lecture 14 c
TRANSCRIPT
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Weve done fluid statics . Now, F luid Dynamics
(fluid flow), which is much more interesting!COURSE THEME: NEWTONS LAWS OF MOTION !
NOW Sects. 14.5 - 14.7: Methods to analyze the dynamics of
fluids in motion . First, we need to discuss F LUI D L ANGUAGE.
Weve introduced a lot of this language while talkingabout fluid statics. But, there is some other terminology we need to discuss before we discuss Newtons Laws (Especially Newtons 2 nd Law!) inFluid Language!
Section 14.5: Fluid Dynamics
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Two main types of fluid flow:
1. Laminar Flow (or Streamline Flow ) Steady flow Each particle of the fluid follows a smooth path The paths of the different particles never cross each other Every fluid particle arriving at a given point has the same velocity
The path taken by the particles is called a streamline
Types of Fluid Flow
Paths of the particles look qualitatively like this!
Well assume this type of flow
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Two main types of fluid flow:
2. Turbulent Flow Irregular flow which has small whirlpool-like regions Its turbulent flow when the particles go above some critical speed
Streamlines can cross each other
Types of Fluid Flow
Paths of the particles canLook qualitatively like this!
Well not discuss this type.
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Viscosity is a measure of the amount of
internal friction in the fluid.
This internal friction or VI SCOUS F ORCE ,
comes from the resistance that two adjacentlayers of fluid have to moving relative to eachother.
Viscosity causes part of the kinetic energy of afluid to be converted to internal energy.
Viscosity
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We make four simplifying assumptions in our treatment of fluid flow to make the analysis easier:
1. The f luid is nonviscous Internal friction is neglected
2. The f low is steady The velocity of each point remains constant
3. The f luid is incompressible
The density remains constant4. The f low is ir rotational
The fluid has no angular momentum about any point
Ideal Fluid Flow
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Streamlines The path a particle takes in steady flow is
a streamline
The velocity of each particle
is tangent to a streamline
A set of streamlines
is called aTUBE OF F L OW
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Consider a fluid moving through a pipe of
nonuniform diameter. The particles movealong the streamlines in steady flow.
The mass m 1 in the small portionof pipe of length x1, crossingarea A1 in some time t , must beexactly the same as the mass m 2 inlength x2, crossing area A2 in thesame time t .
Why? Because no fluid particles
leak out of the pipe! The fluid has
Conservation of Mass!
Equation of Continuity
m 1 = mass of fluidin this volume
m 2 = mass of fluidin this volume
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Conservation of Mass: m 1 = m 2 (1)
For point 1 & point 2, the definition of density in terms of mass m & volume V gives: m = V.
For points1 & 2, use V = Ax (1) gives
r 1A1v1 = r 2A2v2 (2) Fluid is incompressible so, r = constant
(2) gives: A1v1 = A 2v2 (3) (3) is called the EQUATION OF
CONTINUITY FOR FLUIDS The product of the area and the fluid
speed at all points along a pipe isconstant for an incompressible fluid
Av mass f low rate Units: mass per time interval
or kg/s
Av volume f low rate Units: volume per time interval
or m 3/s
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Mass flow rate (mass of fluid passing a point per second) is constant: 1A1v1 = 2A2v2
Equation of Continuity PHYSICS : Conservation of Mass!!
For an incompressible fluid (1 = 2 = )
Then A1v1 = A 2v2 Or: Av = constant
Where cross sectional area A is large, velocityv is small, where A is small, v is large.
Volume flow rate: (V/ t) = A( x/ t) = Av
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Implications of Equation of ContinuityA1v1 = A 2v2
The fluid speed v is low wherethe pipe is wide (large A) The fluid speed v is high wherethe pipe is constricted (small A)
The product, Av , is called thevolume f low rate or f lux .Av = constant says that the
volume that enters one end of the pipe in a given time equals thevolume leaving the other end in thesame time (If no leaks are present!)
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PHYSICS : Conservation of Mass!!
A1v1 = A 2v2 Or Av = constant
Smal l pipe cross section larger v
L arge pipe cross section smaller v
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Example: Estimate Blood Flowr cap = 4 10 -4 cm, r aorta = 1.2 cm
v1 = 40 cm/s, v 2 = 5 10 -4 cm/s Number of capillaries N = ?A2 = N (r cap )2, A 1 = (r aorta )2
A1v1 = A 2v2N = (v
1/v
2)[(r
aorta)2/(r
cap)2]
N 7 10 9
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Example: Heating DuctSpeed in duct:
v1 = 3 m/sRoom volume:
V2 = 300 m 3
Fills room everyt =15 min = 900 sA1 = ?
A1v1 = Volume flow rate = ( V/ t) = V 2/t
A1 = 0.11 m 2