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```Physics 221, March 15
Key Concepts:
•Viscosity
•Turbulence
•Surface tension
Viscosity
Ideal fluids flow without friction and do not dissipate ordered energy.
Real fluids convert ordered energy into disordered energy when flowing.
Viscosity is a measure of the internal friction of a flowing fluid. It is a result of
intermolecular forces.
Ideal flow is always laminar. Viscous flow can be laminar or turbulent.
Fluid density, flow speed, viscosity, and geometry of the flow determine when
viscous flow transitions fro laminar to turbulent flow. A single number, called the
Reynolds number, can be computed for any viscous flow.
R = ρfluidDv/η
Here ρfluid is the density of the fluid, v is its flow speed, η is its viscosity, and D is a
characteristic length that depends on the geometry.
When the Reynolds number R becomes greater than ~2300, the flow becomes
turbulent.
Laminar viscous flow
Definition: η = viscosity (Pa-s)
How can you measure it?
Stokes’ law:
F = 6πηrspherev
Force required to move a sphere through
a viscous fluid with velocity v.
Falling sphere:
vterminal = 2(ρsphere - ρfluid)(rsphere )2g)/(9 η)
Consequences?
Poiseuille’s law gives rate at which a fluid
flows through a hose or a pipe.
Q = π∆Pr4/(8ηL)
Q = volume flow rate
∆P = pressure difference across L of the pipe
r = radius of the pipe
A thin boundary layer of fluid next to
a walls does not move at all.
Why can't you get all the dust off your car by just squirting water from a hose
onto it?
Why can't you simply remove dust just be blowing across a surface?
Why does dust cling to a fast rotating fan?
How can a leaf stay on a car moving at high speed?
What is one common answer to all these questions?
1.
2.
3.
4.
This is a consequence of
Bernoulli’s equation.
This is a consequence of the
formation of stationary
boundary layer.
This is a consequence of
Poiseuille’s law.
This is a consequence of
Stokes’ law.
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Suppose you work in the E.R. and a choking patient comes in with their
trachea half obstructed. (The diameter is half of what it was before the
obstruction.) You manage to correct the problem and send them on their way
safely, explaining that such obstructions are particularly dangerous because
1.
2.
3.
4.
5.
a half – blocked airway passes only
1/4 the regular amount of air.
a half – blocked airway passes only
1/16 the regular amount of air.
a half – blocked airway passes only
1/2 the regular amount of air.
a half – blocked airway passes only
1/32 the regular amount of air.
a half – blocked airway passes only
1/8 the regular amount of air.
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Extra credit:
Two marbles of radii r and √r fall through a high-viscosity fluid.
What is the ratio of their terminal velocities?
1.
2.
3.
4.
5.
1:1
r:1
2:1
1:2
√r :1
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Newtonian and non-Newtonian Fluids
If the viscosity of a fluid depends only on temperature, the fluid
is said to be Newtonian.
If the viscosity depends on flow speed, shear stress, etc, the
fluid is said to be non-Newtonian.
Examples:
Shear thinning materials:
paint, blood, ketchup
(Viscosity decreases with increased stress)
Shear thickening material:
(Viscosity increases with increased stress)
cornstarch dissolved in water
Turbulence
As the flow rate increases, a moving fluid can no longer sustain laminar flow.
Turbulence appears when the Reynolds number is about 2300.
For flow through a smooth cylindrical pipe D is the inner diameter of the pipe
Consequences of turbulence:
Ordered energy is “wasted” (converted into disordered energy.)
Bernoulli’s equation not valid.
Poiseuille’s law not valid, (Q is not proportional to ΔP.)
The equation of continuity still valid for incompressible fluids.
So for flow through a smooth cylindrical pipe
R = ρfluidDv/η = ρfluidDvA/(ηA) = ρfluidDQ/(η π D2/4) = 4 ρfluidQ/(η π D) ∝ Q/D.
A water pump generates a constant flow Q, which is causing a turbulent spray
(Reynolds number R = 3200) out of this cylindrical nozzle:
Which of the following three nozzles could be substituted to
achieve a streamline flow (R < 2000)?
1.
2.
3.
4.
Tube I
Tube II
Tube III
Tube I and III
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Compare the flight of two golf balls given the same impulse by the club. Which ball
will travel a larger distance?
1.
2.
3.
The smooth ball
The dimpled ball
They will travel the same distance
since they have the same initial
velocity.
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Surface tension
Cohesion, or attraction between
molecules of the same kind leads to
surface tension.
The surface tension γ is defined as the
force along a line of unit length.
γ = F/L (N/m)
Laplace's law for a spherical membrane in
equilibrium: Pin - Pout = 2γ/r
The smaller a soap
bubble, the higher
is the pressure
inside.
A soap bubble with radius of 3.0 cm is floating in air. What is the
pressure difference across the two surfaces of the bubble?
The surface tension of soapy water is 69 * 10-3 N/m.
1.
2.
3.
4.
9.2 N/m
2.6 N/m
8.3*10-3 N/m
9.2*10-2 N/m
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Extra credit:
Two bubbles are connected by a hollow tube plus a valve. What will happen
once the valve between the two is opened?
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2.
3.
The small bubble will drain into
the big bubble.
The big bubble will empty into
the small bubble until the two
are equal in size.
Nothing will happen – the
bubbles will stay the same size.
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Capillary action
• Adhesion is the attraction between
unlike molecules.
• Adhesive and cohesive forces
determine the contact angle between
a liquid and a solid surface.
• Adhesion and surface tension together
can produce capillary action.
• The maximum height to which a liquid
will rise through capillary action is
given by
h = 2γcosθ/(ρgr).
– [mg = (F/L) cosθ 2πr (net force = 0)
ρπr2h g = γ cosθ 2πr (solve for h)]
In an experiment water rose by capillary attraction through two
columns of soils, one with coarse grains (column 1) and the other
with fine gains (column 2).
In which column did the water rise faster and in which column
did it rise to greater height?
1. faster:
greater height:
2. faster:
greater height:
3. faster:
greater height:
4. faster:
greater height:
column 1
column 1
column 2
column 2
column 1
column 2
column 2
column 1
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Extra credit:
Blood and water have the same contact angle with glass. The density of blood
is 5% higher than that of water. (ρblood = 1.05 ρwater)
While you make your capillary action measurements you find that water
rises in the same tube a factor 1.69 higher than a blood sample.
The surface tension of water at the same temperature is 0.073 N/m. What is
the surface tension of the blood sample?
1. 0.045 N/m
2. 0.117 N/m
3. 0.073 N/m
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