Fluids in Motion

• Control Volume
• For the purpose of analysing flow processes the region under
investigation, known as a control volume ( or open system) is
Fluids
specifiedinby a boundary called a control surface which may be
any convenient surface, entirely imaginary or partly real and
Motion
partly imaginary. The control volume is of fixed size and shape
but the mass of fluid within it may vary with time or may be
constant and the flow may be uniform or non-uniform, laminar
or turbulent.

2
 Steady flow
• In general, a steady flow process involves no changes in properties,
in mass flow rates or in heat and work transfer rates with respect
to time. However, properties may vary from place to place within
the control volume but are constant at any given place.

m , T1 m , T2

3
 Uniform flow

• Uniform flow takes place such that the properties are the same at
all points within the control volume at any given instant.

a
b
c

• If va = vb = vc, pa = pb = pc, etc at the same instant then the flow is

uniform.

4
Fluids in Motion
Uniform flow

Not parallel

Not straight
5
 One-dimensional flow

• In 1-D flow it is assumed that all properties are uniform over any
plane perpendicular to the flow direction.

2 p2, 2, v2
constant over
1 area A2

p1, 1, v1
constant over
area A1
6
 1-D flow
• 1-D flow is never reproduced exactly in practice. When a fluid flows
along a circular pipe the velocity is zero at the wall and a maximum
at the pipe centre. Nevertheless, the assumption of 1-D flow
simplifies analysis and often produces sufficiently accurate results.

vmax

7
 Streamlines
• At an instant in time, a streamline is an imaginary curve in the fluid across
which, at that instant, no fluid is flowing. Therefore, at that instant, the
velocity of every fluid particle along that line is tangential to it.
• A streamline is a line along which all fluid particles have, at a given instant,
velocity vectors which are tangential to the line.

8
 Streamlines

• As there is no component of velocity across a streamline, streamlines never

cross.

• A number of streamlines give a good indication of the flow pattern.

• For steady flow the streamline pattern is unchanging, for unsteady flow the
streamlines give an instantaneous picture of the flow

9
 Streamlines
• The surface of a solid body is a streamline surface because it is made up of
curves with no flow across them.

• For uniform flow the streamlines must be straight and parallel.

• A streamline cannot be directly visualised

10
 Pathline

• A pathline shows the route taken by a single fluid particle during

a given time interval. It is equivalent to the trace of a single fluid
particle seen on a long exposure photograph.
• Contrast with streamlines which give an instantaneous picture of
the whole flow field.
• Pathlines may cross.

11
Fluids in Motion

streamlines

pathline
12
 Streakline

• In experimental work a dye or other indicator may be injected

into the flow field. The resulting stream of colour is known as a
streakline (or filament line). It gives a picture of the positions of
all fluid particles that have passed through the point of injection.

13
Fluids in Motion Streaklines

14
 Fluids in Motion

• In steady flow (not varying with time), pathlines, streaklines and

streamlines all collapse onto the same trajectory.

15
 Types of flow

• There are essentially two basic types of flow. For example the flow
from a tap at low velocity is smooth. If the flow rate is increased the
flow is no longer smooth and the surface is rough and broken up.
• Osbourne Reynolds (1842-1912) carried out a simple experiment to
illustrate and distinguish the two types of flow.

16
Reynolds’ Experiment

dye

Glass tube
Large tank
 7-27 mm
of water

17
Reynolds’ Experiment
• As the valve was opened
Reynolds observed
different flow patterns.
slow

fast
18
 Reynolds’ Experiment

• With a low flow velocity the filament of dye was smooth and steady.
• As the flow rate was increased a point was reached at which the
filament began to waver.
• Further increase of flow rate increased the filament fluctuations
until at a point along the tube the dye filament suddenly mixed
across the tube
• Further increase did not change the flow pattern but the point at
which mixing occurred moved closer to the inlet.

19
Control Volume Approach
• The method employed for solving most problems in fluid mechanics is
the control volume approach (also known as the Reynolds Transport
Theorem).
• A control volume is a selected volumetric region in space.
• A control surface is the surface enclosing the control volume.
• The mass within the control volume can change with time, and a control
volume can deform with time, and move and rotate in space.
• In contrast with the control volume, a system is defined as a continuous
mass of fluid that always contains the same fluid particles.
Reynolds Transport Theorem
dBsys d
  bdV   bVdA
dt dt cv cs
Lagrangian
Eulerian
Bcv   b dm   b dV
cv cv
 Continuity Equation
• The continuity equation expresses conservation of mass.

b
m2
m1 a
v2

v1 c
d

22
 Continuity Equation
• Control volume abcda contains mass m at time t and mass m + m at
time t + t.
• Conservation of mass:

mass flow in - mass flow out  increase of mass in CV

 m1  m2  m  m   m
Rate of change of mass in CV :
m1 m2 m
 
t t t
23
 Continuity Equation
As δt  0
dm
m 1  m
2
dt
This is the General Continuity Equation.
Under steady flow conditions there is no mass
accumulation and m   0.
Therefore 1  m
m 2
i.e. mass flow rate is constant for steady flow. 24
 Continuity Equation

• Let the inlet area be A1 and the outlet area A2.

