Download 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   udV
dt
dt
dt

F  2k l 2  x 2  l


x
l x

d
p  v dV   udV 
dt
This potential energy change
goes into the energy
conservation equation.
d
dt

1
2
2
2
 
p
v dV  pdV  udV  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  udV 
dt
t
t
2
 1 p 
d 1 2 p
v
p
 2 v     u  

  gz  u  k
dt 

2 
  t t


next