Short Notes: Fluid Kinematics | Short Notes for Mechanical Engineering PDF Download

 Page 1


Fluid Kinematics
Fluid Kinematics deals with the motion of fluids such as displacement, velocity, 
acceleration, and other aspects. This topic is useful in terms of exam and 
knowledge of the candidate.
Kinematics o f Fluids fo r GATE 2019, ESE, & Other Exams
• Kinematics is the branch of classical mechanics that describes the motion of 
bodies and systems without consideration of the forces the cause the motion.
Types of Fluid Flows
Fluid flow may be classified under the following headings;
Steady & Unsteady Flow
S t e a d y f l o w U n s t e a d y f l o w
The flow in which 
characteristics of fluid 
like velocity, pressure, 
density" etc., at a point, 
do not change with 
time is called as steady 
flow.
dv dp dp 
— = 0 — = 0. — = 0 
dt dt dt
If velocity" pressure and 
den sity" changes with 
time then flow is 
unsteady flow.
dv dp dp 
— = = 0.— = = 0. — 0 
dt dt dt
Uniform & Non-uniform Flow
Page 2


Fluid Kinematics
Fluid Kinematics deals with the motion of fluids such as displacement, velocity, 
acceleration, and other aspects. This topic is useful in terms of exam and 
knowledge of the candidate.
Kinematics o f Fluids fo r GATE 2019, ESE, & Other Exams
• Kinematics is the branch of classical mechanics that describes the motion of 
bodies and systems without consideration of the forces the cause the motion.
Types of Fluid Flows
Fluid flow may be classified under the following headings;
Steady & Unsteady Flow
S t e a d y f l o w U n s t e a d y f l o w
The flow in which 
characteristics of fluid 
like velocity, pressure, 
density" etc., at a point, 
do not change with 
time is called as steady 
flow.
dv dp dp 
— = 0 — = 0. — = 0 
dt dt dt
If velocity" pressure and 
den sity" changes with 
time then flow is 
unsteady flow.
dv dp dp 
— = = 0.— = = 0. — 0 
dt dt dt
Uniform & Non-uniform Flow
Uniform Flow Non-uniform Flow
The flow in " w h ich 
velocity at any 
given time does 
not change with 
respect to distance.
M - o
111 this flow, 
velocity at any 
given time changes 
with respect to 
distance.
(&v)
— = = 0
Laminar & Turbulent Flow
Laminar Flowr Turbulent Flow
The flow in which the 
adjacent layers do not 
cross each other and 
move along well 
defined path.
The flow in which 
adjacent layers cross 
each other and do not 
move along well 
defined path.
WUUVW VVW W V\\A V\ V\ V^WWVWW
HE: Err:
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 r
T r r m m n m m m n m m n m
Rotational & Irrotational Flow
Rotational FIow r Irrotational Flow
If the fluid 
particles flowing 
alone stream lines, 
also rotate about 
their own axes, 
then flow is 
rotational.
If fluid particles do 
not rotate about 
their own axes, 
then flow is 
irrotational.
Combining these, the most common flow types are:
• Steady uniform flow
° Conditions do not change with position in the stream or with time.
° E.g. flow of water in a pipe of constant diameter at a constant velocity.
• Steady non-uniform flow
° Conditions change from point to point in the stream but do not change 
with time.
° E.g. Flow in a tapering pipe with constant velocity at the inlet.
• Unsteady uniform flow
o At a given instant in time the conditions at every point are the same but 
will change with time.
° E.g. A pipe of constant diameter connected to a pump pumping at a 
constant rate which is then switched off.
• Unsteady non-uniform flow
° Every condition of the flow may change from point to point and with time 
at every point.
° E.g. Waves in a channel
Flow Pattern
Page 3


Fluid Kinematics
Fluid Kinematics deals with the motion of fluids such as displacement, velocity, 
acceleration, and other aspects. This topic is useful in terms of exam and 
knowledge of the candidate.
Kinematics o f Fluids fo r GATE 2019, ESE, & Other Exams
• Kinematics is the branch of classical mechanics that describes the motion of 
bodies and systems without consideration of the forces the cause the motion.
Types of Fluid Flows
Fluid flow may be classified under the following headings;
Steady & Unsteady Flow
S t e a d y f l o w U n s t e a d y f l o w
The flow in which 
characteristics of fluid 
like velocity, pressure, 
density" etc., at a point, 
do not change with 
time is called as steady 
flow.
dv dp dp 
— = 0 — = 0. — = 0 
dt dt dt
If velocity" pressure and 
den sity" changes with 
time then flow is 
unsteady flow.
dv dp dp 
— = = 0.— = = 0. — 0 
dt dt dt
Uniform & Non-uniform Flow
Uniform Flow Non-uniform Flow
The flow in " w h ich 
velocity at any 
given time does 
not change with 
respect to distance.
