Surfactant - Tissue Functioning and Remodeling in the Circulatory and Ventilatory Systems - page 795

Table 13.1. Flow dimensionless parameters in the presence of a surface tension.

Governing parameter Symbol Expression Meaning

(ratio)

2

ρ gL

Bond number Bo / T s gravity/centrifugal force

2

Δρ gL

E otv os number Eo / T s buoyancy force/surface tension

U / Capillary number Ca μ T s friction force/centrifugal force

2

V gL ) Froude number Fr / ( inertia force/gravity

dT s
dT L

2

1
μ D T

Marangoni number Ma − Δ T surface tension/friction

L V / μ

Reynolds number Re ρ inertia force/friction force

2

L V Weber number We ρ / T s inertia force/centrifugal force

The Bond number is the most common index of the relative magnitude of surface
tension compared to gravity force. A low Bond number ( < 1) indicates that the

surface tension dominates.
The capillary number represents the relative effect of viscous forces vs. surface
or interfacial tension acting across an interface between a liquid and a gas or
between 2 immiscible liquids.
The capillary length (

2 ) prescribes the maximal drop size.
The Froude number is the ratio of inertia forces to gravitational force.
The Marangoni number is proportional to the ratio of thermal gradient-induced
surface tension to viscous forces (

1 /

c =( T s / [ ρ g ])

D T : thermal diffusivity). It is used in thermocap-
illary convection along an interface between 2 fluids due to surface tension gradient
created by a temperature gradient.
The Weber number is a measure of the relative magnitude of inertia forces with
respect to its surface tension.

13.5.3 Surfactant and Marangoni Flows

In general, the air-liquid interface cannot induce Marangoni flow since the air is not
viscous enough with respect to the liquid [ 1617 ].
Surfactant has a molecular structure often characterized by a hydrophylic head
and hydrophobic tail that enables it to reside at interfaces. Insoluble surfactants
remain on the surface, whereas surfactants soluble in the suspending fluid dif-
fuse. Surfactant reduces the surface tension. Therefore, gradients in surfactant
concentration cause gradients in surface tension. The dependence of surface tension
on surfactant concentration ( T s (

c s ) ) requires the coupling of the flow field and

surfactant field.