Page 111 - Optofluidics Fundamentals, Devices, and Applications

92 Cha pte r F i v e

density, U is the characteristic transport speed, a is an appropriate
size scale, and μ is the viscosity. For pure particle transport in a quies-
cent medium, U would be the particle speed and a would be its diam-
eter. In water then a 1-μm particle transported at 100 μm/s would
−4
have a Reynolds number of approximately 10 . If one is considering
an externally induced flow in a microchannel (say by the application
of a pressure difference), U would be the average flow speed in the
channel and a the channel height. In such a case Re can be as high as
approximately 0.1 but is usually much less. In either case, physically
this means that momentum transport occurs via diffusion rather than
convection and that we can ignore the nonlinear terms in the Navier-
Stokes equations. This also implies that the flow will reach its steady
state velocity relatively quickly and that the transient period can be
ignored. Under these assumptions the fluid dynamical equations
reduce to conservation of volume [Eq. (5-1a)] and the Stokes equation
[Eq. (5-1b)].

v
∇⋅ = 0 (5-1a)
2
μ∇ v − ∇ = 0P (5-1b)
where v is the velocity field and P is the pressure.

Hydrodynamic Forces on a Particle in a Flow
Equations (5-1a) and (5-1b) are descriptive of the fluid velocity at
every point in a flow. Generally speaking a particle in a flow will
experience a net pressure force (caused by pressure drop across the
particle) and a friction force (caused by the flow of a viscous liquid
over the surface). In the most general case the net drag force can be
written as
D ∫
F = (T F ⋅ )dS (5-2a)
n
s
where F = drag force
D
T = fluid stress tensor
F
n = normal vector to the surface of the particle.
For an incompressible Newtonian fluid, the stress tensor is written
as
I + ( v + ∇ )
T
T =−P μ ∇ v (5-2b)
F
where I is the isotropic tensor and ∇v is the gradient of the flow
velocity.
The above forms of the hydrodynamic equations are appropriate
for use in numerical simulations, but difficult to manipulate analyti-
cally. Simplified versions of these equations are, however, available