Download Lecture 35 - McMaster Physics and Astronomy

Final Exam
April 18
2 hours long – 30 MC questions
Covers all material with approximately equal weight, up
to and including today’s final lecture on fluids.
Physics 1B03summer-Lecture 13
Fluid Dynamics
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Equation of Continuity
Bernoulli’s equation and examples
Physics 1B03summer-Lecture 13
Fluid Dynamics
Approximations:
1) no viscosity (frictionless flow)
2) steady, “laminar” flow. If the flow is turbulent, mechanical
energy is lost (converted to thermal energy).
3) “incompressible” fluid. Sufficiently accurate for gases if
pressure differences are small.
Physics 1B03summer-Lecture 13
Streamlines
-the paths followed by particles in steady flow
-velocity is parallel to the streamline
- particles never cross streamlines; the streamlines mark out
imaginary “tubes of flow”
area A1
speed v1
area A2
speed v2
Physics 1B03summer-Lecture 13
Equation of Continuity
“Volume flow rate” (volume per unit time)
= (cross-sectional area)  (linear velocity)
“Mass flow rate” (mass per unit time)
= (density)  (volume flow rate)
So, if mass in = mass out, then
rAv = mass flow rate = constant
or
r1A1v1 = r2A2v2
for steady flow.
“Incompressible” fluids (density remains uniform): cancel out
density to get
Volume flow rate = constant
or
A1v1 = A2v2
Physics 1B03summer-Lecture 13
A fluid if flowing through a pipe of 10mm radius at a velocity of
10m/s. How fast will it be flowing if the pipe narrows to 5mm in
radius ?
radius
r1 = 10mm
radius
r2 = 5mm
Physics 1B03summer-Lecture 13
Bernoulli’s Equation: work and energy in fluids
Conditions: steady flow, incompressible fluid.
Look at energy balance along a streamline:
Change in (kinetic energy/volume)
+ change in (potential energy/volume)
= (net work by pressure)/volume
then,
or,
P1  12 rv12  rgy1  P2  12 rv22  rgy2
P  12 rv 2  rgy  a constant along a streamline
Note: the above equation looks similar what we have
seen before if we replace ρ by m.
Physics 1B03summer-Lecture 13
Example
a) What is the velocity of the water
leaving the little hole
b) How far (horizontally) from the hole
does the water hit the ground?
d
h
x
Physics 1B03summer-Lecture 13
Example
What is the speed of the water leaving
the hole in the tank?
gauge pressure P0
h
v
Physics 1B03summer-Lecture 13
Example
Water moving at 10m/s through a 1m radius pipe at a pressure
of 50kPa. It then falls 50m and goes into a 0.3m radius
pipe. What is the water pressure at the bottom ?
h
Physics 1B03summer-Lecture 13