BIOMECHANICS of LOCOMOTION through FLUIDS€¦ · BIOMECHANICS of LOCOMOTION through FLUIDS ... (at...

BIOP2203 Biomechanics S1

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BIOMECHANICS of LOCOMOTION through FLUIDS

Questions: - Explain the biomechanics of different modes of locomotion through fluids

(undulation, rowing, hydrofoils, jet propulsion). - How does size influence the mode and speed of swimming ? - What determines the energy cost of swimming and how does it compare to

running and flying ? FluidMechanics: - Fluid density, hydrostatic pressure, buoyancy, Bernoulli principle, viscosity. - Boundary layer, laminar and turbulent flow, Reynolds number, hydrodynamic

drag and lift. §3.1 FLUID STATICS Motion through gases and liquids such as air and water for animal locomotion. Density of fluid

- Mass per unit volume ( kg / m-3) Vm

=ρ

Water ρ = 1000 kg m-3 Seawater ρ = 1026 kg m-3 (due to dissolved minerals) Air ρ = 1.21 kg m-3 (at sea level and 20°C) Pressure in fluid

- Force per unit area (N m-2 or Pa ) AFP =


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Hydrostatic pressure: - Increase with depth in fluid (The fluid must support its own weight) hgpp o ρ+= where po = pressure at surface (atmospheric pressure) . h = depth (m). - Pressure in water increases (1 atmosphere per 10.3 m increase in depth). Atmospheric pressure: - Atmospheric pressure decreases with altitude. yb

aepp −= where y = altitude (m) pa = pressure at sea level (1.013 × 105 N m-2) . b = coefficient (1.16 × 10-4 m-1) .

Buoyant force: - A body partially or fully immersed in a fluid experiences an upward buoyant

force. - The magnitude of the force is equal to weight of the fluid displaced by the

body (Archimedes' principle). - Weight acts at the centre of gravity. - The buoyant force acts at the geometrical centre of body immersed in fluid

(centre of buoyancy).


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§3.2 FLUID DYNAMICS Steady flow: - Velocity, density and pressure at each point do not vary with time. Equation of continuity: - Assume an ideal fluid (incompressible, non-viscous and not turbulent). - Fluid flow through a pipe of uniform size. Av = constant where A = cross sectional area of pipe (m2). v = speed of fluid (m/s). - The product of the area and fluid speed at all points along the pipe is constant

for an incompressible fluid. - If the fluid is compressible (ρ ≠ constant) ρ A v = constant

Example: Fluid flows through a rigid tube with no leaks. If the radius increases by 20%

what is the change in the fluid speed v?

Since v ∝ A-1 ∝ (radius)-2 2

2

1

2

1

1

2rr

AA

vv

⎟⎟⎠

⎞⎜⎜⎝

⎛==

so r2 = 1.20 r1 ⇒ =⎟⎠⎞

⎜⎝⎛=

2

1

220.11

vv 0.694

speed decreases by 31%


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Bernoulli equation:

The conservation of mass and energy applied to a fluid.

=ρ+ρ+ ygv21p 2 constant

where y = height of fluid (m). - The work done on a fluid per unit volume (W = F ∆x = p ∆V) is equal to the

changes in kinetic energy (KE) and potential energy (PE) per unit volume. - The sum of the pressure (p), the kinetic energy per unit volume (½ ρv2) and the

gravitational potential energy per unit volume (ρgy) as the same value at all points along a streamline.

- The fluid in the section (∆x1) moves to the section of length (∆x2). The volume

of fluid in the two sections are equal.

- In horizontal flow (y = constant) =ρ+ 2v21p constant

- Pressure is less where velocity is high. - Pressure of an incompressible fluid can be measured with Venturi tubes.


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Flow around an asymmetrical body:

- The Bernoulli effect states that the increased velocity of the air above the wing

compared to the air beneath the wing causes the decreased pressure above the wing.

- This is NOT the primary effect producing lift on wings. Viscosity:

- Property of a real fluid due to inertial friction. - Consider the force between two plates (area A, separation d, relative velocity

of plates, v). - The force required to overcome the viscosity of the fluid:

d

vAFviscousµ

=

Where µ = coefficient of viscosity (Pa.s, temperature dependent). Air (20°C) µ = 1.82 × 10-5 Pa.s Water (20°C) µ = 1.00 × 10-3 Pa.s Example: A square glass plate (density 2800 kg m-3) 1.0 mm thick slides on a 1.0 mm

film of water across a surface held at 2.0o to the horizontal. What is the maximum (terminal) speed?

