Lagrangian method determines the behavior of fluid particles. To describe the motion of every particle of fluid flow, it is necessary to observe and study the motion of each particle.

Let us consider S be the position vector of one particle w.r.t a fixed point.  The particle travels from initial spatial coordinates at the initial time 0 to reach certain point in the given time t0. To identify the position of the particle, the equation can be given by-

 = S (0 , t)

For the rectangular Cartesian coordinate system in terms of x, y and z, the above equation can be written as-

x = x (x0, y0, z0, k)                                 (For scalar components)

Similarly,

y= y (x0, y0, z0, k)

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

Where,

x0 = Initial component of x

y0= Initial component of y

z0= Initial component of z

k = Unit vector

Calculate the velocity:

Suppose u be the scalar component of velocity in x direction

v be the scalar component of velocity in y direction

w be the scalar component of velocity in z direction

The equation can be written as-

u = [dx/dt](x0, y0, z0)
v = [dy/dt](x0, y0, z0)
w = [dz/dt](x0, y0, z0)

Where,

(x0, y0, z0) = Reference (fixed) point at t = t0

Calculate the acceleration:

Suppose ax be the scalar component of velocity in x direction

ay be the scalar component of velocity in y direction

az be the scalar component of velocity in z direction

The equation can be written as-

ax = [d2x/ dt2](x0, y0, z0)

ay = [d2y/ dt2](x0, y0, z0)

az = [d2z/ dt2](x0, y0, z0)

Note:

Lagrangian method is of limited use because of the difficulties in solving large numerical. Thus, its practical application is limited.

 

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