Fluid Kinematics - Notes | Study Fluid Mechanics for Mechanical Engineering - Mechanical Engineering

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.

Main Aspects
The subject has three main aspects:

1. Development of methods and techniques for describing and specifying the motions of fluids.
2. Determination of the conditions for the kinematic possibility of fluid motions.
3. Characterization of different types of motion and associated deformation rates of any fluid element.

Scalar & Vector Fields

➢ 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.

Description of Fluid Motion
(a) Lagrangian Method

• 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.
Therefore, (6.2a)

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)and hence,

where a_x, a_y, a_z are accelerations in x, y, z directions respectively.

➢ Advantages of Lagrangian Method

• Since the motion and trajectory of each fluid particle are known, its history can be traced.
• Since particles are identified at the start and traced throughout their motion, conservation of mass is inherent.

➢ Disadvantages of Lagrangian Method

•  The solution of the equations presents appreciable mathematical difficulties except for certain special cases and therefore, the method is rarely suitable for practical applications.