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Dynamic Forces
Fig. from Warrick et al. 2005. 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 = GF/S = (/t)
Dynamic viscosity
Rate of shearShear modulus
SF
Shear strain
No-slip condition & Boundary layer
FS
U
z(F/S) = (U/z)
Lingulodinium polyedrum
Viscosity
F SU
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
A1V1 = A2V2
Principle of Continuity: example
200 mm s-1
1 mm s-15.7 x 104 mm2
200 mm2
Total Area Flow speedAorta
Capillaries
Principle of Continuity: example
Continuity in flow fields
The principle of continuity must apply between a pair of streamlines
Narrow streamlines
A1V1 = A2V2
Static Pressure
• Pressure exerted in all directions by a fluid at rest
• Measurable pressure when fluid is brought to a halt
Δp = ½ρU2
Dynamic Pressure
Δp
U
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
Bowhead Whale:Continuous filter feeder
(Werth 2004, J. Exp. Biol. )
Anterior Opening
Posterior Opening
Venturi manometerShowing pressure drop
ΔP = ρu2/(1-A12/A2
2)
A2A1
A1A2
A1 A2
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 ρU2
1.0
0.5
0Cp
-0.5
Cp = Measured pressureDynamic pressure
ΔP → Drag
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
Vortex shedding caused by flow past a flat plate
Vortex Street
st = StU/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
Δt
Δx
Δx
Why drag is important
Thrust = Drag
Drag: resists motion through fluid
D = Fh + Fs + Fw + Fi
Total Drag Hydrodynamic or Pressure Drag
Skin Friction Wave Drag Induced Drag
Total DragThrust
Drag: resists motion through fluid
Fh = ½ CDSρU2
D = Fh + Fs + Fw + Fi
Total Drag Hydrodynamic or Pressure Drag
Skin Friction Wave Drag Induced Drag
US = 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
Fs = ½ CDSρU2
D = Fh + Fs + Fw + Fi
Total Drag Hydrodynamic or Pressure Drag
Skin Friction Wave Drag Induced Drag
A = “wetted” surface areaU
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 = Fh + Fs + Fw + Fi
Total Drag Hydrodynamic or Pressure Drag
Skin Friction Wave Drag Induced Drag
Wave drag occurs near the sea surface
Drag: resists motion through fluid
D = Fh + Fs + Fw + Fi
Total Drag Hydrodynamic or Pressure Drag
Skin Friction Wave Drag Induced Drag
Drag generated by the oscillation of appendages
DragLift
What is the drag coefficient, CD?
Fh = CdS½ρU2
Dynamic pressure
ProjectedArea
Drag
U
Drag depends on shape:D = CdS½ρU2
0.2 0.4 0.6 0.8 1.0 1.2 1.4Drag coefficient (Cd)
Dynamic pressure
Flow-wise projected area
drag
CD can get even bigger during unsteady motion
CD → 2.0
(Potvin, SLU)
Drag also depends on scaleReynolds number (Re) =
v = viscosity
νρ = density U = velocityL = length
ρLU
Re
Cd
Low Re Intermediate Re
High Re
D = CdS½ρU2
Life at low Reynolds numbers
• Viscous• Difficult to produce vortices• Efficiency of locomotion
decreases• Many organisms resort to
dragging themselves through the medium
• High Cd
D = CdS½ρU2
Drag reduction: Convergence on streamlined form
What happens when animals deviate from this ideal form?
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