Biomedical Engineering Reference
In-Depth Information
Table1.1
Scaling Law of Typical Physical Quantities Intervening in Microfluidics
Physical Quantity
Scale
Reference
Area
L
2
Volume
L
3
Velocity
L
L
0
Time
Gravity force
L
3
Inertia
L
3
Hydrostatic pressure
L
Hydraulic resistance
L
−4
Chapter 2
Stokes drag
L
Chapter 6
Diffusion constant
L
−1
Chapter 5
Reynolds number
L
2
(1.9)
Péclet number
L
2
(1.25)
Diffusion time (mass or temperature)
L
2
Chapter 5
Laplace pressure
L
-1
Chapter 3
Bond number
L
2
(1.20)
Capillary rise
L
-1
Chapter 3
Capillary number
L
(1.10)
Weber number
L
3
(1.11)
Ohnesorge number
L
-1/2
(1.12)
Deborah number
L
-3/2
(1.17)
Elastocapillary number
L
-2
(1.19)
Marangoni number
L
(1.23)
Marangoni force
L
-1
Chapter 3
Knudsen number
L
-1
(1.4)
Electric field
L
-1
Chapter 7
At the same time, the use of the theorem is very powerful because it does not
involve the form of the equation or system of equations, just the variables interven-
ing in the problem. Also, because the choice of dimensionless parameters is not
unique, it only provides a way of generating sets of dimensionless parameters. The
user still has to determine the meaningful dimensionless parameters corresponding
to the specific problem.
More formally, in mathematical terms, if we have an equation such as
(
)
=
f q q q
(1.1)
where the
q
i
are the
n
physical variables, expressed in terms of
k
independent physi-
cal units, (1.1) can be restated as
,
,...,
0
1
2
n
(
)
F
π π π
=
,
,...,
0
(1.2)
where the
π
i
are dimensionless parameters constructed from the
q
i
by
p
=
n
−
k
equations of the form
1
2
p
m
m
m
q q q
(1.3)
where the exponents
m
i
are integer numbers. For example, if we consider the
Navier-Stokes equations for a flow around an obstacle, the variables are the ob-
stacle dimension
L
, the flow velocity far from the obstacle
U
, the fluid density
π
=
1
2
....
n
i
n
1
2