Let w be the thickness of the glass, ρ its density:

( )d

vAsingm µ=θ ⇒ ( ) ( )

µθρ

=µ

θ=

sindgwAsindgmv = 0.96 m/s


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The Motion of a Body through a Fluid - Is equivalent to the motion of a fluid around a body. - Drag force acts to oppose the motion of a body through a fluid. - The drag force depends on: - The velocity of the body relative to the fluid. - The properties of the fluid. - The size and shape of the body. Boundary layer: - Viscous fluid interacts with surface of body, sticks to surface forming a very

thin 'boundary layer' that is carried along with the body. - Velocity of fluid gradually diminishes with distance from the body (velocity

gradient). - Forms retarding drag force, called viscous drag (also called friction drag,

surface drag).

Types of Fluid Flow The types of fluid flow depends on: - The size, shape and roughness of the object. - The viscosity of the fluid and the fluid velocity. Fluid flow around a body is characterised by the Reynolds number,

µ

ρ=

lvRe

Where v = relative velocity of body and fluid (m/s) l = geometric length perpendicular to the flow (m) ρ = density of fluid (kg m-3) µ = coefficient of viscosity of fluid (Pa.s) small and slow → low Re large and fast → high Re


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Flow type depends on Reynolds number: Re < 1 laminar flow (viscous drag only). Re ≈ 1 transition to partially turbulent flow. 1 < Re < 103 turbulent wake grows (Viscous and pressure drag). 103 < Re < 106 turbulent wake grows (Pressure drag dominates). Re ≈ 106 transition to fully turbulent flow (pressure drag decreases). Re > 106 fully turbulent flow (pressure drag dominates). Laminar (Stokes flow):

- Fluid moves around body in uniform layers of differing speeds (boundary layer). - Viscous drag force exerted on sides of body due to viscosity of fluid. vFv µκ= l Where v = relative velocity of body and fluid (m/s). l = characteristic length perpendicular to the flow (m). µ = coefficient of viscosity of fluid (Pa.s) (Measure of resistance of fluid to flow). κ = constant, depends on shape of body. (κ = 3π for sphere). Note that Fv ∝ v, independent of fluid density and area of body. Example: What is the terminal speed of a glass sphere of 2.0 mm diameter falling

through glycerol (µ ≈ 1.0 Pa.s)? Let r be the sphere's radius ρ its density, then l = 2r.

The drag force: vr6Fv µπ= must equal the weight gr34gmF 3

g ρπ==

Equating ⇒ µ

ρ=

9rg2v

2 = 6.1 mm/s


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Partially turbulent:

- The fluid is unable to follow surface contours and the boundary layer separates

from surface. - Bernoulli effect, wake forms (region of turbulence and low pressure behind

object). ⇒ net pressure on the front of the object. - Pressure drag is given by

2Dp vCS

21F ρ=

Where ρ = density of fluid (kg m-3). S = characteristic area perpendicular to flow (m2). CD = drag coefficient (depends on shape of body and Reynolds number). Note that Fp ∝ v2, again independent of fluid density and area of body. As a result, pressure drag usually dominates viscous drag. Example: A basketball of 76 cm diameter with a mass of 0.60 kg falls from a plane.

What is its terminal velocity if the drag coefficient is 1.0?

2D

2 vCr21gm πρ= ⇒

D2 Cr

gm2vπρ

= = 4.6 m/s


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Fully turbulent flow:

- Boundary layer becomes turbulent. - Reduces the tendency of the boundary layer to separate from body. - Separation point moves forward. ⇒ sharp decrease in drag force, then continues to increase. - The size of the wake decreases. - Characteristic areas: Sw = wetted area (total surface area of body) (m2) Sf = frontal (cross sectional area) (m2). Sp = planar area (hydrofoils and aerofoils) (m2). Drag Coefficient Wind tunnel measurements can empirically determine drag forces and coefficients.


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Drag coefficient vs Reynolds number (bluff bodies):

When the drag is dominated by viscous drag, we say the body is streamlined, and when it is dominated by pressure drag, we say the body is a bluff body.

- CD approximately constant over a wide range of Re (i.e. velocity) (Re ≈ 103→105) - Minimum value of CD at Re ≈ 2 × 105 to 2 × 106. - Abrupt drop in drag at critical velocity (change to turbulent boundary layer

makes wake smaller and reduces pressure drag). ⇒ net drag decreases at onset ofturbulence. For Re = 106 and: l = 0.1 m; v= 10 m/s in water and 150 m/s in air. l = 1.0 m; v= 1 m/s in water and 15 m/s in air. Streamlined body:

- Rounded at front, tapers gradually to point at back. - Low drag coefficient (CD ≈ 0.05) (mostly surface friction). - Designed to reduced turbulence in wake. Surface roughness: - Shifts transition of partially turbulent to fully turbulent to lower Reynolds

number (lower velocity). - Lower drag, CD. - e.g. Fuzzy tennis balls, dimples on golf balls.


