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NNPC FSTP ENGINEERS
Physics
Course Code:
Lesson 7
Contents
Fluid Mechanics
How Motion Takes Place
in a Fluid
Topics
• General Characteristics of Fluid Motion
• Bernoulli’s Equation
General Characteristics of Fluid
Flow
• Fluid flow can be steady or nonsteady. When the fluid
velocity v at any given point is constant in time, the
fluid motion is said to be steady. That is, at any given
point in a steady flow the velocity of each passing
fluid particle is always the same. These conditions
can be achieved at low flow speeds; a gently flowing
stream is an example.
• Fluid flow can be rotational or irrotational. If the
element of fluid at each point has no net angular
velocity about that that point, the fluid is irrotational.
• Fluid flow can be compressible or incompressible
General Characteristics of Fluid
Flow
• Fluid flow can be viscous or nonviscous. Viscosity in fluid
motion is the analogue of friction in the motion of solids
We shall confine our discussion of fluid dynamics to steady,
irrotational, incompressible, nonviscous flow.
In steady flow, the velocity v at a given point is constant in
time. If we consider the point p within the fluid, since v at p
does not change in time, every particle arriving at p will pass
on with the same speed in the same direction. The same is
true about the point Q and R. Hence if we trace out the path
of the particle, that curve will be the path of every particle
arriving at p. This curve is called a streamline.
R
P
Q
General Characteristics of Fluid
Flow
If we assume a steady flow and select a finite number
of streamlines to form a bundle, like the streamline
pattern shown below, the tubular region is called a
tube of flow.
Q
A1,V1
P
A2,V2
General Characteristics of Fluid
Flow
• Let the speed be v1 for fluid particle at p and v2 for
fluid particles at Q. Let A1 and A2 be the crosssectional areas of the tubes perpendicular to the
streamlines at the points p and Q respectively. At
the time interval  t a fluid element travels
approximately the distance v  t. Then the mass of
fluid  m1 crossing A1 in the time interval  t is
approximately,
m1  1 A1v1t
Or the mass flux m1
t
is approximately
1 A1v1
General Characteristics of Fluid
Flow
• The mass flux at p is
• The mass flux at Q is
1 A1v1
2 A2v2
Where 1 and 2 are the fluid densities at p and Q
respectively. Because no fluid can leave through the
walls of the tube and there are no sources or sinks
where fluids can be created or destroyed in the tube.
The mass crossing each section of the tube per unit
time must be the same. In particular the mass flux at p
must equal that at Q.
1 A1v1  2 A2v2
vA = constant
General Characteristics of Fluid
Flow
If the fluid is incompressible, as we shall henceforth
assume, then A v  A v
1 1
2 2
or
Av = constant.
3
The product Av gives the volume flux or flow rate in m
s
Bernoulli’s Equation
According to the continuity equation, the speed
of fluid flow can vary along the paths of the
fluid. The pressure can also vary; it depends
on height as in the static situation and it also
depends on the speed of flow.
We shall derive an important relationship
called Bernoulli’s equation that relates the
pressure, flow speed, and height
Bernoulli’s Equation
For flow of an ideal incompressible fluid.
The equation is an ideal tool for analysing plumbing systems,
hydroelectric generating stations and the flight of aeroplanes.
The dependence of pressure on speed follows from the continuity
equation. When an incompressible fluid flows along a flow tube,
with varying cross section, its speed must change, and so an
element of fluid must have an acceleration. If the tube is
horizontal, the force that causes this acceleration has to be
applied by the surrounding fluid. This means that the pressure
must be different in regions of different cross section; if it were the
same every where, the net force on every fluid element must be
zero. When a horizontal flow tube narrows, and a fluid element
speeds up, it must be moving towards a region of lower pressure
in order to have a net forward force to accelerate it.
Bernoulli’s Equation
Let us compute the work done by a fluid element during a
time dt. The pressure at the two ends are p1 and p2; the force
on the cross section at a is p1A1 and the force at c is p2A2.
The net work dW done on the element by the surrounding
fluid during this displacement is therefore
dW  p1 A1ds1  p2 A2ds2   p1  p2 dv
The second term has a negative sign because the force
at c opposes the displacement of the fluid.
The work done would be equal to the change in the total
mechanical energy (kinetic + gravitational potential
energy) associated with the fluid element.
Bernoulli’s Equation
The mechanical energy between b and c does not change.
At the beginning of dt, the fluid between a
and b has
volume A1ds1, mass A1ds1, and kinetic energy, 1   A1ds1 v12 .
2
At the end of dt, the fluid between c and d has kinetic
energy
1
  A2 ds2 v22
2
The net change in kinetic energy dK during time dt is

1
dK  dv v22  v12
2

Bernoulli’s Equation
For the change in gravitational potential energy, at the
beginning of dt, the potential energy for the mass between
a and b is
dmgy1  dvgy1
dmgy2  dvgy2
For the mass between c and d,
The net change in potential energy dU during dt is
dU  dvg y2  y1 
Using the energy equation dW = dK + dU we obtain
1
( p1  p2 )dv  dv(v22  v12 )  dvg( y2  y1 )
2
1
p1  p2   (v22  v12 )  g ( y2  y1 )
2
Bernoulli’s Equation
This is Bernoulli's equation. It states that the work done on
a unit volume of fluid by the surrounding fluid is equal to
sum of the changes in kinetic and potential energies per
unit volume that occur during the flow.
The first term on the right is the pressure difference
associated with the change of speed of the fluid. The
second term on the right is the additional pressure
difference caused by the weight of the fluid and the
difference in the elevation of the two ends. In a more
convenient form,we can state the equation as
1
1
2
2
p1  gy1  v1  p2  gy2  v2
2
2
Bernoulli’s Equation
The subscripts 1 and 2 refer to any two points along the
flow tube so we can also write;
1 2
p  gy  v  const
2
Note that when the fluid is not moving, v1 = v2 = 0 and the
equation reduces to the pressure relation we met before.
APPLICATIONS OF
BERNOULLIS EQUATIONS
Bernoulli's equation is useful for describing a variety of phenomena.
First we observe that if a fluid is at rest so that
v = 0, P + gh = constant.
This is just the hydrostatic equation between pressure and height.
For flow in a horizontal, constant height tube,
1 2
  v  K
2
Thus pressure must be lower in a region in which a fluid is moving faster.
Consider the flowmeter below:- A device designed to measure the speed
of fluids in a pipe
A2
P2, v2
P1,v1,
A1
Applying Bernoulli's equation,
1 2
1 2
p1  v1  p2  v2
2
2
1
p1  p2   v22  v12
2

Since
v2 

v1 A1
A2
2


1 2  A1 
  1
p1  p2  v1 
2
 A2 

The pressure difference = gh, so we can solve for v1
EXAMPLE 2
Water flowing from a tank
A1
P1=PA
h
A2
Consider points just outside and inside the hole and apply Bernoulli's
equation
1
1
2
p A   fluid gh   fluidu  p A  v 2
2
2
1
1
2
 f gh   f u   f v 2
2
2
1 2 1 2
gh  u  v
2
2
2
2
2 gh  u  v
hence
v  u 2  2 gh
If A1 A2
A2
u
v
A1
U2 term could be ignored
v  2 gh
v
ASSIGNMENT
• GRP 4: WE CAN HARNESS ENERGY
OF THE EARTH – GEOTHERMAL
ALTERNATIVE! (15 mins)