Page 384 - Introduction to Continuum Mechanics

368 Streamline, Pathline, Streakline, Steady, Unsteady, Laminar and Turbulent Flow

Similarly from dx^ /dt = *2, we have

thus

and obviously, x$ = X^

(m)Streakline
A streakline through a fixed point x^, is the line at time t formed by all the particles which
passed through ^ at T < t.
Let X = X(x, t) denote the inverse of x = x(X, t), then the particle which was at \, at time r,
has the material coordinates given by X = X(X O,T); this same particle is then at
x = x (X(x 0, t), t) at time t. Thus, the streakline at time t is given by

Example 6.10.3
Given the dimensionless velocity field

find the streakline formed by the particles which passed through the spatial position
(a 1,a 2,a 3).

Solution. The pathline equations for this velocity field was obtained in Example 6,10.2 to
u~

From which we obtain the inverse equations

Thus, the particle which passes through a lt a 2, a 3 at time T is given by