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

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


Fh = ½ CDSρU2




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


Fs = ½ CDSρU2




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





Wave drag occurs near the sea surface





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?

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

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