• Assume 1-D flow so that the density and velocity at inlet
are 1 and v1 respectively, and 2 , v2 at outlet.

m1 x1
m1  1 A1x1 and  1 A1
t t
m1
as t  0  m 1 and m 1  1 A1v1
t
m 2   2 A2v2
25
 Continuity Equation

 1  m 2
For steady flow m
 1 A1v1   2 A2v2
and for incompressible flow (constant density)
A1v1  A2v2
For several inflows and outflows and steady conditions :
 m in   m out
or   ρAv    ρAv
in out
26
- Discharge or Volume flow rate, Q {m3/s}

Q  V.A
The dot product means that the velocity component normal to the area
For a fluid with variable velocity:
Q   V.dA
- Mass Flow Rate, , {kg/s}
Constant velocity m   V.A
Variable velocity and
constant density m   V.dA   Q

- Mean Velocity V=Vavg =Q /A

• Example: Find the volume and mass flow rate of
water.

• Solution: π 2 π
Q  VA  V  D  10  0 .012  7 .85  10 4 m 3 /s
4 4

m  ρQ  1000 7 .85  10 4  0.785 kg/s
• Example: Find the volume and mass flow rate of
water.
• Solution:
Q  VA cos 30o  6.798  104 m 3 /s


m  ρQ  0.679 kg/s
• Example: steady, incompressible flow of water through the device.
Given:
A1=0.2 m2
A2=0.2 m2
A3=0.15 m2
V1=5 m/s
V3=12 m/s
Q4=0.1 m3/s
 = 999 kg/m3

Find V2 and it’s direction.

• Solution:

 ρV.dA  0.0  ρV.A  0

d
cs ρV.dA   dt cv ρ dV cs

since the problem is incompressible: =const

V1 . A1  V2 . A2  V3 . A3  V4 . A4  0.0
 V1 A1  V2 A2  V3 A3  V4 A4  0.0
 V1 A1  V2 A2  V3 A3  Q4  0.0
 5  0.2  V2  0.2  12  0.15  0.1  0.0 V2  4.5 m / s

Because it is –ve, the assumption is not valid, so the V2 goes inward the cv.
Worked Example
• Air of density 1.2 kg/m3 flows along a pipe of circular cross-section at
the rate of 7.2 kg/hour. The diameter of the pipe is 20 mm.
a) Assuming 1-D flow, what is the air velocity in the pipe? Is the flow
laminar or turbulent? (the viscosity of air is 1.8x10-5 kg/m s).
b)The flow expands smoothly into a larger pipe, where the new
Reynolds number is 1000. Assuming the flow is incompressible, what is
the velocity is this new pipe?

31
Worked Example

Mass flow rate is related to velocity by the expression:

m  Av
 7.2 
 
 v
m
  3600   5.31 m / s
A 1.2 x   0.022 
 
4 

32
Worked Example

For the pipe, Reynolds number is:

vd 1.2 x 5.31 x 0.02
Re    7080
 1.8 x10 5

7080 is significantly greater than 2000, therefore the

flow is turbulent.

33
Worked Example
Apply continuity: 1 A1v1   2 A2 v2
But 1   2
v2 A1 d12
   2
Also, v1 A d 2
v2 d 2 v1d1
Re 2   1000 and Re1   7080
 
v2 d 2 v1d1 d1 v2
    7.08
1000 7080 d2 v1
34
Combining:
2
v2  v2  1
  7.08   v2   0.14 m / s
v1  v1  7.08

35
Description of Fluid Motion
Langrangian Method
 The Euler Equation

For a fluid in motion, pressure depends on density, elevation, acceleration (net

force), viscosity (shear stresses) and shaft work done on or by the fluid (pump or
turbine). The Euler equation applies where the following can be assumed:

steady flow (properties constant at a given point

ideal (frictionless) flow. No shear stresses

No shaft work.

