Transcript Fluid Flow
Fluid Flow
Pressure Force
Each volume element in a
fluid is subject to force due
to pressure.
• Assume a rectangular box
• Pressure force density is the
gradient of pressure
dz
p
dV
dx
dy
p
p
dx dydz dV
x
x
dFx
p
p
p
dF xˆ dV yˆ dV zˆ dV
x
y
z
dF pdV
Equation of Motion
A fluid element may be
subject to an external force.
• Write as a force density
• Assume uniform over small
element.
dF fdV
dv
dm dF
dt
x
dv F 2k l x l l x
dV
fdV pdV
dt
dv
p f
dt
dv
P f
dt
2
The equation of motion uses
pressure and external force.
• Write form as force density
• Use stress tensor instead of
pressure force
This is Cauchy’s equation.
2
2
2
Euler’s Equation
Divide by the density.
• Motion in units of force
density per unit mass.
The time derivative can be
expanded to give a partial
differential equation.
• Pressure or stress tensor
This is Euler’s equation of
motion for a fluid.
dv 1
f
p
dt
1
v
f
v v p
t
1
v
f
v v P
t
Momentum Conservation
dv
dV
pdV fdV
dt
d
v dV f p dV
dt
v dV
• Euler equation with force
density
• Mass is constant
d
v dV fdV pdV
dt V
V
V
d
v dV fdV nˆpdS
dt V
V
S
The momentum is found for
a small volume.
Momentum is not generally
constant.
• Effect of pressure
The total momentum change
is found by integration.
• Gauss’ law
Energy Conservation
The kinetic energy is related
to the momentum.
d
dt
• Right side is energy density
2
2
2
• Total time derivative
• Volume change related to
velocity divergence
1
2
v dV v f p dV
2
d
pdV dp dV p ddV
dt
dt
dt
x
p F 2k l x l l x
v pdV p v dV
t
v pdV
Some change in energy is
related to pressure and
volume.
d
dt
1
2
2
d
pdV p p v dV
dt
t
p
v dV pdV v fdV p v dV
t
2
Work Supplied
The work supplied by
expansion depends on
pressure.
dW
ddV
p
p v dV
dt
dt
• Potential energy associated
with change in volume
dW
d
d
udm udV
dt
dt
dt
F 2k l 2 x 2 l
x
l x
d
p v dV udV
dt
This potential energy change
goes into the energy
conservation equation.
d
dt
1
2
2
2
p
v dV pdV udV v fdV
t
2
Bernoulli’s Equation
f gzˆ
Gravity is an external force.
• Gradient of potential
• No time dependence
v fdV v dV
d
dV
dt t
d
dV
dV
dt
t
The result is Bernoulli’s
equation.
• Steady flow no time change
• Integrate to a constant
d 1 2
p
dV dV
2 v dV pdV dV udV
dt
t
t
2
1 p
d 1 2 p
v
p
2 v u
gz u k
dt
2
t t
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