Surfactant - Tissue Functioning and Remodeling in the Circulatory and Ventilatory Systems - page 794

13.5.1 Interfacial Tension and Marangoni Stress

The surface tension is approximately equal to [ 1616 ]:

2 L 2

T s ∼ U / ( ) , (13.4)

where U is the total cohesive energy per molecule and L the characteristic size of
the molecule. The surface tension increases when the intermolecular attraction rises
and molecular size lowers.
When the surface tension is uniform at the interface, the net surface tension is
nul and the fluid remains static. However, surface tension gradients generate a net
interfacial force that can drive flow and distort surface. Flows within thin viscous
layers due to Marangoni stresses, film thickness, and material concentration can be
derived from lubrication theory. Surface tension acts as a boundary condition in the
governing equations of fluid dynamics.
The normal stress at the free surface is in equilibrium with the curvature force
associated with the surface tension:

n · T · n = T s ( ∇ · n ) , (13.5)

where n is the unit outward normal to the surface and

T

T = − p I + μ ( ∇ v +[ ∇ v ] ) (13.6)

the stress tensor ( v : velocity vector).
The tangential component of the hydrodynamic stress at the surface balance the
tangential Marangoni stress associated with surface tension gradient that results
itself from gradients in chemical composition at the interface, in the absence of
temperature changes:

n · T · t = ∇ T s · t , (13.7)

where t is the unit tangent vector to the interface.

13.5.2 Dimensionless Governing Parameters

The motion of a fluid (density ) with a free surface at which is

exerted a surface tension ( T s ) under a gravitation field (gravitational acceleration g ),
characterized by characteristic length ( L ) and velocity ( V ), can be defined by a
set of 6 physical quantities (

ρ and viscosity μ

, L , V , g ,and T s ). This set gives rise to a set of
dimensionless parameters (Table 13.1 )[ 1616 ].
When the Marangoni stress has a characteristic magnitude

ρ , μ

Δ T s / L , the dimen-

sionless parameter L Δ

T s / ( LT s ) characterizes the relative effects of the Marangoni

stress to the curvature stress.