Biomedical Engineering Reference
In-Depth Information
2
γ
(4.66)
D
P
=
N
( cos
-
θ
+
cos
θ
)
>
P P
-
cap
a
r
i
o
R
and the flow will come to a stop ( P i and P o are the inlet and outlet pressures). Let us
introduce now the linearized Hoffman-Tanner law to find a more workable expres-
sion of the capillary pressure drop [38]
æ
ö
1
ACa
θ θ
=
1
+
(4.67)
ç
÷
a
s a
,
3 ,
3
θ
è
ø
s a
and
æ
ö
1
ACa
θ θ
=
1
-
(4.68)
ç
÷
r
s r
,
3 ,
3
θ
è
ø
s r
with
U
µ
γ
1
Ca
=
(4.69)
where the index s stands for the static contact angle, and q s,r and q s,a are the two
static contact angles. They are equal if there is no static hysteresis (i.e., if the surface
is perfectly smooth). The minus sign in (4.68) derives from the fact that we con-
sider Ca as positive. After a substitution of (4.68) and (4.67) in (4.66), using some
algebra, and keeping the higher order terms only, the capillary pressure drop can
be cast under the form
æ
ö
sin
θ
sin
θ
2
γ
2
ANU
µ
s a
,
s r
,
1
D
P
@
N
( cos
-
θ
+
cos
θ
)
+
+
(4.70)
ç
÷
cap
s a
,
s r
,
2
2
R
3
R
θ
θ
è
ø
a s
,
r s
,
4.3.7  Two-Phase Flow Pressure Drop
In the preceding section, the pressure drop for plug flow has been derived. An ex-
tremely important factor is the pressure drop due to droplet flow, which is complex
and still a subject of investigation. The theories developed for macrofluidic applica-
tions, like that of flow homogeneization, do not apply to microscopic two-phase
flows since the pressure drop depends of the precise number of droplets circulating
in the microchannel. In this problem, many parameters intervene: relative size of
the droplets ( r / r cyl ), surface tension of the droplets (gas bubbles or solid spheres do
not behave similarly), the viscosity of the carrier fluid and that of the droplet, the
frequency (number of droplets per unit time), the spacing between droplets, and so
fourth. In a general manner, two-phase flow hydraulic resistance is larger than that
of the single-phase flow. If we assume a droplet regime, we follow the approach of
Engl et al. [39], and the pressure drop is equal to the single phase pressure drop plus
a corrective term to take into account the effect of the droplets
η
π
LQ
8
æ
L
ö
sp
d
D =
P
+
(4.71)
1
ç
÷
è
ø
4
D
R
 
 
Search WWH ::




Custom Search