Solving (9.92) subject to (9.93) gives the particle coordinates as a function of time:
We see that for t x, the particle trajectory is described by x(t) = Ux and y(t) = Ut. Thus the particle eventually ends up at a distance Ux to the right of its original fluid streamline. Larger particles with high relaxation times will move, because of their inertia, significantly to the right, while small particles, with x —► 0, will follow closely their original streamline. For example, for a 2-pm-diameter particle moving with a speed U = 20 m s_1 and having density pp = 2gem"3, we find that the displacement is 0.48mm, while for a 20-pm-diameter particle Ux = 4.83 cm.
The flow depicted in Figure 9.13 is the most idealized one representing the stagnation flow of a fluid toward a flat plane (see also Figure 9.14). If we imagine that in Figure 9.14 there is a flat plate at x = 0 and that y = 0 is the line of symmetry, then all the particles that initially are a distance smaller than xq = Ux from the line of symmetry will collide with x(t) = Ux[ 1 - exp(—i/x)] y(t) = -Ux[ 1 - exp(—i/x)] + Ut
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