M - o
111 this flow, 
velocity at any 
given time changes 
with respect to 
distance.
(&v)
— = = 0
Laminar & Turbulent Flow
Laminar Flowr Turbulent Flow
The flow in which the 
adjacent layers do not 
cross each other and 
move along well 
defined path.
The flow in which 
adjacent layers cross 
each other and do not 
move along well 
defined path.
WUUVW VVW W V\\A V\ V\ V^WWVWW
HE: Err:
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 r
T r r m m n m m m n m m n m
Rotational & Irrotational Flow
Rotational FIow r Irrotational Flow
If the fluid 
particles flowing 
alone stream lines, 
also rotate about 
their own axes, 
then flow is 
rotational.
If fluid particles do 
not rotate about 
their own axes, 
then flow is 
irrotational.
Combining these, the most common flow types are:
• Steady uniform flow
° Conditions do not change with position in the stream or with time.
° E.g. flow of water in a pipe of constant diameter at a constant velocity.
• Steady non-uniform flow
° Conditions change from point to point in the stream but do not change 
with time.
° E.g. Flow in a tapering pipe with constant velocity at the inlet.
• Unsteady uniform flow
o At a given instant in time the conditions at every point are the same but 
will change with time.
° E.g. A pipe of constant diameter connected to a pump pumping at a 
constant rate which is then switched off.
• Unsteady non-uniform flow
° Every condition of the flow may change from point to point and with time 
at every point.
° E.g. Waves in a channel
Flow Pattern
Three types of fluid element trajectories are defined: Streamlines, Pathlines, and 
Streaklines.
• Pathline is the actual path travelled by an individual fluid particle over some 
time period.The pathline of a fluid element A is simply the path it takes 
through space as a function of time. An example of a pathline is the trajectory 
taken by one puff of smoke which is carried by the steady or unsteady wind.
• Timeline is a set of fluid particles that form a line at a given instant.
• Streamline is a line that is everywhere tangent to the velocity field. 
Streamlines are obtained analytically by integrating the equations defining 
lines tangent to the velocity field as illustrated in the figure below:
dy _ V 
dx U
where u,v, and w are the velocity components in x, y and z directions respectively as 
sketched •
• Streakline is the locus of particles that have earlier passed through a 
prescribed point.A streakline is associated with a particular point P in space 
which has the fluid moving past it. All points which pass through this point are 
said to form the streakline of point P . An example of a streakline is the 
continuous line of smoke emitted by a chimney at point P , which will have 
some curved shape if the wind has a time-varying direction
• Streamtube: The streamlines passing through all these points form the 
surface of a stream-tube. Because there is no flow across the surface, each 
cross-section of the streamtube carries the same mass flow. So the 
streamtube is equivalent to a channel flow embedded in the rest of the 
flowfield.
Page 4


Fluid Kinematics
Fluid Kinematics deals with the motion of fluids such as displacement, velocity, 
acceleration, and other aspects. This topic is useful in terms of exam and 
knowledge of the candidate.
Kinematics o f Fluids fo r GATE 2019, ESE, & Other Exams
• Kinematics is the branch of classical mechanics that describes the motion of 
bodies and systems without consideration of the forces the cause the motion.
Types of Fluid Flows
Fluid flow may be classified under the following headings;
Steady & Unsteady Flow
S t e a d y f l o w U n s t e a d y f l o w
The flow in which 
characteristics of fluid 
like velocity, pressure, 
density" etc., at a point, 
do not change with 
time is called as steady 
flow.
dv dp dp 
— = 0 — = 0. — = 0 
dt dt dt
If velocity" pressure and 
den sity" changes with 
time then flow is 
unsteady flow.
dv dp dp 
— = = 0.— = = 0. — 0 
dt dt dt
Uniform & Non-uniform Flow
Uniform Flow Non-uniform Flow
The flow in " w h ich 
velocity at any 
given time does 
not change with 
respect to distance.
M - o
111 this flow, 
velocity at any 
given time changes 
with respect to 
distance.
(&v)
— = = 0
Laminar & Turbulent Flow
Laminar Flowr Turbulent Flow
The flow in which the 
adjacent layers do not 
cross each other and 
move along well 
defined path.
The flow in which 
adjacent layers cross 
each other and do not 
move along well 
defined path.
WUUVW VVW W V\\A V\ V\ V^WWVWW
HE: Err:
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 r
T r r m m n m m m n m m n m
Rotational & Irrotational Flow
Rotational FIow r Irrotational Flow
If the fluid 
particles flowing 
alone stream lines, 
also rotate about 
their own axes, 
then flow is 
rotational.
