Detailed Notes: Fluid Kinematics | Fluid Mechanics for Mechanical Engineering PDF Download

 Page 1


 
 
 
 
FLUID KINEMATICS 
Introduction 
Kinematics is the geometry of Motion. 
Kinematics of fluid describes the fluid motion and its consequences without consideration of the 
nature of forces causing the motion. 
 
The fluid kinematics deals with description of the motion of the fluids without reference to the 
force causing the motion. 
Thus it is emphasized to know how fluid flows and how to describe fluid motion. This concept 
helps us to simplify the complex nature of a real fluid flow. 
When a fluid is in motion, individual particles in the fluid move at different velocities. Moreover 
at different instants fluid particles change their positions. In order to analyze the flow behavior, a 
function of space and time, we follow one of the following approaches 
1. Lagarangian approach 
2. Eularian approach 
In the Lagarangian approach a fluid particle of fixed mass is selected. We follow the fluid 
particle during the course of motion with time  
 
The fluid particles may change their shape, size and state as they move. As mass of fluid 
particles remains constant throughout the motion, the basic laws of mechanics can be applied to 
them at all times. The task of following large number of fluid particles is quite difficult. 
Therefore this approach is limited to some special applications for example re-entry of a 
spaceship into the earth's atmosphere and flow measurement system based on particle imagery. 
Page 2


 
 
 
 
FLUID KINEMATICS 
Introduction 
Kinematics is the geometry of Motion. 
Kinematics of fluid describes the fluid motion and its consequences without consideration of the 
nature of forces causing the motion. 
 
The fluid kinematics deals with description of the motion of the fluids without reference to the 
force causing the motion. 
Thus it is emphasized to know how fluid flows and how to describe fluid motion. This concept 
helps us to simplify the complex nature of a real fluid flow. 
When a fluid is in motion, individual particles in the fluid move at different velocities. Moreover 
at different instants fluid particles change their positions. In order to analyze the flow behavior, a 
function of space and time, we follow one of the following approaches 
1. Lagarangian approach 
2. Eularian approach 
In the Lagarangian approach a fluid particle of fixed mass is selected. We follow the fluid 
particle during the course of motion with time  
 
The fluid particles may change their shape, size and state as they move. As mass of fluid 
particles remains constant throughout the motion, the basic laws of mechanics can be applied to 
them at all times. The task of following large number of fluid particles is quite difficult. 
Therefore this approach is limited to some special applications for example re-entry of a 
spaceship into the earth's atmosphere and flow measurement system based on particle imagery. 
 
 
 
In the Eularian method a finite region through which fluid flows in and out is used. Here we do 
not keep track position and velocity of fluid particles of definite mass. But, within the region, the 
field variables which are continuous functions of space dimensions ( x , y , z ) and time ( t ), are 
defined to describe the flow. These field variables may be scalar field variables, vector field 
variables and tensor quantities. For example, pressure is one of the scalar fields. Sometimes this 
finite region is referred as control volume or flow domain. 
For example the pressure field 'P' is a scalar field variable and defined as 
                     
Velocity field, a vector field, is defined as  
Similarly shear stress is a tensor field variable and defined as 
                  
Note that we have defined the fluid flow as a three dimensional flow in a Cartesian co-ordinates 
system. 
Advantages of Lagrangian Method: 
1. Since motion and trajectory of each fluid particle is known, its history can be traced. 
2. Since particles are identified at the start and traced throughout their motion, conservation 
of mass is inherent. 
Disadvantages of Lagrangian Method: 
1. The solution of the equations presents appreciable mathematical difficulties except 
certain special cases and therefore, the method is rarely suitable for practical applications. 
Types of Fluid Flow 
Uniform and Non-uniform flow : If the velocity at given instant is the same in both magnitude 
and direction throughout the flow domain, the flow is described as uniform. 
Page 3


 
 
 
 
FLUID KINEMATICS 
Introduction 
Kinematics is the geometry of Motion. 
Kinematics of fluid describes the fluid motion and its consequences without consideration of the 
nature of forces causing the motion. 
 
