Page 383 - Introduction to Continuum Mechanics

Newtonian Viscous Fluid 367

Similarly, from dx2/ch = x,^-, we have

s
Thus, X2 — a^ . Obviously, XT, = #3

(ii) Pathline
A pathline is the path traversed by a fluid particle. To photograph a pathline, it is
necessary to use long time exposure of a reflecting particle. Mathematically, the pathline of
a particle which was at X at time t 0 can be obtained from the velocity field v(x,/) as follows:
Let x = x(f) be the pathline, then

For the velocity field of the previous example, find the pathline for a particle which was at
(XiJC^s) at time *o
Solution From

we have

Thus,

i.e.,