Ice cube in a
glass of water
Water level, h?
Vwater + Vsub
h=
A
Vsub
mice g
mice
=
=
! water g ! water
After the piece
of ice melts:
Barge with
steel beams:
Vsub = 0
#Vsub
!Vwater
msteel
="
! water
!mwater
mice
=
=
= Vsub
" water
" water
"Vdisp = Vsteel
msteel
=
! steel
Steady flow in a river.
Velocity in each point is shown
by a vector with the length
proportional to the velocity.
Velocity gets higher, where the
river gets narrower.
Flow represented by streamlines,
that are everywhere tangent to flow
direction. Higher density of the
streamlines corresponds to higher
flow velocity.
In a steady flow there are no variations in velocity and pattern of flow in
time. Nevertheless, the actual fluid elements flowing past any particular
point at different times are always different. The fluid elements also get
accelerated and decelerated as they move along the streamlines.
Motion of fluids obeys the standard laws of mechanics.
Newton’s second law:
r
r
F = ma
Becomes Navier-Stokes equation:
Newton’s second law is actually a
complicated differential equation!
r
r
d r
F =m 2
dt
Any way to make our life easier?!
Let’s try to use the laws of conservation!!
2
Motion of fluids obeys the standard laws of mechanics.
m = const
r
r
Conservation of momentum:
mv1 + mv2 = const
Conservation of mass:
Conservation of energy:
2
mv
KE + PE =
+ mgh = const
2
Using the laws of conservation means doing
appropriate bookkeeping and doing algebra
instead of solving differential equations!
Steady flow
Flow tube A small tubelike region
bounded on its sides
by a continuous set of
streamlines
and on its ends by
small areas at right
angles to the
streamlines.
Cross-section areas on
the left and right ends
are:
A1 and A2.
Densities and
velocities are:
ρ1, ρ2 and v1, v2
Mass of fluid entering the tube from
the left over the time interval Δt
!x1 = v1!t
Steady flow
!V1 = A1v1!t
m = "1!V1 = "1 A1v1!t
By mass conservation, over the
time interval Δt, the same mass is
exiting the tube from the right
m = " 2 A2 v2 !t
Therefore
"1 A1v1!t = " 2 A2 v2 !t
!1 A1v1 = ! 2 A2 v2
!vA = const
everywhere along a flow tube
If the fluid is incompressible and its
density, ρ, is constant, we have
vA = const
vA = const
Does it work for traffic?
Once you pass the spot of accident there are more lanes
available (larger A) and the traffic speeds up (higher v).
What is the matter?
Traffic is highly compressible.
You have got to use
!vA = const
How does the total energy of a small fluid element change, as it
moves inside the flow tube from cross-section 1 to cross-section 2?
Kinetic energy:
1
"KE = m(v22 ! v12 )
2
Potential energy:
"PE = mg (h2 ! h1 )
How does this change in the
total energy become possible?
There are external forces
originating from pressure of
the liquid outside the tube,
which do work on the fluid
element!
Positive work as it enters from the left
Negative work as it exits from the right
The total energy balance
W1 = F1!x1 = P1 A1!x1
W2 = F2 !x2 = " P2 A2 !x2
!KE + !PE = W1 + W2
1
2
2
m(v2 " v1 ) + mg ( h2 " h1 ) = P1 A1!x1 " P2 A2 !x2
2
The total energy balance
1
2
2
m(v2 " v1 ) + mg ( h2 " h1 ) = P1 A1!x1 " P2 A2 !x2
2
V
Incompressible fluids – constant density and volume V = A !x = A !x
1
1
2
2
1 2
1 2
P1 + !v1 + !gh1 = P2 + !v2 + !gh2
2
2
1 2
P + !v + !gh = const
2
Bernoulli’s equation
Bernoulli’s equation
1 2
P + !v + !gh = const
2
Inertial jet
The fire truck pump is generating a pressure P.
What is the maximal velocity at the nozzle?
What is the maximal height the jet can reach?
Venturi Flowmeter
The venturi flowmeter is a practical instrument which makes use
of the Bernoulli effect and a manometer pressure gauge.