Transcript PHYS 342: Modern Physics

XIII. Fluid Mechanics
A. Introduction & Definitions
1. Density: mass per unit volume
r = M/V; units = kgm-3
e.g.,
2.
(XIII.A.1)
r(H2O) = 1000 kgm-3;
r(Fe) ~ 8000 kgm-3
Pressure: Perpendicular Force per unit area
P = Fperp/A; units = Nm-2 = Jm-3 = Pa
3. “Gauge Pressure”
(XIII.A.2)
XIII. Fluid Mechanics B. Hydrostatic Equilibrium
1.
Pressure & gravity in uniform fluids
Assuming equilibrium, then the force due to pressure must be
equal to the force due to gravity: “Hydrostatic Equilibrium”
h
-(p)A
dh
dW = rgdV = rgAdh
(p+dp)A
-pA + (p + dp)A – rgAdh = 0;
dp/dh = rg (XIII.B.1)
XIII. Fluid Mechanics B. Hydrostatic Equilibrium
1.
Pressure & gravity in uniform fluids
Assuming equilibrium, then the force due to pressure must be
equal to the force due to gravity: “Hydrostatic Equilibrium”
h
-(p)A
dh
dW = rgdV = rgAdh
(p+dp)A
For uniform density,
∫dp = rg∫dh
p = p0 + rgh
(XIII.B.2)
XIII. Fluid Mechanics B. Hydrostatic Equilibrium
2.
Pascal’s Law
Pressure applied to an enclosed fluid is transmitted
undiminished to every portion of the fluid and the walls
containing the fluid.
Consider two connected cylinders with different areas
A1 and A2:
P = F1/A1 = F2/A2 => F2 = (A2/A1)F1.
(B.3)
Pascal’s law demonstrates how force can be multiplied, e.g.,
hydraulics.
XIII. Fluid Mechanics C. Buoyancy
1.
Archimedes Principle: When a body is completely or
partially immersed in a fluid, the fluid exerts an
upward force on the body equal to the weight of the
fluid displaced by the body.
a)
FB
Consider raising a marble statue from the sea floor with
V = 5 m3 and r ~ 3000 kgm-3. What is the tension in the
rope? Assume sea water.
T
S F = 0 = T + FB – W;
T = W - FB.
W
XIII. Fluid Mechanics C. Buoyancy
1.
FB
Archimedes Principle: When a body is completely or
partially immersed in a fluid, the fluid exerts an
upward force on the body equal to the weight of the
fluid displaced by the body.
T
m = rmV = 15,000 kg.
|W| = mg ~ 150,000 N.
|WSW|= rswVg = 50,000 N = |FB|.
T = W - FB = 100,000 N;
|T/W| ~ 67%
W
XIII. Fluid Mechanics C. Buoyancy
b) In terms of densities:
T = W - FB = rmVg – rswVg;
T = mstatueg(1- rsw/rm).
FB
W
T
(XIII.C.1)
XIII. Fluid Mechanics D. Bernoulli’s Eqn.
1.
2.
BE: Relates pressure, flow speed and height for an ideal fluid.
Derivation: Apply work-energy theorem to a section of fluid in
a flow tube.
a) Work
ds1 = v1dt
ds2 = v2dt
dW = Fds.
V= Ads.
F1
F2
y1
y2
dW= (p1 - p2)dV.
(D.1)
XIII. Fluid Mechanics D. Bernoulli’s Eqn.
1.
2.
BE: Relates pressure, flow speed and height for an ideal fluid.
Derivation: Apply work-energy theorem to a section of fluid in
a flow tube.
b) Kinetic Energy
ds1 = v1dt
ds2 = v2dt
dK = d(1/2mv2)
F1
F2
y1
y2
= 1/2rdV(v22 - v12).
(D.2)
XIII. Fluid Mechanics D. Bernoulli’s Eqn.
1.
2.
BE: Relates pressure, flow speed and height for an ideal fluid.
Derivation: Apply work-energy theorem to a section of fluid in
a flow tube.
c) Gravitational Potential Energy
ds1 = v1dt
ds2 = v2dt
dU = d(mgh)
F1
F2
y1
y2
= rdVg(y2 - y1)
(D.3)
XIII. Fluid Mechanics D. Bernoulli’s Eqn.
1.
2.
BE: Relates pressure, flow speed and height for an ideal fluid.
Derivation: Apply work-energy theorem to a section of fluid in
a flow tube.
WET:
dW = dK + dU =>
Dp = 1/2rdV(v22 - v12) + rdVg(y2 - y1), or
(D.4)
p + rgy + 1/2rv2 = constant.
(D.5)
Example Problem #12
Suppose that air streams past the wings such that the speed is
100 m/s over the top surface and 90 m/s past the bottom
surface. If the plane has a mass of 1500 kg, and the wings are 2
m wide, what is the wingspan required for lift?
Note: Density of air ~ 1 kgm-3.
p + rgy + 1/2rv2 = constant.
Note: rgy = constant.
pt + 1/2rvt2 = pb + 1/2rvb2;
Dp = 1/2r(vb2 - vt2) ~ - 950 Pa.
(pressure below is higher than pressure above)
Example Problem #12
3.
Example: Lift. Suppose that air streams past the wings such
that the speed is 100 m/s over the top surface and 90 m/s past
the bottom surface. If the plane has a mass of 1500 kg, and the
wings are 2 m wide, what is the wingspan required for lift?
Minimum lift is L = mg ~ 15,000 N.
L = pA = p(Width)(Wingspan) = (950 Pa)(2 m)(WS);
WS = 15,000 N/(1900 Pa-m) ~ 8 m.