Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 22

where c p is the specific heat at constant pressure, b ¼ 1 =rðvr=vT Þ p is the thermal expansion

coefficient, k is the thermal conductivity, and F is the dissipation function. For Newtonian fluid, the

dissipation function caused by viscous stress is:

m 2 vu
vx

2

2 vy
vy

2

2 vw
vz

2 vy

vx þ

2 vw

vy þ

2 vu

vz þ vx x 2

vu
vy

vy
vz

vw

F ¼ þ þ þ þ þ

(2.20)

3 m vu

2

2 vy

vy þ

vw
vz

þ

vx þ

:

Assuming an incompressible flow, a constant thermal conductivity, and ignoring the kinetic energy
change, the energy equation can be simplified to heat-convection equation:

D T
D t ¼

2 T

rc p k V : (2.21)

2.1.2.4 Conservation of species
The conservation of species leads to the diffusion/convection equation:
D c
D t ¼

2 c

D V þ r g ; (2.22)

where c is the concentration of the species, D is the diffusion coefficient of the species (solute) in the
carrier fluid (solvent), and r g is the generation rate of the species per volume. The above equation
assumes a constant, isotropic diffusion coefficient. The left-hand side of (2.22) represents the accu-
mulation and convection of species. The first term on the right-hand side represents molecular
diffusion, while the last term is the generation of species. The above conservation equations can also be
formulated for the cylindrical and the spherical coordinate systems.

2.2 MOLECULAR DIFFUSION
2.2.1 Random walk and Brownian motion
A random walk is the path traced by a particle taking successive steps, each in a random direction. The
construction of a simple random walk follows the three basic rules:

The particle starts at a predefined point,

The distance done by each step is equal, and

The direction from one point to the next is random.

Following these rules, random walk of a particle can be realized with a simple program.
Considering a one-dimensional random walk on a line, the particle has a random choice of two
directions for each of its steps. The distance done by each step is assumed to be s . The position x ( n )at
a step ( n ) can be described as

X

n

x ð n Þ¼ s i (2.23)

i ¼ 1