Download fluid - GEOCITIES.ws

FLUID
Characteristics of Fluid Flow (1)
Steady flow (lamina flow, streamline flow)
The fluid velocity (both magnitude and
direction) at any given point is constant in
time
The flow pattern does not change with time
Non-steady flow (turbulent flow)
Velocities vary irregularly with time
e.g. rapids, waterfall
Characteristics of Fluid Flow (2)
Rotational and irrotational flow
The element of fluid at each point has a net
angular velocity about that point
Otherwise it is irrotational
Example: whirlpools
Compressible and incompressible fluid
Liquids are usually considered as
incompressible
Gas are usually considered as highly
compressible
Characteristics of Fluid Flow (3)
Viscous and non-viscous fluid
Viscosity in fluid motion is the analog of
friction in the motion of solids
It introduces tangential forces between layers of
fluid in relative motion and results in
dissipation of mechanical energy
Streamline
A streamline is a curve whose tangent at
any point is along the velocity of the fluid
particle at that point
It is parallel to the velocity of the fluid
particles at every point
No two streamlines can cross one another
In steady flow the pattern of streamlines in
a flow is stationary with time
Change of speed of flow with
cross-sectional area
If the same mass of fluid is to pass through
every section at any time, the fluid speed
must be higher in the narrower region
Therefore, within a constriction the
streamlines must get closer together
Kinematics (1)
Mass of fluid flowing past area Aa
=avatAa
Mass of the fluid flowing past area Ab =
bvbtAb
Kinematics (2)
In a steady flow, the total mass in the
bundle must be the same
 avaAa t= bvbAb t
i.e. avaAa = bvbAb
or vA = constant
The above equation is called the continuity
equation
For incompressible fluids
vA = constant
Further reading
Static liquid pressure
The pressure at a point within a liquid acts
in all directions
The pressure depends on the density of the
liquid and the depth below the surface
P = gh
Further reading
Bernoulli’s equation
Bernoulli’s equation
This states that for an incompressible, nonviscous fluid undergoing steady lamina flow,
the pressure plus the kinetic energy per unit
volume plus the potential energy per unit
volume is constant at all points on a streamline
i.e. p  12 v 2  gh  constant
Derivation of Bernoulli’s
equation (1)
The pressure is the same
at all points on the same
horizontal level in a fluid
at rest
In a flowing fluid, a
decrease of pressure
accompanies an increase
of velocity
Derivation of Bernoulli’s
equation (2)
In a small time interval t, fluid XY has
moved to a position X’Y’
At X, work done on the fluid XY by the
pushing pressure
= force  distance moved
= force  velocity  time
= p1A1  v1  t
figure
Derivation of Bernoulli’s
equation (3)
At Y, work done by the fluid XY emerging
from the tube against the pressure
= p2A2  v2  t
Net work done on the fluid
W = (p1A1  v1 - p2A2  v2)t
For incompressible fluid, A1v1= A2v2
 W = (p1 - p2)A1 v1 t
figure
Derivation of Bernoulli’s
equation (4)
Gain of p.e. when XY moves to X’Y’
= p.e. of X’Y’ - p.e. of XY
= p.e. of X’Y + p.e. of YY’ - p.e. of XX’ - p.e. of
X’Y
= p.e. of YY’ - p.e. of XX’
= (A2 v2 t)gh2 - (A1 v1 t)gh1
= A1 v1 tg(h2 - h1)
figure
Derivation of Bernoulli’s
equation (5)
Gain of k.e. when XY moves to X’Y’
= k.e. of YY’ - k.e. of XX’
1
A2 v2t  v2   A1v1t  v12

