Description of Fluid Motion
A. Lagrangian Method
Using Lagrangian method, the fluid motion is described by tracing the kinematic behaviour 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 :
 |  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 | (6.1) |
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) |
| (where, x0,y0,z0 are the initial coordinates and x, y, z are the coordinates at a time t of the particle.) |
| (6.1a) |
| y = y(x0,y0,z0,t) | (6.1b) |
| z = z(x0,y0,z0,t) | (6.1c) |
Hence in can be expressed as
|
|  ,  , and  are the unit vectors along x, y and z axes respectively. |
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,
 | (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 ax, ay, az are accelerations in x, y, z directions respectively.
Advantages of Lagrangian Method:
- Since motion and trajectory of each fluid particle is 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 certain special cases and therefore, the method is rarely suitable for practical applications.
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