Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 37

If the particles are idealized so that they are small enough not to disturb the flow and large enough
so that molecular diffusion is neglected, they can move passively with the flow. The particle transport
mechanism can simply be described by the advection equations:
d x

8
<

= d t ¼ u ð x ; y ; z ; t Þ

d y

=

d t

¼ y ð x ; y ; z ; t Þ :

(2.85)

:

d z

=

d t

¼ w ð x ; y ; z ; t Þ

Mathematically, pathlines or trajectories can be obtained by solving (2.85) with the initial
condition at t z 0 ). Numerical integration methods, such as the Runge-Kutta

method, can be used for determining the positions of the particles.
In a two-dimensional flow, streamlines are given by the solution of:

¼ 0( x ¼ x 0 , y ¼ y 0 , z ¼

(

d x = d s ¼ u ð x ; y ; z ; t Þ

(2.86)

d y = d s ¼ y ð x ; y ; z ; t Þ

where t is treated as a constant and s is the independent variable. Streamlines build the image of the flow
field at a time instant. In experiments, streamlines can be constructed from the two images recorded with
particle image velocimetry (PIV). The flow is traced with fluorescent particles. Particle images are
recorded at two successive time instances t and t

þ D t . The particle velocities are determined by the

recorded particle displacement and the time delay t . The streamlines are tangential to the velocity at

each point. The streamlines can be depicted as the level sets of the stream function j , which is defined as:

D

(

d x = d s ¼ u ð x ; t Þ¼ vj = vy

(2.87)

d y = d s ¼ y ð x ; t Þ¼ vj = vx :

A streakline through a point ( x , y , z ) at a time instance is the curve formed by all particles, which

s

( t ) passed through this point previously. In experiments, the streakline is the tracing curve of

a nondiffusive tracer injected into the flow at the given point.
Example 2.9 ( Trajectories and streamlines ). An Eulerian velocity field is given as:

< s

(

d x = d t ¼ x sin ð at Þ

d y = d t ¼ y sin ð bt þ c Þ:

All variables and constants are dimensionless. Determine the trajectories and streamlines of
particles initially at ( x 0 ¼ 3).

Solution. Solving the differential equations with the initial condition ( x 0 , y 0 ) results in the position
of the fluid particle as a function of time:

1, y 0 ¼ 1), ( x 0 ¼ 1, y 0 ¼ 2), and ( x 0 ¼ 1, y 0 ¼

<

x 0 cosh 1
a

sinh 1
a

exp

1
a cos

x ¼ þ ð at Þ

y ¼ y 0 cosh cos þ sinh cos

exp

:

ð c Þ ð c Þ 1

b cos ðbt þ cÞ

:

b b

Figure 2.13 shows graphically the above pathlines, with the three initial particle positions, a ¼ 1,

b ¼ 2, and c ¼ p /2.