The fluid kinematics deals with description of the motion of the fluids without reference to the 
force causing the motion. 
Thus it is emphasized to know how fluid flows and how to describe fluid motion. This concept 
helps us to simplify the complex nature of a real fluid flow. 
When a fluid is in motion, individual particles in the fluid move at different velocities. Moreover 
at different instants fluid particles change their positions. In order to analyze the flow behavior, a 
function of space and time, we follow one of the following approaches 
1. Lagarangian approach 
2. Eularian approach 
In the Lagarangian approach a fluid particle of fixed mass is selected. We follow the fluid 
particle during the course of motion with time  
 
The fluid particles may change their shape, size and state as they move. As mass of fluid 
particles remains constant throughout the motion, the basic laws of mechanics can be applied to 
them at all times. The task of following large number of fluid particles is quite difficult. 
Therefore this approach is limited to some special applications for example re-entry of a 
spaceship into the earth's atmosphere and flow measurement system based on particle imagery. 
 
 
 
In the Eularian method a finite region through which fluid flows in and out is used. Here we do 
not keep track position and velocity of fluid particles of definite mass. But, within the region, the 
field variables which are continuous functions of space dimensions ( x , y , z ) and time ( t ), are 
defined to describe the flow. These field variables may be scalar field variables, vector field 
variables and tensor quantities. For example, pressure is one of the scalar fields. Sometimes this 
finite region is referred as control volume or flow domain. 
For example the pressure field 'P' is a scalar field variable and defined as 
                     
Velocity field, a vector field, is defined as  
Similarly shear stress is a tensor field variable and defined as 
                  
Note that we have defined the fluid flow as a three dimensional flow in a Cartesian co-ordinates 
system. 
Advantages of Lagrangian Method: 
1. Since motion and trajectory of each fluid particle is known, its history can be traced. 
2. Since particles are identified at the start and traced throughout their motion, conservation 
of mass is inherent. 
Disadvantages of Lagrangian Method: 
1. The solution of the equations presents appreciable mathematical difficulties except 
certain special cases and therefore, the method is rarely suitable for practical applications. 
Types of Fluid Flow 
Uniform and Non-uniform flow : If the velocity at given instant is the same in both magnitude 
and direction throughout the flow domain, the flow is described as uniform. 
 
 
 
 
Mathematically the velocity field is defined as , independent to space dimensions 
( x , y , z ). 
When the velocity changes from point to point it is said to be non-uniform flow. Fig. shows 
uniform flow in test section of a well designed wind tunnel and describing non uniform velocity 
region at the entrance. 
Steady and unsteady flows 
The flow in which the field variables don't vary with time is said to be steady flow. For steady 
flow, 
                               Or            
It means that the field variables are independent of time. This assumption simplifies the fluid 
problem to a great extent. Generally, many engineering flow devices and systems are designed to 
operate them during a peak steady flow condition. 
If the field variables in a fluid region vary with time the flow is said to be unsteady flow. 
                                   
 
 
Page 4


 
 
 
 
FLUID KINEMATICS 
Introduction 
Kinematics is the geometry of Motion. 
Kinematics of fluid describes the fluid motion and its consequences without consideration of the 
nature of forces causing the motion. 
 
The fluid kinematics deals with description of the motion of the fluids without reference to the 
force causing the motion. 
Thus it is emphasized to know how fluid flows and how to describe fluid motion. This concept 
helps us to simplify the complex nature of a real fluid flow. 
When a fluid is in motion, individual particles in the fluid move at different velocities. Moreover 
at different instants fluid particles change their positions. In order to analyze the flow behavior, a 
function of space and time, we follow one of the following approaches 
1. Lagarangian approach 
2. Eularian approach 
In the Lagarangian approach a fluid particle of fixed mass is selected. We follow the fluid 
particle during the course of motion with time  
 
The fluid particles may change their shape, size and state as they move. As mass of fluid 
particles remains constant throughout the motion, the basic laws of mechanics can be applied to 
them at all times. The task of following large number of fluid particles is quite difficult. 
Therefore this approach is limited to some special applications for example re-entry of a 
spaceship into the earth's atmosphere and flow measurement system based on particle imagery. 
 
