Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 13

10 3 , the fluid is a continuum. For 10 3 10 1 , a continuum transport phenomena. For Kn < < Kn <

model with modified boundary conditions is appropriate. For Kn > 10, the fluid can only be described

by a free molecular flow model.
Kinetic theory can be applied to liquids as well. In this model, the motion of liquid molecules is
confined within a space limited by its neighboring molecules. Based on this theory, the viscosity of
a liquid can be estimated as:

exp 3

N A Z
y

8 T b
T

m ¼ : (2.8)

10 23

where N A ¼ 6.023 is the Avogadro number o r the number of molecules per mole;

10 34 m 2 kg/s is the Planck constant; y is the molar volume; T b is the boiling
temperature; and T is the temperature of the liquid.
The models of viscosity for gas (2.4) and for liquid (2.8) show opposite temperature dependency.
While the viscosity of gases increases with higher temperature, the viscosity of liquids decreases.
Example 2.2 ( Dynamic viscosity of water ).

Z ¼ 6.626068

the molar volume of water at 25 Cis

If

10 6 m 3 /mol, determine the viscosity of water at this temperature.
Solution. The boiling temperature of water under atmospheric pressure is assumed to be 100 C.
According to (2.8) , the viscosity of water can be estimated as:

18

exp 3

N A h
y

8 T b
T

m ¼ :

10 23 10 34

6 : 023 6 : 626068

¼ exp ½ 3 : 8 ð 100 þ 273 Þ=ð 25 þ 273 Þ

10 6 18

10 3 Pa

¼ 2 : 58 $ sec :

The equation overestimates the viscosity of water.
In the previous discussion, continuum properties are derived from the molecular model using
statistic methods. If there are not enough molecules for good statistics, numerical tools are used for
modeling transport phenomena at the molecular level. There are two numerical methods: molecular
dynamics (MD) and direct simulation Monte Carlo (DSMC). While MD is a deterministic method,
DSMC is a statistical method.
Molecular dynamics is a numerical method for modeling the motion of single molecules. The
interactions between the molecules can be described by the classical second law of Newton. The
simplest model of a molecule is a hard sphere of a mass m . The binary interaction between two
molecules is determined by the Lennard-Jones force (2.2) :

f ij ðrÞ¼f ji ðrÞ (2.9)

where r is the distance between the molecules. The bold letter indicates a vector variable.

The dynamics of molecule i can be described by Newton's second law:

¼jr i r j j

X

N

d r i
d t ¼

m ¼ f ij

(2.10)

j ¼ 1 ; j s 1