If fluid particles do 
not rotate about 
their own axes, 
then flow is 
irrotational.
Combining these, the most common flow types are:
• Steady uniform flow
° Conditions do not change with position in the stream or with time.
° E.g. flow of water in a pipe of constant diameter at a constant velocity.
• Steady non-uniform flow
° Conditions change from point to point in the stream but do not change 
with time.
° E.g. Flow in a tapering pipe with constant velocity at the inlet.
• Unsteady uniform flow
o At a given instant in time the conditions at every point are the same but 
will change with time.
° E.g. A pipe of constant diameter connected to a pump pumping at a 
constant rate which is then switched off.
• Unsteady non-uniform flow
° Every condition of the flow may change from point to point and with time 
at every point.
° E.g. Waves in a channel
Flow Pattern
Three types of fluid element trajectories are defined: Streamlines, Pathlines, and 
Streaklines.
• Pathline is the actual path travelled by an individual fluid particle over some 
time period.The pathline of a fluid element A is simply the path it takes 
through space as a function of time. An example of a pathline is the trajectory 
taken by one puff of smoke which is carried by the steady or unsteady wind.
• Timeline is a set of fluid particles that form a line at a given instant.
• Streamline is a line that is everywhere tangent to the velocity field. 
Streamlines are obtained analytically by integrating the equations defining 
lines tangent to the velocity field as illustrated in the figure below:
dy _ V 
dx U
where u,v, and w are the velocity components in x, y and z directions respectively as 
sketched •
• Streakline is the locus of particles that have earlier passed through a 
prescribed point.A streakline is associated with a particular point P in space 
which has the fluid moving past it. All points which pass through this point are 
said to form the streakline of point P . An example of a streakline is the 
continuous line of smoke emitted by a chimney at point P , which will have 
some curved shape if the wind has a time-varying direction
• Streamtube: The streamlines passing through all these points form the 
surface of a stream-tube. Because there is no flow across the surface, each 
cross-section of the streamtube carries the same mass flow. So the 
streamtube is equivalent to a channel flow embedded in the rest of the 
flowfield.
z<
3-D streamtube 2-D streamtube
Note:
• The figure below illustrates streamlines, pathlines, and streaklines for the 
case of a smoke being continuously emitted by a chimney at point P , in the 
presence of a shifting wind.
• In a steady flow, streamlines, pathlines, and streaklines all coincide.
• In this example, they would all be marked by the smoke line.
Velocity of Fluid Particle
• Velocity of a fluid along any direction can be defined as the rate of change of 
displacement of the fluid along that direction
• Let V be the resultant velocity of a fluid along any direction and u, v and w be 
the velocity components in x, y and z directions respectively.
• Mathematically the velocity components can be written as 
u = f ( x, y, z, t )
v = f ( x, y, z, t ) 
w = f ( x, y, z, t )
• Let V r is resultant velocity at any point in a fluid flow.
• Resultant velocity Vr = ui + vj + wk
Where u=dx/dt, v=dy/dt and w=dz/dt are the resultant vectors in X, Y and Z 
directions, respectively.
Acceleration of Fluid Particle
instantam 
wind velo
pathline of fluid element A 
{smoke puff]
S ’t) streakline at 
successive times
U‘ + V* + I f *
Acceleration of a fluid element along any direction can be defined as the rate 
of change of velocity of the fluid along that direction.
Page 5


Fluid Kinematics
Fluid Kinematics deals with the motion of fluids such as displacement, velocity, 
acceleration, and other aspects. This topic is useful in terms of exam and 
knowledge of the candidate.
Kinematics o f Fluids fo r GATE 2019, ESE, & Other Exams
• Kinematics is the branch of classical mechanics that describes the motion of 
bodies and systems without consideration of the forces the cause the motion.
Types of Fluid Flows
Fluid flow may be classified under the following headings;
Steady & Unsteady Flow
S t e a d y f l o w U n s t e a d y f l o w
The flow in which 
characteristics of fluid 
like velocity, pressure, 
density" etc., at a point, 
do not change with 
time is called as steady 
flow.
dv dp dp 
— = 0 — = 0. — = 0 
dt dt dt
If velocity" pressure and 
den sity" changes with 
time then flow is 
unsteady flow.
dv dp dp 
— = = 0.— = = 0. — 0 
dt dt dt
Uniform & Non-uniform Flow
Uniform Flow Non-uniform Flow
The flow in " w h ich 
velocity at any 
given time does 
not change with 
respect to distance.
M - o
111 this flow, 
velocity at any 
given time changes 
with respect to 
distance.
(&v)
— = = 0
Laminar & Turbulent Flow
Laminar Flowr Turbulent Flow
The flow in which the 
adjacent layers do not 
cross each other and 
move along well 
defined path.
