StU/D Frequency of Vortex Shedding Diameter Strouhal Number(non-dimensional)

  20. Why drag is important 1. Drag = mass x acceleration Thrust = Drag 2. Work against drag = drag x distance Δx 3. Energy expenditure or Power = drag x velocity Δx Δt

  21. Drag: resists motion through fluid D = Fh + Fs + Fw + Fi Total Drag Skin Friction Wave Drag Induced Drag Hydrodynamic or Pressure Drag Thrust Total Drag

  22. Drag: resists motion through fluid D = Fh + Fs + Fw + Fi Total Drag Skin Friction Wave Drag Induced Drag Hydrodynamic or Pressure Drag U S = flow-wise area Fh = ½ CDSρU2 Due to separation & ΔP, E is invested in moving fluid around object, but isn’t being returned back to the fluid

  23. Drag: resists motion through fluid D = Fh + Fs + Fw + Fi Total Drag Skin Friction Wave Drag Induced Drag Hydrodynamic or Pressure Drag U A = “wetted” surface area Fs = ½ CDSρU2 A direct consequence of the interlamellar stickiness of fluid

  24. Skin friction vs. Pressure drag Flow remains attached, drag dominated by skin friction Early separation,high pressure drag relative to skin friction

  25. Drag: resists motion through fluid D = Fh + Fs + Fw + Fi Total Drag Skin Friction Wave Drag Induced Drag Hydrodynamic or Pressure Drag Wave drag occurs near the sea surface

  26. Drag: resists motion through fluid D = Fh + Fs + Fw + Fi Total Drag Skin Friction Wave Drag Induced Drag Hydrodynamic or Pressure Drag Drag Lift Drag generated by the oscillation of appendages

  27. What is the drag coefficient, CD? U Fh = CdS½ρU2 Dynamic pressure Projected Area Drag

  28. D = CdS½ρU2 Drag depends on shape: Dynamic pressure drag Flow-wise projected area 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Drag coefficient (Cd)

  29. CD can get even bigger during unsteady motion CD → 2.0 (Potvin, SLU)

  30. ρLU Reynolds number (Re) = ν ρ = density U = velocity L = length v = viscosity Drag also depends on scale D = CdS½ρU2 Low Re Intermediate Re High Re Cd Re

  31. Life at low Reynolds numbers D = CdS½ρU2 • Viscous • Difficult to produce vortices • Efficiency of locomotion decreases • Many organisms resort to dragging themselves through the medium • High Cd

  32. Drag reduction: Convergence on streamlined form What happens when animals deviate from this ideal form?

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