 
 
In the Eularian method a finite region through which fluid flows in and out is used. Here we do 
not keep track position and velocity of fluid particles of definite mass. But, within the region, the 
field variables which are continuous functions of space dimensions ( x , y , z ) and time ( t ), are 
defined to describe the flow. These field variables may be scalar field variables, vector field 
variables and tensor quantities. For example, pressure is one of the scalar fields. Sometimes this 
finite region is referred as control volume or flow domain. 
For example the pressure field 'P' is a scalar field variable and defined as 
                     
Velocity field, a vector field, is defined as  
Similarly shear stress is a tensor field variable and defined as 
                  
Note that we have defined the fluid flow as a three dimensional flow in a Cartesian co-ordinates 
system. 
Advantages of Lagrangian Method: 
1. Since motion and trajectory of each fluid particle is known, its history can be traced. 
2. Since particles are identified at the start and traced throughout their motion, conservation 
of mass is inherent. 
Disadvantages of Lagrangian Method: 
1. The solution of the equations presents appreciable mathematical difficulties except 
certain special cases and therefore, the method is rarely suitable for practical applications. 
Types of Fluid Flow 
Uniform and Non-uniform flow : If the velocity at given instant is the same in both magnitude 
and direction throughout the flow domain, the flow is described as uniform. 
 
 
 
 
Mathematically the velocity field is defined as , independent to space dimensions 
( x , y , z ). 
When the velocity changes from point to point it is said to be non-uniform flow. Fig. shows 
uniform flow in test section of a well designed wind tunnel and describing non uniform velocity 
region at the entrance. 
Steady and unsteady flows 
The flow in which the field variables don't vary with time is said to be steady flow. For steady 
flow, 
                               Or            
It means that the field variables are independent of time. This assumption simplifies the fluid 
problem to a great extent. Generally, many engineering flow devices and systems are designed to 
operate them during a peak steady flow condition. 
If the field variables in a fluid region vary with time the flow is said to be unsteady flow. 
                                   
 
 
 
 
 
Four possible combinations 
 
One, two and three dimensional flows 
 
Although fluid flow generally occurs in three dimensions in which the velocity field vary with 
three space co-ordinates and time. But, in some problem we may use one or two space 
components to describe the velocity field. For example consider a steady flow through a long 
straight pipe of constant cross-section. The velocity distributions shown in figure are 
independent of co-ordinate x and  and a function of r only. Thus the flow field is one 
dimensional. 
Page 5


 
 
 
 
FLUID KINEMATICS 
Introduction 
Kinematics is the geometry of Motion. 
Kinematics of fluid describes the fluid motion and its consequences without consideration of the 
nature of forces causing the motion. 
 
The fluid kinematics deals with description of the motion of the fluids without reference to the 
force causing the motion. 
Thus it is emphasized to know how fluid flows and how to describe fluid motion. This concept 
helps us to simplify the complex nature of a real fluid flow. 
When a fluid is in motion, individual particles in the fluid move at different velocities. Moreover 
at different instants fluid particles change their positions. In order to analyze the flow behavior, a 
function of space and time, we follow one of the following approaches 
1. Lagarangian approach 
2. Eularian approach 
In the Lagarangian approach a fluid particle of fixed mass is selected. We follow the fluid 
particle during the course of motion with time  
 
The fluid particles may change their shape, size and state as they move. As mass of fluid 
particles remains constant throughout the motion, the basic laws of mechanics can be applied to 
them at all times. The task of following large number of fluid particles is quite difficult. 
Therefore this approach is limited to some special applications for example re-entry of a 
spaceship into the earth's atmosphere and flow measurement system based on particle imagery. 
 
