2.3 Water in Motion: Hydrodynamics
25
v
a
v
v
b
Fig. 2.5: Patterns of the flow: a streamlines, b streaklines
time. A pathline can be found by long exposures of a single marked particle
moving with the flow.
As streamlines, pathlines and streaklines are dependent on time in various
ways, their shapes are different for flow which is dependent on time (unsteady
flow) and flow which is independent of time (steady flow). There are two basic
methods for studying fluid motion: the Lagrangian method and the Eulerian
method.
Lagrangian Method. This method is named after the Italian-French mathematician Joseph-Louis Lagrange (1736-1813). The approach consists offollowing fluid particles during the course of time and giving paths, velocities, and
pressures in terms of the original position of the particles, and the time elapsed
since the particles occupied their original positions.
If the initial position of a given particle at time to is (xo, Yo, zo), the Lagrangian
method gives the position (x, y, z) at the instant t as:
x = F1(xo, Yo, Zo, t - to) }
y = F2(xo, Yo, Zo, t - to) ,
Z = F3(XO, Yo, Zo, t - to)
(2.13)
where functions F1(), F2() and F3() provide the 'recipes' for changing the
particle's x, y, Z coordinates, respectively, during the elapsed time (t - to). The
Lagrangian method, seldom used in fluid mechanics, is more appropriate to
solid mechanics.