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?