40
The Euler Equation
• Consider a small element (p+p) A
moving in the s direction s
with velocity v and
a,v
acceleration a.
• (ie along a streamline) pA gm
• Apply F=ma (Newton) to the z
element.
s-direction


41
The Euler Equation

Apply F  ma to the fluid element

Net force in s - direction is :
pA   p  p A  gm cos    pA  gm cos 
z
also, m  As and cos  
s
z
  pA  gAs  As a
s
1 p z
 g a0
 s s
42
The Euler Equation

Particle acceleration :
In general, as a fluid particle moves from A to B its
velocity changes due to time and location. This can
be expressed as :
v v δv v δs v
δv  δs  δt   
s t δt s δt t

43
The Euler Equation

In the limit, as δt  0, this becomes

dv v ds v ds
  and as  v, then
dt s dt t dt
dv v v
acceleration   v 
dt s t
For steady flow (not varying with time)
v v
 0 and a  v
t s
44
 The Euler Equation

substituting for a, as δt  0
1 dp dz dv
 g v 0
 ds ds ds
• This is Euler’s equation. It applies along a streamline (along a
streamline the velocity is tangential to the streamline), and in steady
flow.

45
The Bernouilli Equation

Integrate Euler' s equation :

 dp 
    gdz  vdv   0 ds
 
dp
   g  dz   vdv  0

dp v2
and   gz   constant1
 2

46
The Bernouilli Equation
If density is constant (liquid or gas at small
variations of pressure, then :
p v2
 gz   constant 2
 2
This is the Bernouilli equation and only applies for :
ideal flow, steady flow, zero shaft work, and
along a streamline (because these assumption s were
necessary to derive the Euler equation).
It also only applies for incompress ible flow.
47
 The Bernouilli Equation
• In this form each component has the units energy/unit mass and is a
special statement of the law of conservation of energy.
• Energy can be present as “pressure” energy (work done if a fluid
particle moves from pressure p to pressure 0), potential energy
(elevation) or kinetic energy.

48
The Bernouilli Equation
Multiply all the terms in the Bernouilli equation
by  :
p  gz  12 v 2  constant
In this alternate form of the Bernouilli equation
all the terms have the units of pressure.
In this case the constant represents the total pressure
for the system. Therefore, total pressure is constant for
steady, incompress ible, inviscid flow along a streamline ,
and no shaft work.
49
The Bernouilli Equation
• p is often referred to as ps
the static pressure. It can be
measured by attaching a pa
tube 2 to a small
2 v
1
perpendicular hole in the
side of the pipe. p1 z2
• is often referred to as
the dynamic pressure. z1
• gz is the elevation piezometer :
pressure.
p1, gauge  gz
50
The Bernouilli Equation

Divide all the terms in the Bernouilli equation

by g :
ps v2
z  constant (HT )  Total head
g 2g
In this form of the Bernouilli equation all the
terms have the unit of height. In fluid mechanics
this is referred to as head. Total head is constant for
inviscid, incompress ible, steady flow along a
streamline with no shaft work.
51
 The Bernouilli Equation

ps
• is called the static head or the pressure head
g
•z is the elevation head

v2
• is the dynamic head or the velocity head
2g

52
The Bernouilli Equation
Pressure form :
ps  gz  12 v 2  total pressure
ps  gz may be referred to as the piezometri c
pressure.
" Head" form :
ps v2
z  Total head
ρg 2g
ps
 z may be referred to as the piezometri c head.
ρg
53
 Flow along a pipe

• Measurements are relative to pipe centreline, the central streamline,

as is usual for 1-D flow, although in general each streamline has its
own values of Hs, HD and z.

54
 TOTAL HEAD

HD2=v22/2g HD1=v12/2g

HS1=pS1/g
HS2=pS2/g

1
z1
z2 2

DATUM
Flow along a pipe
Apply Bernouilli between 1 and 2 :
For steady flow of an ideal fluid along a streamline ,
with no shaft work, Total head is constant.
p1 v12 p2 v22
HT   z1    z2 
g 2 g g 2g
Apply continuity between 1 and 2 : A1v1  A2v2
NB The Bernouilli equation only applies
for constant density (incompressible fluid).
56
Applications of the Bernouilli Equation
Flow along a pipe

In this case A1  A2
v12 v22
 
2g 2g
and substituting back into Bernouilli :
p1 p2
 z1   z2
g g
[ie H PZ 1  H PZ 2 ]

57
Worked Example
• A pipe carrying water tapers from a cross-section of 0.3 m2 at A to
0.15 m2 at B. At A the velocity, assumed uniform, is 1.8 m/s and the
pressure is 117 kPa gauge. If frictional effects are negligible,
determine the pressure at B which is 6m above the level of A.

58
Worked Example (15)
AB =
0.15m2

B
6m
A

AA = 0.3 m2
pA = 117 kPa gauge
vA = 1.8 m/s
Worked Example (15)
Apply continuity between A and B :
ρvA AA  ρvB AB
v A AA 1.8 x 0.3
 vB    3.6 m / s
AB 0.15
Apply Bernouilli between A and B (central streamline ) :
p A  gz A  12 v A2  pB  gzB  12 vB2
 pB  p A  g  z A  z B   12 v A2  vB2 
pB  117 x 103  1000 x 9.81 x  6   5001.82  3.62 
 53.3 kPa
60