When a fluid is at rest (where the mean velocity is zero) it exerts a force perpendicular to any surface with which it is in contact with. At any point within the fluid the pressure must be the same in all directions or else the fluid would move.
Figure 4.1 - Pressure in a Static Fluid
The (atmospherical) pressure on the surface is p0 . The mass of the cylinder of height (z0-z) is
m=Ar (z0-z)
where A is the cross sectional area of the cylinder. Therefore the weight it A(z0-z)r g .
The pressure at z is
p(z)=p0+(z0-z)r g
the pressure at a level (z+dz) is
p(z+dz)=p0+(z0-(z+dz))r g
therefore the pressure difference is
dp=p(z+dz)-p(z)
dp=-gr dz Þ
dp
dz
=-gr (4.1)
This is known as the Hydrostatic Equation.
Pascal's Law:
pressure applied to an enclosed fluid is transmitted undiminised to all parts of the fluid and to the walls of the container
This equation has applications in hydraulic lifts and brake systems (see Problem Sheet 3).
Figure 4.2 - Pascal's Barrel
Figure 4.2 shows Pascal's Barrel which demostrates his law. If the pipe is long enough, fill it with water, therefore the pressure becomes very large and is transmitted to the barrel, therefore the barrel explodes!
4.2 Buoyancy
SEE CLASSWORK 3
4.3 Variation of Atmospheric Pressure With Heat
SEE CLASSWORK 3
4.4 Continuity Equation
Now we consider moving fluids, this is known as hydrodynamics.
At any point in the fluid there are particular values of
pressure (x,y,z,t)
density (x,y,z,t)
velocity (x,y,z,t)
which can also vary with time, consider an elecment of volume V within a fluid moving along flowlines
Figure 4.3 - A Moving Element of Fluid
In a time D t the mass of fluid flowing into V across A1 is r1A1v1D t . The mass of fluid moving out of V across A2 is r2A2v2D t . Therefore the change in mass in the volume V in a time D t is
-D m=(r2A2v2-r1A1v1)D t
dm
dt
=D (r Av) (4.2)
This is known as the Continuity Equation. If the flow is steady then at any point r v does not change with time and the flowlines become streamlines
dm
dt
=0 Þ r Av=constant (4.3)
If the flow is incompressible then r is also constant so
Av=constant
and so we can see where the phase still waters run deep comes from.
4.5 Bernoulli's Equation
From the continuity equation we know that if an incompressible fluid flows along a tube of varying cross-sectional area then its speed varies. Therefore it must undergo an acceleration and this must be (according to Newtons Second Law) due to an applied force.
If the flow is horizontal there is no gravitational acceleration and the force must be due to a pressure gradient.
Figure 4.4 - A Steady Non-Viscous Flow Along Streamlines
Properties of the fluid between X and Y do not change so we only need to consider the elements at each end.
A1v1D t =A2v2D t=dV
the masss of each element is r dV and the change in potential energy is
dU=r gdV(z2-z1)
the change in kinetic energy is
dK=
1
2
r dV(v22-v12)
conservation of energy states that if the total energy ( U+K ) changes then the fluid must have had work done on it. Work is done by the fluid surrounding the elecment. The force on the lower end is p1A1 so the work done is p1A1v1D t . The work done against the motion on the upper end is p2A2v2D t .
