Dynamic ForcesFig. from Warrick et al. 2005. Nature

Solids vs. FluidsSolids resist shear deformation: they care how far they are deformedFluids care about how rapidly they are deformed: rate of shearF/S = GF/S = (/t)Dynamic viscosityRate of shearShear modulusSShear strain

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

ViscosityFSUzF/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 5C)

Temperature effects on viscosityQ = CJT* More viscous blood should flow more slowly* Slower mass flow can minimize heat loss

Temperature effects on viscosityLeopard SealWeddell SealCrabeater SealBlood viscosity triples as temperature decreases (38 - 3C)

Principle of ContinuityA1V1 = A2V2

Principle of Continuity: example200 mm s-11 mm s-15.7 x 104 mm2200 mm2Total AreaFlow speedAortaCapillaries

Principle of Continuity: example

Continuity in flow fieldsThe principle of continuity must apply between a pair of streamlinesNarrow streamlinesA1V1 = A2V2

Static PressurePressure exerted in all directions by a fluid at restMeasurable pressure when fluid is brought to a haltp = U2Dynamic PressurepU

Bernoullis PrinciplePressure Drop = ghP1+ V1 = P2 + V2Fluid has a local static and dynamic pressure: the sum of these is constant

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

Venturi manometerShowing pressure dropP = u2/(1-A12/A22)A2A1A1A2A1A2Bernoulli:Pressure drop(Werth 2004, J. Exp. Biol. )

Difference in dynamic pressure creates drag

Flow is redirected around the body, flow is robbed of its momentumMomentum lost in wake (eventually in the form of heat)

Momentum = mUDynamic pressure = U21.00.50Cp-0.5Cp = Measured pressureDynamic pressureP Drag

No lift, little dragDifference in dynamic pressure can also create liftLift, Little DragStall: Lift & DragFL = CLSU2Fluid separates from surface: vortices & eddies are generated

Vortex shedding caused by flow past a flat plate

Vortex Streetst = StU/DFrequency of Vortex SheddingStrouhal Number(non-dimensional)Diameter

1. Drag = mass x acceleration3. Energy expenditure or Power = drag x velocity2. Work against drag = drag x distancetxxWhy drag is importantThrust = Drag

Drag: resists motion through fluidD = Fh + Fs + Fw + FiTotal DragHydrodynamic or Pressure DragSkin FrictionWave DragInduced DragTotal DragThrust

Drag: resists motion through fluidFh = CDSU2D = Fh + Fs + Fw + FiTotal DragHydrodynamic or Pressure DragSkin FrictionWave DragInduced DragUS = flow-wise areaDue to separation & P, E is invested in moving fluid around object, but isnt being returned back to the fluid

Drag: resists motion through fluidFs = CDSU2D = Fh + Fs + Fw + FiTotal DragHydrodynamic or Pressure DragSkin FrictionWave DragInduced DragA = wetted surface areaUA direct consequence of the interlamellar stickiness of fluid

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

Drag: resists motion through fluidD = Fh + Fs + Fw + FiTotal DragHydrodynamic or Pressure DragSkin FrictionWave DragInduced DragWave drag occurs near the sea surface

Drag: resists motion through fluidD = Fh + Fs + Fw + FiTotal DragHydrodynamic or Pressure DragSkin FrictionWave DragInduced DragDrag generated by the oscillation of appendagesDragLift

What is the drag coefficient, CD?Fh = CdSU2Dynamic pressureProjectedAreaDragU

Drag depends on shape:D = CdSU20.20.40.60.81.01.21.4Drag coefficient (Cd)Dynamic pressureFlow-wise projected areadrag

CD can get even bigger during unsteady motionCD 2.0(Potvin, SLU)

Drag also depends on scaleReCdLow ReIntermediate ReHigh ReD = CdSU2

Life at low Reynolds numbersViscousDifficult to produce vorticesEfficiency of locomotion decreasesMany organisms resort to dragging themselves through the mediumHigh Cd

D = CdSU2

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

Fluid sticks to itself. If a tiny bit of fluid moves, it tends to bring other bits of fluid with it. The magnitude of that tendency is viscosity. It determines how rapidly it is deformed from a given F/A.Fluids also stick to solids just as well. In fact, the velocity of the fluid at the solid/fluid interface is exactly the velocity of the solid.

No Slip condition. Boundary LayerKinematic Viscosity = practical gooiness of a fluid how easily it flows, how likely it is to for vortices, how steep its velocity gradient will beDynamic Viscosity = interlamellar stickiness of the fluid, how likely a parcel of fluid is influenced by movement of particles nearbyIncuding density with kinematic viscosity accounts loosely for the inertia of a given parcel of fluidPrinciple of continuity -> as cross-sectional area decreases, flow velocity Increases and pressure decreasesHydraulic & hydrodynamic pressureImprove filtering efficiencyAnd not create an anterior compressive bow wave that may push prey awayrapid viscous dissipation of kinetic energy imparted to the surrounding fluid efficiency of flapping locomotion declines because bound circulation decreases and viscous drag increases, resulting in impractical lift-to-drag ratios