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Lagrangian Method for Describing Fluid Motion

In Lagrangian method the behaviour of each and every individual particle of fluid flow is described. The observer follows the particle and travels with the particle under study. Indentities of the particle are made by specifying their initial spatial coordinates at a given time. The position of this particle at any other instant of time is a function of its identity and time. Mathematically,

where S is the position vector of a particle with reference to a fixed point of reference at time t, and 0 is its initial position at a given time t= to . Thus  identifies the identity of the particle. Equation (4.1) can be written into scalar components of x, y, z rectangular cartesian coordinates system.

 

x = x (x0, y0, z0, t)

y = y (x0, y0‘ z0, t)

z = z (x0, y0, z0, t)

 

where x0, y0, z0 are the initial coordinates and x, y, z are the coordinates at time t of the

particle.

Now    where  k the unit vector along x , y and z axes respectively

 

If u, v and ware the scalar components of velocity in x, y and z directions respectively, we can write

u = [ d x / d t] x0 , y0 , z0

v = [ d y / d t] x0 , y0 , z0

w = [ d z / d t] x0 , y0 , z0

If ax, n y, and a z are the scalar components of acceleration in x, y and z directions respectively, we can write

a x = [ d2x / d t2] x0 , y0 , z0

a y = [ d2y / d t2] x0 , y0 , z0

a z = [ d2z / d t2] x0 , y0 , z0

Here (x0, y0, z0) are fixed point in fluid field at t = t0

This method is rarely suitable for practical applications because the mathematical

difficulties of solving equations ( 4.3) and ( 4.4).