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.
Links of Previous Main Topic:-
- Vapour compression refrigeration cycle introduction
- Basic fluid mechanics and properties of fluids introduction
- Fluid statics introduction
- Manometers measurement pressure
- Fluid kinematics
- Lagrangian method for describing fluid method
Links of Next Mechanical Engineering Topics:-
- Eulerian method for describing fluid method
- Lagrangian relationship from eulerian equations
- Steady and unsteady flows
- Uniform and non uniform flows
- Stream line
- Path lines
- Streak lines
- Acceleration of a fluid particle
- Continuity equation
- Continuity equation in three dimensions in differential form
- Continuity equation in a cylindrical polar coordinate system
- Bernoullis equation
- Basics and statics of particles introduction
- Equilibrium of rigid bodies introduction
