Lecture 5
Pressure distribution in fluids
Pressure and pressure gradient
Hydrostatic pressure
1
Chapter 2
• Concept of pressure and pressure gradient.
• Hydrostatics and centripetal force.
• Newton’s law: acceleration is 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)
• Friction forces are always parallel to velocity:
– This is why they are often called “tangential
forces”.
– 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, i.e. could
pressure be a vector?
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In a fluid at rest:
This conclusion is independent of 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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Resultant of Pressure force
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Resultant Force
(including friction and weight)
 
yx y  dy
 
yx y
 u 

  
 y  y
 u 

  
 y  y  dy
Weight  g dxdydz 
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Summarize
p
pressure  
x i

  u i
dxdz 
  x j

Friction 

 u i

 

 x j
 x j dx j 



x j
dxdydz




2
    ui
x 2j
weight   i g dxdydz 
du i
 2u i
p



 g i
2
dt
x i
x j
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General equation
du i
 2u i
p



 g i
2
dt
x i
x j
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 its derivatives are also null.
The pressure is hydrostatic.
If there is velocity, but the vertical acceleration is null the pressure still is hydrostatic.
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Hydrostatic pressure
p
 g z
z
• If the vertical axes (zz) points downwards gravity
acceleration is positive (g z  9.8) otherwise (g z  9.8)
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Hydrostatics
This is a simplification of Earth: it
assumes it is spherical and has
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homogeneous specific mass
If  was constant:
This image is a snapshot from White’s book. It is a traditional engineer vision of Earth valid only
very close to the mean sea level.
In fact vertical variability of density in lakes&oceans and in the atmosphere is critical for their
dynamics and for life on Earth.
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In case of gas
In a isothermal atmosphere (with uniform temperature) the pressure would decay
exponentially with altitude. In fact the temperature decreases with altitude.
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Pressure variation in the
atmosphere
(See more in White, chapter 2)
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15
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Hydrostatic Forces
• Objective:
• Compute the hydrostatic force over flat and curve
surfaces and its application point (pressure centre).
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Hydrostatic Force
• What is the value of
the force applied on
the vertical wall?
h
F 
 pdA b gz dz
A

0
h
z2 
h

F  bg     g  A
2

 2 0
F  PGC A
In this case the force is the pressure at the gravity center times the area of the surface.
And on the gate?
18
We will see that it is identical.
Thus
it would
bedosinteresting
to derive generic formulas.
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Hydrostatic Force
F  pa A  hCG A
F  pa A P CG A
But hCG depends on the inclination
. It is convenient to calculate it as a
function of the surface, i.e. it is
convenient to attach the reference
to the surface.
h   sin 
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Computing using former
knowledge
F  pa A  hCG A
F  pa A     sin  dA
F  pa A   sin   dA
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OR
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Pressure Centre
(Measured from the gravity centre)
   C .G.  y 
The pressure centre is always below
the gravity centre!
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Pressure Centre
(x coordinate)
In case of surfaces with vertical symmetry CP lays on the axis.
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Summary
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