Miniaturized Detection Systems - Waterborne Pathogens: Detection Methods and Applications - page 312

gases, and how they interact with their surroundings. The equation describ-
ing the motion of incompressible Newtonian fluids is the Navier-Stokes
equation

(

∂ ν
∂ t

)

=−∇ p + η ∇ 2 v ρ + ν · ∇ ν

(10.1)

this is, in principle, a representation of Newton's 2nd law of mechanics for a
fluid. 21 The left hand side comprises the density, ρ , multiplied by the accel-
eration. The two terms in the acceleration are the local acceleration and the
convective acceleration, respectively. The convective acceleration arises from
spatial variations in the velocity field and is nonlinear; this explains why
analytical solutions to general flow using the Navier-Stokes equation are
difficult to obtain. The right hand side of the equation represents the forces
acting upon the flow where ∇ p is the pressure and η ∇ 2 v is the viscous force.
From the Navier-Stokes equation it is possible to derive a dimensionless
parameter known as the Reynolds number, defined as

Re = ρ vl

η

(10.2)

where ρ is the fluid density, v the velocity, l the typical length scale of the
system, and η the viscosity. The Reynolds number is an estimate of the
moment of inertia versus the viscous forces in a fluid system. 21 As the length
scales and transport velocities of a system decrease, so does the Reynolds
number. When the Reynolds number is very small the nonlinear terms
in the Navier-Stokes equation disappear, resulting in linear and predict-
able Stokes flow. This is the case in typical microfluidic systems; e.g. for
water, velocities of 1 µm s −1 -1 cm s −1 and channel radii of 1-100 µm, the
Reynolds numbers are in the range 10 −6 -10. Such low Reynolds numbers
means that the viscous forces dominate the system and that, as mentioned
above, flows are laminar and turbulence-free, with diffusion as the primary
method of mixing. 22 This enables the design of systems with functionalities
that are virtually impossible to obtain at the macroscopic scale.
Diffusion is the transport of matter as a result of random molecular
motion which ultimately acts to neutralize concentration gradients. The
proportionality of the flux of molecules to the concentration gradient, can
be described using Fick's first law, which in one dimension is given by

J =− D dc
dx

(10.3)