Kinematics of Fluid
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 subject has three main aspects:
Scalar and Vector Fields
Scalar: Scalar is a quantity which can be expressed by a single number representing its magnitude.
Example: mass, density and temperature.
Scalar Field
If at every point in a region, a scalar function has a defined value, the region is called a scalar field.
Example: Temperature distribution in a rod.
Vector: Vector is a quantity which is specified by both magnitude and direction.
Example: Force, Velocity and Displacement.
Vector Field
If at every point in a region, a vector function has a defined value, the region is called a vector field.
Example: velocity field of a flowing fluid.
Flow Field
The region in which the flow parameters i.e. velocity, pressure etc. are defined at each and every point at any instant of time is called a flow field.
Thus a flow field would be specified by the velocities at different points in the region at different times.
Description of Fluid Motion
A. Lagrangian Method
(i) Using Lagrangian method, the fluid motion is described by tracing the kinematic behaviour of each particle constituting the flow.
(ii) Identities of the particles are made by specifying their initial position (spatial location) at a given time. The position of a particle at any other instant of time then becomes a function of its identity and time.
Analytical expression of the last statement :
is the position vector of a particle (with respect to a fixed point of reference) at a time t.
is its initial position at a given time t =t0 (1)
Equation (1) can be written into scalar components with respect to a rectangular Cartesian frame of coordinates as:
x = x(x0,y0,z0,t) (where, x0,y0,z0 are the initial coordinates and x, y, z are the coordinates at a time t of the particle.) (1a)
y = y(x0,y0,z0,t) (1b)
z = z(x0,y0,z0,t) (1c)
Hence 
can be expressed as
Velocity and acceleration
The velocity 
and acceleration 
of the fluid particle can be obtained from the material derivatives of the position of the particle with respect to time. Therefore,
In terms of scalar components,
where u, v, w are the components of velocity in x, y, z directions respectively.
Similarly, for the acceleration,
and hence,
where ax, ay, az are accelerations in x, y, z directions respectively.
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.
B. Eulerian Method
The method was developed by Leonhard Euler.
This method is of greater advantage since it avoids the determination of the movement of each individual fluid particle in all details.
It seeks the velocity and its variation with time t at each and every location ( ) in the flow field.
In Eulerian view, all hydrodynamic parameters are functions of location and time.
Mathematical representation of the flow field in Eulerian method:
Where
Therefore,
u = u (x, y, z, t)
v = v (x, y, z, t)
w = w (x, y, z, t)
Services: - Kinematics of Fluid Homework | Kinematics of Fluid Homework Help | Kinematics of Fluid Homework Help Services | Live Kinematics of Fluid Homework Help | Kinematics of Fluid Homework Tutors | Online Kinematics of Fluid Homework Help | Kinematics of Fluid Tutors | Online Kinematics of Fluid Tutors | Kinematics of Fluid Homework Services | Kinematics of Fluid