Biomedical Engineering Reference
In-Depth Information
The Lipmann-Young law [16] relates the change of contact angle to the electrowet-
ting number: cos
θ
−
cos
θ
0
=
η
ew
/2 (Figure 1.11).
The Marangoni number determines the magnitude of convective motions in
a droplet. Even if they do not move, free-standing droplets are seldom at rest.
Internal motions are frequent. These motions are essentially caused by interfacial
forces, especially by a gradient of interfacial tension. The most common case is
that of a gradient of interfacial temperature, which can be due to surrounding
temperature conditions or to evaporation [17] (Figure 1.12). Another cause of
Marangoni convection is a gradient of concentration on the interface, which
may occur when surfactants are added to the liquid. For example, a gradient of
temperature results in a gradient of surface tension according to
γ
=
γ
0
(1 +
β
/
T
)
where
β
=
−
1/
T
C
,
T
C
being the critical temperature in Kelvin. Marangoni convec-
tion occurs if the gradient of the surface tension force dominates the viscosity
forces. A dimensionless number—the Marangoni number—determines the strength
of the convective motion [18]
D
γ
ρ ν α
R
(1.23)
Ma
=
In the domain of bioreactions, like DNA immobilization [19], it is essential to deter-
mine whether the process is limited by the reaction time or by the time of transport
of the species involved in the reaction. The Damköhler number is defined as the
ratio of these two characteristic times
Da
(1.24)
where
τ
R
and
τ
Tr
are, respectively, the times taken by the reaction and the transport
of species. For a purely diffusive situation, the transport time is of the order of
τ
Tr
≈
L
2
/
D
, where
D
is the diffusion coefficient. In such a case, the Damköhler num-
ber can be written as
Da
=
Dτ
R
/
L
2
.
The Peclet number is a fundamental dimensionless number in convective trans-
port as well at the macroscale as at the microscale. It relates the rate of advection of
a flow to its rate of diffusion or thermal diffusion. It is equal to the product of the
Reynolds number with the Prandtl number
ν
/
α
in the case of thermal conduction,
and the product of the Reynolds number with the Schmidt number
ν
/
D
in the case
of mass dispersion
=
τ τ
R Tr
Pe
=
V L
α
th
(1.25)
Pe
=
V L D
mass
In the general problem of mass transfer at a solid wall, the Sherwood number
represents the ratio of convective to diffusive mass transport. It is defined by the
expression
=
kL
Sh
(1.26)
D
where
L
is a characteristic distance,
D
is the diffusion coefficient, and
k
is a mass
transfer coefficient (unit m/s). Usually the Sherwood number is correlated to the
