Lecture 4
Pressure distribution in fluids.
Pressure and pressure gradient.
Hydrostatic pressure
1
Basic Principles of Fluid
Mechanics
• Mass conservation
•
dm
0
dt
=>


v j 

t
x j
• Lei de Newton:
•


dv
Fm
dt
=>
v
v
p

 i  v j i  

t
x j
xi x j
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 vi

 x
j


  g z

xi

2
Methods for Fluid Mechanics
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Chapter 2
• Concept of pressure and pressure gradient,
• Hydrostatics and centripetal force.
• Lei de Newton: Acceleration as the result of
the applied forces.
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Forces in Fluids
• Surface or volumetric (or mass)
• Surface forces can be normal (pressure) or
tangential (friction)
• Tangential forces are always parallel to
velocity. Their equation can be complex if
velocity is not parallel to any reference axis.
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Pressure is a scalar
• Could
px , p y , pn be different in one point?
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In a fluid at rest:
Esta conclusão é independente de θ.
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Pressure Gradient
• It will be shown that the pressure gradient can
generate acceleration, but not the pressure by
itself.
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Pressure force resultant
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Force resultant (including friction and
weight)
 
yx y  dy
 
xy y
 u 
   
 y  y
 u 
   
 y  y  dy
Weight  g dxdydz 
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Summing up
p
pressure  
xi
  u 




u
dxdz   
    
  y 

2

y



u
y

dy
y


Friction 
 2
dxdydz
y
weight  g dxdydz 
dui
 ui
p


  2  gi
dt
xi
x j
2
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General equation
dui
 ui
p


  2  gi
dt
xi
x j
2
2


dui
ui
ui 
 ui
p



uj

  2  g i
 t

dt

x
xi
x j
j 

At rest the fluid has null velocity and so are its derivatives. The pressure is
hydrostatic.
Se a aceleração vertical for nula a pressão continua a ser hidrostática.
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Hydrostatic pressure
p
 g z
z
• If the vertical axes (zz) points downwards
gravity acceleration is positive, otherwise is
negative.
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Hydrostatics
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If  was constant:
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In case of gas
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Hydrostatic Forces
• Objectif:
• Compute the hydrostatic force over flat and curve
surfaces and the application point (pressure centre).
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Força Hidrostática
h   sin 
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Computing using former knowledge
F  pa A     sin  dA
F  pa A   sin   dA
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OU
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Pressure Centre (Measured from the
gravity centre)
   C .G.  y 
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Pressure Centre (x coordinate)
On surfaces with vertical simetry CP lays on the axis.
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Resumo
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