 
 
In the Eularian method a finite region through which fluid flows in and out is used. Here we do 
not keep track position and velocity of fluid particles of definite mass. But, within the region, the 
field variables which are continuous functions of space dimensions ( x , y , z ) and time ( t ), are 
defined to describe the flow. These field variables may be scalar field variables, vector field 
variables and tensor quantities. For example, pressure is one of the scalar fields. Sometimes this 
finite region is referred as control volume or flow domain. 
For example the pressure field 'P' is a scalar field variable and defined as 
                     
Velocity field, a vector field, is defined as  
Similarly shear stress is a tensor field variable and defined as 
                  
Note that we have defined the fluid flow as a three dimensional flow in a Cartesian co-ordinates 
system. 
Advantages of Lagrangian Method: 
1. Since motion and trajectory of each fluid particle is known, its history can be traced. 
2. Since particles are identified at the start and traced throughout their motion, conservation 
of mass is inherent. 
Disadvantages of Lagrangian Method: 
1. The solution of the equations presents appreciable mathematical difficulties except 
certain special cases and therefore, the method is rarely suitable for practical applications. 
Types of Fluid Flow 
Uniform and Non-uniform flow : If the velocity at given instant is the same in both magnitude 
and direction throughout the flow domain, the flow is described as uniform. 
 
 
 
 
Mathematically the velocity field is defined as , independent to space dimensions 
( x , y , z ). 
When the velocity changes from point to point it is said to be non-uniform flow. Fig. shows 
uniform flow in test section of a well designed wind tunnel and describing non uniform velocity 
region at the entrance. 
Steady and unsteady flows 
The flow in which the field variables don't vary with time is said to be steady flow. For steady 
flow, 
                               Or            
It means that the field variables are independent of time. This assumption simplifies the fluid 
problem to a great extent. Generally, many engineering flow devices and systems are designed to 
operate them during a peak steady flow condition. 
If the field variables in a fluid region vary with time the flow is said to be unsteady flow. 
                                   
 
 
 
 
 
Four possible combinations 
 
One, two and three dimensional flows 
 
Although fluid flow generally occurs in three dimensions in which the velocity field vary with 
three space co-ordinates and time. But, in some problem we may use one or two space 
components to describe the velocity field. For example consider a steady flow through a long 
straight pipe of constant cross-section. The velocity distributions shown in figure are 
independent of co-ordinate x and  and a function of r only. Thus the flow field is one 
dimensional. 
 
 
 
 
But in the case of flow over a weir of constant cross-section (), we can use two co-ordinate 
system x and z in defining the velocity field. So, this flow is a case of two dimensional flow. The 
reduction of independent space variable in a fluid flow problem makes it simpler to solve. 
 
Laminar and Turbulent flow 
In fluid flows, there are two distinct fluid behaviors experimentally observed. These behaviors 
were first observed by Sir Osborne Reynolds. He carried out a simple experiment in which water 
was discharged through a small glass tube from a large tank (the schematic of the experiment 
shown in Fig.). A colour dye was injected at the entrance of the tube and the rate of flow could 
be regulated by a valve at the out let. 
When the water flowed at low velocity, it was found that the die moved in a straight line. This 
clearly showed that the particles of water moved in parallel lines. This type of flow is called 
laminar flow, in which the particles of fluid moves along smooth paths in layers. There is no 
exchange of momentum from fluid particles of one layer to the fluid particles of another layer. 
This type of flow mainly occurs in high viscous fluid flows at low velocity, for example, oil 
flows at low velocity. Fig. shows the steady velocity profile for a typical laminar flow. 
When the water flowed at high velocity, it was found that the dye colour was diffused over the 
whole cross section. This could be interpreted that the particles of fluid moved in very irregular 
paths, causing an exchange of momentum from one fluid particle to another. This type of flow is 
known as turbulent flow. The time variation of velocity at a point for the turbulent flow is shown 
in Fig