The flow in which 
adjacent layers cross 
each other and do not 
move along well 
defined path.
WUUVW VVW W V\\A V\ V\ V^WWVWW
HE: Err:
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 r
T r r m m n m m m n m m n m
Rotational & Irrotational Flow
Rotational FIow r Irrotational Flow
If the fluid 
particles flowing 
alone stream lines, 
also rotate about 
their own axes, 
then flow is 
rotational.
If fluid particles do 
not rotate about 
their own axes, 
then flow is 
irrotational.
Combining these, the most common flow types are:
• Steady uniform flow
° Conditions do not change with position in the stream or with time.
° E.g. flow of water in a pipe of constant diameter at a constant velocity.
• Steady non-uniform flow
° Conditions change from point to point in the stream but do not change 
with time.
° E.g. Flow in a tapering pipe with constant velocity at the inlet.
• Unsteady uniform flow
o At a given instant in time the conditions at every point are the same but 
will change with time.
° E.g. A pipe of constant diameter connected to a pump pumping at a 
constant rate which is then switched off.
• Unsteady non-uniform flow
° Every condition of the flow may change from point to point and with time 
at every point.
° E.g. Waves in a channel
Flow Pattern
Three types of fluid element trajectories are defined: Streamlines, Pathlines, and 
Streaklines.
• Pathline is the actual path travelled by an individual fluid particle over some 
time period.The pathline of a fluid element A is simply the path it takes 
through space as a function of time. An example of a pathline is the trajectory 
taken by one puff of smoke which is carried by the steady or unsteady wind.
• Timeline is a set of fluid particles that form a line at a given instant.
• Streamline is a line that is everywhere tangent to the velocity field. 
Streamlines are obtained analytically by integrating the equations defining 
lines tangent to the velocity field as illustrated in the figure below:
dy _ V 
dx U
where u,v, and w are the velocity components in x, y and z directions respectively as 
sketched •
• Streakline is the locus of particles that have earlier passed through a 
prescribed point.A streakline is associated with a particular point P in space 
which has the fluid moving past it. All points which pass through this point are 
said to form the streakline of point P . An example of a streakline is the 
continuous line of smoke emitted by a chimney at point P , which will have 
some curved shape if the wind has a time-varying direction
• Streamtube: The streamlines passing through all these points form the 
surface of a stream-tube. Because there is no flow across the surface, each 
cross-section of the streamtube carries the same mass flow. So the 
streamtube is equivalent to a channel flow embedded in the rest of the 
flowfield.
z<
3-D streamtube 2-D streamtube
Note:
• The figure below illustrates streamlines, pathlines, and streaklines for the 
case of a smoke being continuously emitted by a chimney at point P , in the 
presence of a shifting wind.
• In a steady flow, streamlines, pathlines, and streaklines all coincide.
• In this example, they would all be marked by the smoke line.
Velocity of Fluid Particle
• Velocity of a fluid along any direction can be defined as the rate of change of 
displacement of the fluid along that direction
• Let V be the resultant velocity of a fluid along any direction and u, v and w be 
the velocity components in x, y and z directions respectively.
• Mathematically the velocity components can be written as 
u = f ( x, y, z, t )
v = f ( x, y, z, t ) 
w = f ( x, y, z, t )
• Let V r is resultant velocity at any point in a fluid flow.
• Resultant velocity Vr = ui + vj + wk
Where u=dx/dt, v=dy/dt and w=dz/dt are the resultant vectors in X, Y and Z 
directions, respectively.
Acceleration of Fluid Particle
instantam 
wind velo
pathline of fluid element A 
{smoke puff]
S ’t) streakline at 
successive times
U‘ + V* + I f *
Acceleration of a fluid element along any direction can be defined as the rate 
of change of velocity of the fluid along that direction.
If ax, ay and az are the components of acceleration along x, y and z direction 
respectively, they can be mathematically written as ax = du/ dt.
S im ila rly
3w d x thv ifi thv d z
; 3 a - dt by dt 3; dt i)i
But u = (dx/dt), v = (dy/dt) and w = (dz/dt) 
I lence
[f A is the resultant acceleration vector, it is given by 
A = a J + ttJ + ii_k
I 2 ^ 2
= ^ + a y + a.
For steady flow, the local acceleration will be zero
Stream Function
• The partial derivative of stream function with respect to any direction gives 
the velocity component at right angles to that direction. It is denoted by ip.
du du
— = v.— = -u
dx du
• Continuity equation for two-dimensional flow is
Equations of Rotational Flow
• As qj satisfies the continuity equation hence if ip exists then it is a possible 
case of fluid flow.
• Rotational components of fluid particles are:
dxd\' d\dx
11d x dx
= 2l ^ y~ dz
1 dx du
2 dx dx
Equation of Irrotational Flow