2
2
1
2
2
A
v

t

v

v

=
1 1
2
1 
2
= 1
2
figure
Derivation of Bernoulli’s
equation (6)
For non-viscous fluid
net work done on fluid = gain of p.e. + gain of
k.e.
(p1 - p2)A1 v1 t = A1 v1 tg(h2 - h1) +
1
A1v1t  v2 2  v12 
2
1
p1  p2  g (h2  h1 )   (v2 2  v12 )
2
figure
Derivation of Bernoulli’s
equation (7)
1
1
2
p1  h1 g  v 1  p2  h2 g  v 22
2
2
or
1 2
p  hg  v  constant
2
figure
Derivation of Bernoulli’s
equation (8)
Assumptions made in deriving the equation
Negligible viscous force
The flow is steady
The fluid is incompressible
There is no source of energy
The pressure and velocity are uniform over any
cross-section of the tube
Further reading
Applications of Bernoulli
principle (1)
Jets and nozzles
Bernoulli’s equation suggests that for fluid flow
where the potential energy change hg is very
small or zero, as in a horizontal pipe, the
pressure falls when the velocity rises
The velocity increases at a constriction and this
creates a pressure drop. The following devices
make use of this effect in their action
Applications of Bernoulli
principle (2)
Bunsen burner
The coal gas is made to pass a constriction
before entering the burner
The decrease in cross-sectional area causes a
sudden increase in flow speed
The reduction in pressure causes air to be
sucked in from the air hole
The coal gas is well mixed with air before
leaving the barrel and this enables complete
combustion
Applications of Bernoulli
principle (3)
Carburettor of a car engine
The air first flows through a filter which
removes dust and particles
It then enters a narrow region where the flow
velocity increases
The reduced pressure sucks the fuel vapour
from the fuel reservoir, and so the proper airfuel mixture is produced for the internal
combustion engine
Applications of Bernoulli
principle (4)
Filter pump
The velocity of the running water increases at
the constriction
The surrounding air is dragged along by the
water jet and this causes a drop in pressure
Air is then sucked in from the vessel to be
evacuated
Spinning ball
If a tennis ball is `cut’ it spins as it travels
through the air and experiences a sideways
force which causes it to curve in flight
This is due to air being dragged round by
the spinning ball, thereby increasing the air
flow on one side and decreasing it on the
other
A pressure difference is thus created
figure
Further reading
Aerofoil
A device which is shaped so that the relative
motion between it and a fluid produces a
force perpendicular to the flow
Fluid flows faster over the top surface than
over the bottom. It follows that the pressure
underneath is increased and that above
reduced. A resultant upwards force is thus
created, normal to the flow
e.g. aircraft wings, turbine blades, sails of a
yacht
Pitot tube (1)
a device for measuring flow velocity and in
essence is a manometer with one limb parallel to
the flow and open to the oncoming fluid
The pressure within a flowing fluid is measured at
two points, A and B. At A, the fluid is flowing
freely with velocity va. At B where the Pitot tube
is placed, the flow has been stopped
Pitot tube (2)
By Bernoulli’s
equation:
1
Pa  va 2  Pb  0
2
1
2
Po  gha  va  Po  ghb
2
where P0 = atmospheric pressure
Pitot tube (3)
 v a  2g hb  ha 
Note:
• In real cases, v varies across the diameter of the pipe
carrying the fluid (because of the viscosity) but if the
open end of the Pitot tube is offset from the axis by
0.7  radius of the pipe, then v is the average flow
velocity
• The total pressure can be considered as the sum of
two components: the static and dynamic pressures
Pitot tube (4)
PT  ( p  gh ) 12 v 2
Total
Static
Dynamic
pressure pressure pressure
A moving fluid exerts its total pressure in the
direction of flow. In directions at right angles to
the flow, the fluid exerts its static pressure only
figures
Further reading: paragraph of ‘Pitot Static
System’ near the bottom of the page
Venturi meter (1)
This consists of a horizontal tube with a
constriction. Two vertical tubes serving as
manometers are placed perpendicular to the
direction of flow, one in the normal part and the
other in the constriction
In steady flow the liquid level in the manometer
connected to the wider part of the tube is higher
than that in the narrower part
figure
Venturi meter (2)
From Bernoulli’s principle
1 2
1
2
P1  v1  P2  v2
2
2
(h1 = h2)
2
1
1 2 1 2 v2 2
2
P1  P2  v2  v1  v1 (( 2 )  1)
2
2
2
v1
For an incompressible fluid,
A1v1 = A2v2
 v2  A1
v1
A2
Venturi meter (3)
Hence
1
A1 2
2
P1  P2  v1 [( )  1]
2
A2
 v1 can be deduced
Streamline

vP
P

vQ
Q
Change of speed in a constriction
Streamlines are
closer when the
fluid flows faster
Derivation of Bernoulli’s
equation
v2
Y’
v1
X
Y
X’
v2t
Area A2
p1A1
Area A1
p2A2
v1t
h2
h1
Bunsen burner
Carburettor
air
filter
fuel
to engine cylinder
Filter pump
Spinning ball
Aerofoil
Pitot tube (1)
Pitot tube (2)
Pitot is here
Pitot tube: fluid velocity
measurement (1)
Static pressure
holes
Fast moving air,
lower pressure
inside chamber
Stagnant air,
higher pressure
inside tube
Flow of air
Static tube
P1= total pressure
P2= static pressure
P2 – P1 = ½(v2)
Total tube
Pitot tube: fluid velocity
measurement (2)
Ventri meter (1)
Venturi meter (2)
Venturi meter (3)
Density of
liquid = 
v1
A1
v2
A2