Kinematics of Fluid (Practice) Mechanical Engineering Notes | EduRev

Mechanical Engineering : Kinematics of Fluid (Practice) Mechanical Engineering Notes | EduRev

The document Kinematics of Fluid (Practice) Mechanical Engineering Notes | EduRev is a part of Mechanical Engineering category.
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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.Kinematics of Fluid (Practice) Mechanical Engineering Notes | EduRev

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 and 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 

Kinematics of Fluid (Practice) Mechanical Engineering Notes | EduRev
whereKinematics of Fluid (Practice) Mechanical Engineering Notes | EduRevis the position vector of a particle (with respect to a fixed point of reference) at a time t. (6.1)
Kinematics of Fluid (Practice) Mechanical Engineering Notes | EduRevis 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.
HenceKinematics of Fluid (Practice) Mechanical Engineering Notes | EduRevcan be expressed as:

 Kinematics of Fluid (Practice) Mechanical Engineering Notes | EduRev   Kinematics of Fluid (Practice) Mechanical Engineering Notes | EduRev are the unit vectors along x, y and z axes respectively.

The velocityKinematics of Fluid (Practice) Mechanical Engineering Notes | EduRevand acceleration of the fluid particle can be obtained from the material derivatives of the position of the particle with respect to time.
Therefore,Kinematics of Fluid (Practice) Mechanical Engineering Notes | EduRev(6.2a)

In terms of scalar components:
Kinematics of Fluid (Practice) Mechanical Engineering Notes | EduRev

where u, v, w are the components of velocity in x, y, z directions respectively.

Similarly, for the acceleration,Kinematics of Fluid (Practice) Mechanical Engineering Notes | EduRev(6.3a)
and hence,
Kinematics of Fluid (Practice) Mechanical Engineering Notes | EduRev
where ax, ay, az 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.
(b) Eulerian Method
  • The method was developed by Leonhard Euler.
  • This method is of a greater advantage since it avoids the determination of the movement of each individual fluid particle in all details.
  • It seeks the velocityKinematics of Fluid (Practice) Mechanical Engineering Notes | EduRevand its variation with time t at each and every locationKinematics of Fluid (Practice) Mechanical Engineering Notes | EduRevin of the flow field.
  • In the Eulerian view, all hydrodynamic parameters are functions of location and time.

➢ Mathematical representation of the flow field in the Eulerian method
Kinematics of Fluid (Practice) Mechanical Engineering Notes | EduRev= v (Kinematics of Fluid (Practice) Mechanical Engineering Notes | EduRev, t) (6.4), whereKinematics of Fluid (Practice) Mechanical Engineering Notes | EduRevandKinematics of Fluid (Practice) Mechanical Engineering Notes | EduRev
Therefore:
► u = u (x, y, z, t)
► v = v (x, y, z, t)
► w = w (x, y, z, t) 

(c) The relation between Eulerian and Lagrangian Method

The Eulerian description can be written as:
Kinematics of Fluid (Practice) Mechanical Engineering Notes | EduRev(6.5)
or
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:
Kinematics of Fluid (Practice) Mechanical Engineering Notes | EduRev
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.


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