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Surface Friction - Surface friction contributes to total drag. - Significant for streamlined bodies and flat surfaces (i.e. hydrofoils and

aerofoils).

2fwfriction vCS

21F ρ=

Where Cf = surface friction coefficient. Sw = wetted surface area (m2).

- For laminar flow in boundary layer (Re < 106): 2/1f Re33.1C =

- For turbulent flow in boundary layer (Re > 106): 5/1f Re075.0C =

Bluff bodies: - At onset of the turbulent boundary layer, increased friction drag, but reduced wake ⇒ reduced pressure drag. Hydrofoils and Aerofoils - Produce lift when moving through the water/air from the asymmetrical motion

of the body through the fluid.


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Hydrodynamic forces: - Hydrodynamic force acts at angle to direction of motion of body. - Resolve force on body into components: Drag Lift

2DpD vCS

21F ρ= 2

LpL vCS21F ρ=

where: CD = drag coefficient CL = lift coefficient backwards along direction of motion perpendicular to direction of motion Lift and drag depend on angle of attack, α Drag:- FD increases with increasing α FD ≠ 0 at α = 0°. Lift:- FL ≈ 0 for α = 0°. FL reaches maximum, then decreases rapidly (stalling).

Aspect ratio:

- Aspect ratio, chordspanA =

- For foils with same planar area ( Sp = span.chord ), high aspect ratio give the same lift for less drag.

Wind tunnel testing: - Hydrofoils and aerofoils of same shape and angleof attack produce same lift at

same Reynolds number. (Re = ρvl/µ, l is foil chord) ⇒ Test scale model of aerofoil in wind tunnel with appropriate wind speed.


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§3.3 BUOYANCY Animal density: - Swimming animals without special adaptations for buoyancy are more dense

than water. Most fish ρ ≈ 1080 kg m-3 Water ρ = 1000 kg m-3 Seawater ρ = 1026 kg m-3 Muscle ρ = 1060 kg m-3 Bone ρ = 2000 kg m-3 Dense animals avoid sinking by - Swimming upwards (e.g. plankton). - Swimming horizontally with fins at +ve angle of attack to produce lift. (must exceed minimum swimming speed to stay afloat)

Low density animals: - Part of animal consists of low density material. → Animal density ≈ water

density. - Fat, blubber (ρ ≈ 930 kg m-3) (e.g. seals, whales) - Wax esters (ρ ≈ 860 kg m-3). - Low density body fluids (ion depleted fluids) (e.g. deep sea squids). - Gas (ρ ≈ 0) - Air-filled lungs (swimming mammals and reptiles). - Swim bladder (many bony fish). - Gas-filled floats (cuttlefish, nautilus). Swimming animals: - Buoyancy of animal essentially negates effect of gravity. - Swim by - Undulation (fish, eels, spermatozoon) - Rowing (water beetle, duck). - Hydrofoils (dolphin, penguin). - Jet propulsion (squid).


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§3.4 SWIMMING by UNDULATION

Swim with wave-like motion. - Waves travel backwards along the body. - Pushes the organism forward.


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High Reynolds Number High Reynolds number (Re= ρvl/µ>>1) - Large and fast. - Movement through the water controlled by inertia (Neglect viscosity of surrounding water). - If the animal stops undulating,forward motion Example: - Fish, eels v ≈ 12 cm/s, l ≈ 15 cm µwater ≈ 1 × 10-3 Pa.s ρwater ≈ 1 × 103 kg m-3 ⇒ Re ≈ 1.8 × 104 ( >>1 ) Swimming (acceleration) - Fish of mass m accelerates to velocity v by driving mass of water M

backwards at velocity V. - Fish required to do work W to accelerate (from rest). - Fish will attain a higher velocity for the same work if it pushes a larger mass of

water backwards.

