Dynamic Forces Fig. from Warrick et al Nature

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

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

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

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

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

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

Principle of Continuity A 1 V 1 = A 2 V 2

Principle of Continuity: example 200 mm s -1 1 mm s x 10 4 mm mm 2 Total AreaFlow speed Aorta Capillaries

Principle of Continuity: example

Continuity in flow fields The principle of continuity must apply between a pair of streamlines Narrow streamlines A 1 V 1 = A 2 V 2

Static Pressure Pressure exerted in all directions by a fluid at rest Measurable pressure when fluid is brought to a halt Δp = ½ρU 2 Dynamic Pressure ΔpΔp U

Bernoulli’s Principle Pressure Drop = ρgh P 1 + ½ ρV 1 = P 2 + ½ρV 2 Fluid has a local static and dynamic pressure: the sum of these is constant

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

Venturi manometer Showing pressure drop ΔP = ρu 2 /(1-A 1 2 /A 2 2 ) A2A2 A1A1 A1A1 A2A2 A1A1 A2A2 Bernoulli: Pressure drop (Werth 2004, J. Exp. Biol. )

Difference in dynamic pressure creates drag Flow is redirected around the body, flow is robbed of its momentum Momentum lost in wake (eventually in the form of heat) Momentum = mU Dynamic pressure = ½ ρU CpCp -0.5 C p = Measured pressure Dynamic pressure ΔP → Drag

Difference in dynamic pressure can also create lift No lift, little dragLift, Little DragStall: Lift & Drag F L = ½ C L SρU 2 Fluid separates from surface: vortices & eddies are generated

Vortex shedding caused by flow past a flat plate

Vortex Street  st = S t U/D Frequency of Vortex Shedding Strouhal Number (non-dimensional) Diameter

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

Drag: resists motion through fluid D = F h + F s + F w + F i Total Drag Hydrodynamic or Pressure Drag Skin FrictionWave DragInduced Drag Total DragThrust

Drag: resists motion through fluid F h = ½ C D SρU 2 D = F h + F s + F w + F i Total Drag Hydrodynamic or Pressure Drag Skin FrictionWave DragInduced Drag U S = flow-wise area Due to separation & ΔP, E is invested in moving fluid around object, but isn’t being returned back to the fluid

Drag: resists motion through fluid F s = ½ C D SρU 2 D = F h + F s + F w + F i Total Drag Hydrodynamic or Pressure Drag Skin FrictionWave DragInduced Drag A = “wetted” surface area U A direct consequence of the interlamellar stickiness of fluid

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

Drag: resists motion through fluid D = F h + F s + F w + F i Total Drag Hydrodynamic or Pressure Drag Skin FrictionWave DragInduced Drag Wave drag occurs near the sea surface

Drag: resists motion through fluid D = F h + F s + F w + F i Total Drag Hydrodynamic or Pressure Drag Skin FrictionWave DragInduced Drag Drag generated by the oscillation of appendages Drag Lift

What is the drag coefficient, C D ? F h = C d S½ρU 2 Dynamic pressure Projected Area Drag U

Drag depends on shape: D = C d S½ρU Drag coefficient (C d ) Dynamic pressure Flow-wise projected area drag

C D can get even bigger during unsteady motion C D → 2.0 (Potvin, SLU)

Drag also depends on scale Reynolds number (Re) = v = viscosity ν ρ = densityU = velocity L = length ρLU Re CdCd Low ReIntermediate Re High Re D = C d S½ρU 2

Life at low Reynolds numbers Viscous Difficult to produce vortices Efficiency of locomotion decreases Many organisms resort to dragging themselves through the medium High C d D = C d S½ρU 2

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