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Biology 427 BiomechanicsLecture 16. Basic fluid dynamics II: continuity and Bernoulli’s principle.•Recap definition of a fluid and Newton’s law of viscosity and a coevolutionary story.
•Conservation of mass and continuity•An application of continuity•Conservation of energy and Bernoulli’s principle•Applications of Bernoulli’s principle
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2 rules govern flow•Newton’s Law of viscosity
•No slip condition (fluid adheres to solid surfaces)
τ = µ du/dy = ν d(ρ u)/dy
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Blood viscosity depends non-linearly on red blood cell concentration ( hematocrit) and Temperature
5
4
3
2
10.1 0.2 0.3 0.4 0.5 0.6
0 10
20
3040
Relativebloodviscosity(relative to water)
O2 carrying capacity
Flow rate
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2 more rules govern flow-- and applications of those rules to biology:•Conservation of mass -- the Principle of Continuity•Conservation of energy -- Bernoulli’s Principle
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Volumein/timeuin Ain
Volumeout/timeuout Aout
Conservation of mass -- the “principle of continuity”: rate of mass flow in must equal rate of mass flow out.
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Volumein/timeuin Ain
Conservation of mass -- the “principle of continuity”: rate of mass flow in must equal rate of mass flow out.
Σuin Ain = Σ uout Aout
Aout
Aout
Ain
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Area 1 cm2 Blood velocity 10 cm/s
Aorta
Capillary Area 2 10-7 cm2
Blood velocity 0.1 cm/s
How many capillaries do we have?
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Conservation of Energy: what is the relationship between fluid motion and pressure?
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Conservation of Energy: what is the relationship between fluid motion and pressure?
Like mass, energy cannot appear or disappear unaccountably -- it is conserved. We will follow this along a streamline.
3 assumptions:flow is steady (no acceleration at a point in space, du/dt = 0fluid is incompressible (ρ = constant)flow is “inviscid” (µ= 0)
u1,h1,P1u2,h2,P2
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Conservation of Energy: what is the relationship between fluid motion and pressure?
Along a stream line we consider three forms of energyPotential energy (PE = m g h)Kinetic energy (KE = m u2/2)Mechanical work (W = F d = P Area d = P V = P m/ ρ)
Along a streamline PE + KE + W = constant
u1,h1,P1u2,h2,P2
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Conservation of Energy: what is the relationship between fluid motion and pressure?
Along a stream line we consider three forms of energyPotential energy (PE = g h)Kinetic energy (KE = u2/2)Mechanical work (W = F d = P Area d = P V = P / ρ)
Along a streamline (PE + KE + W)/m = constant
u1,h1,P1u2,h2,P2
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Conservation of Energy: what is the relationship between fluid motion and pressure?
Along a streamline (PE + KE + W)/m = constant
u1,h1,P1u2,h2,P2
P2/ ρ + gh2 + u22/2 = P1/ ρ + gh1 + u1
2/2 (P2 - P1) /ρ + g(h2 -h1) + (u2
2- u12 )/2 = 0
(P2 - P1) /ρ + (u22- u1
2 )/2 = 0
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Conservation of Energy: what is the relationship between fluid motion and pressure? (P2 - P1) /ρ + (u2
2- u12 )/2 = 0
Why is fluid faster over the top?
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Conservation of Energy: what is the relationship between fluid motion and pressure? (P2 - P1) /ρ + (u2
2- u12 )/2 = 0
u1 = 0.2 m/s ; u2 = 3 m/s What is (P2 - P1) ?
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Conservation of Energy: what is the relationship between fluid motion and pressure? (P2 - P1) /ρ + (u2
2- u12 )/2 = 0
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Conservation of Energy: what is the relationship between fluid motion and pressure? (P2 - P1) /ρ + (u2
2- u12 )/2 = 0
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Conservation of Energy: what is the relationship between fluid motion and pressure? (P2 - P1) /ρ + (u2
2- u12 )/2 = 0
Is the direction drawn here correct?
http://www.asknature.org/strategy/e27b89ebcdec8c9b5b2cd9ac84b8f8a0
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Area 1 cm2 Blood velocity 10 cm/s
Aorta
Capillary Area 2 10-7 cm2
Blood velocity 0.1 cm/s
How many capillaries?
u1 = 0.2 m/s ; u2 = 3 m/s What is (P2 - P1) ?
Worksheet