⎟⎠⎞

⎜⎝⎛ +=+=

Mm1vm

21VM

21vm

21W 222

Swimming (constant velocity):

- Tail mostly moves water sideways (transversely) (The net momentum of the water in the transverse direction = 0). - Fish with tail area Atail swimming at velocity vswim - Water is given a sideways velocity vwater


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Power output of the tail:

tailtail vFP = where t

pF watertail =

momentum of water: pwater = Mwater vwater

waterswimtailwaterwater

tail vvAtvMF ρ==

⇒ ( ) tailwaterswimtail vvvAP ρ= Metabolic Energy Cost

Metabolic energy cost of transport (swimming) - Determined from oxygen consumption (subsurface swimming of fish). Energy cost of transport (J/kg.m) depends on - Swimming speed (vswim) - Body size (The cost of transport decreases with increasing mass). - Environmental temperature. (cost of transport is lower at higher temperatures). Net cost of transport decreases with increasing size.


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Low Reynolds number (Re = ρvl/µ < 1) - Small and slow. - Movement through the water controlled by the viscosity of the fluid µ (Neglect inertia of surrounding water). - If the animal stops undulating forward motion stops (almost) instantaneously. Examples: - Spermatozoon - Flagellates v ≈ 100 µm/s, l ≈ 60 µm Wave velocity ≈ 12 µm/s µwater ≈ 1 × 10-3 Pa.s ≈ 20 body length/s ρwater ≈ 1 × 103 kg m-3 ⇒ Re ≈ 0.006 ( < 1 ) Wave frequency ≈ 50 Hz


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Forces generated during undulation of tail: - Viscous drag force on sperm tail (pressure drag is negligible) vFv µκ= l for Re < 1 - Model sperm tail as a cylindrical rod. Power required for swimming with a flagellum: - Consider only work done to overcome drag on tail. - Longitudinal motion of flagellum: P = Fdrag v = κa µlν2 ≈ µlv2 (κa ≈ 1 for a long slender rod). - Side to side motion of the flagellum: side to side velocity >> forward velocity P ≈ 50 µlν2 >> longitudinal motion power. §3.5 ROWING Rowing underwater or on the surface of the water. - Use drag on oars to provide forward thrust. Water beetle has middle and hind legs with hinged hair-like bristles. - Spread for the power stroke. - Trail for the recovery stroke.


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Mechanics of rowing a boat on surface of the water: - Oars used to drive mass of water backwards. - Produces a wake of forward moving water behind boat (Water in the boundary

layer is dragged along by the hull). - KE left behind in wake supplied by work of rowing. - Streamlined hull is designed to reduce wake.

Rowing at constant velocity: rate of transmission rate of transmission of momentum = of forward momentum to water by oars to water in wake by hull - Oars with large blades are the most efficient. - Less power is used to accelerate a large mass of water at low speed than

accelerate a small mass of water at high speed. Surface Waves (bow and stern waves): - Water is given PE when raised in a wave since energy must be imparted from

an external force, i.e. muscles. - Additional drag on body (wave drag). - Limits the speed of swimming on the surface of the water. Gravity is important in dynamics of water waves. - For dynamically similar wave patterns, body must travel with equal Froude

number (Fr = v2/gl), where l = hull length). Power required becomes large when Fr ≈ 0.16, i.e. at lg16.0v = For duck: - Webbed feet are spread during power stroke when rowing. - Hull length l = 0.33 m. - Maximum speed of swimming (Fr ≈0.16), vmax = 0.7 m/s.


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§3.6 HYDROFOILS Types of hydrofoils - Wings (penguins). - Flippers (turtles). - Flukes (whales, dolphins). - Tails (tuna). Use lift on hydrofoil to generate thrust. - Mainly large and fast animals (Re >> 1). Example: Penguin (swims by beating its wings).

Up / down component cancels over one complete cycle. ⇒ COG moves forward only.

upstroke downstroke

-ve angle of attack +ve angle of attack R forward and down R forward and up

Upstroke different sign of angle of attack to flying. - Not required to generate vertical force to overcome gravity. - Only require forward force for propulsion.


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Hydrofoils: - Greatest lift (for same drag and area) for high aspect ratio ( A = span/chord). ⇒ Long and narrow tails. - Some fish use tails as vertical hydrofoils.

- Most fish use a combination of undulation and hydrofoil motion for

locomotion. - Dolphins and whales use tails as horizontal hydrofoils.

Leaping dolphins and penguins: - Less drag in air than water. - Avoids the high drag at surface (bow wave) when breathing. §3.7 JET PROPULSION Locomotion is performed by squirting

water out of a cavity in their body.

- Not a steady velocity (series of jerks).

Examples - Squids - Jellyfish - Scallops (Open and close their hinged shells).


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BIOMECHANICS of LOCOMOTION through FLUIDS€¦ · BIOMECHANICS of LOCOMOTION through FLUIDS ... (at...

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