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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.

__The subject has three main aspects:__

**Development of methods and techniques**for describing and specifying the motions of fluids.**Determination of the conditions**for the kinematic possibility of fluid motions.**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 =t_{0}__Equation (6.1) can be written into scalar components with respect to a rectangular cartesian frame of coordinates as:__

► x = x(x_{0},y_{0},z_{0},t) **(6.1a)**

► y = y(x_{0},y_{0},z_{0},t) **(6.1b**)

► z = z(x_{0},y_{0},z_{0},t) **(6.1c)**

where, x_{0},y_{0},z_{0} 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)**

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, (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.

- 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 velocityand its variation with time t at each and every locationin 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**

= v (, t) (6.4), whereand__Therefore:__

► u = u (x, y, z, t)

► v = v (x, y, z, t)

► w = w (x, y, z, t)

Try yourself:In which method of describing fluid motion, the observer remains stationary and observes changes in the fluid parameters at a particular point only?

View Solution

__The Eulerian description can be written as:__**(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:__

or x = x(x_{0},y_{0},z_{0},t)

y = y(x_{0},y_{0},z_{0},t)

z = z(x_{0},y_{0},z_{0},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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