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: Scalar is a quantity that 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 that 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.
Using the Lagrangian method, the fluid motion is described by tracing the kinematic behavior of each particle constituting the flow.
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
whereis the position vector of a particle (with respect to a fixed point of reference) at a time t. (6.1)
is its initial position at a given time, t =t0
Equation (6.1) can be written into scalar components with respect to a rectangular cartesian frame of coordinates as:
► x = x(x0,y0,z0,t) (6.1a)
► y = y(x0,y0,z0,t) (6.1b)
► z = z(x0,y0,z0,t) (6.1c)
where, x0,y0,z0 are the initial coordinates and x, y, z are the coordinates at a time t of the particle.
Hencecan be expressed as:
|are the unit vectors along x, y and z axes respectively.|
The velocityand acceleration of the fluid particle can be obtained from the material derivatives of the position of the particle with respect to time.
In terms of scalar components:
where u, v, w are the components of velocity in x, y, z directions respectively.
Similarly, for the acceleration,(6.3a)
where ax, ay, az are accelerations in x, y, z directions respectively.
➢ Advantages of Lagrangian Method
➢ Disadvantages of Lagrangian Method
➢ Mathematical representation of the flow field in the Eulerian method
= v (, t) (6.4), whereand
► u = u (x, y, z, t)
► v = v (x, y, z, t)
► w = w (x, y, z, t)
The Eulerian description can be written as:
dx/dt = u(x,y,z,t)
dy/dt = v(x,y,z,t)
dz/dt = w(x,y,z,t)
The integration of Eq. (6.5) yields the constants of integration which are to be found from the initial coordinates of the fluid particles.
Hence, the solution of Eq. (6.5) gives the equations of Lagrange as:
or x = x(x0,y0,z0,t)
y = y(x0,y0,z0,t)
z = z(x0,y0,z0,t)
The above relation is the same as the Lagrangian formulation.
In principle, the Lagrangian method of description can always be derived from the Eulerian method.