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\begin{center}

{\bf SURFACE TENSION MODULE} 

\vspace*{.1in}

by John W. M. Bush

Department of Mathematics, MIT



\end{center}

This set of notes has been developed as supporting material for the Surface Tension
module in the 1.63J/2.21J Fluid Dynamics course, which will be presented in a series of
6 lectures at the end of the spring term. These lecture notes have been drawn from many
sources, including text books, journal articles, and lecture notes from courses taken by the
author as a student. These notes are not intended as a complete discussion of the subject, 
or as a scholarly work in which all relevant references are cited. Rather, they are intended
as an introduction that will hopefully motivate the interested  student to learn more about the
subject. Topics have been chosen according to their perceived value in developing the
physical insight of the students.

\vspace*{.2in}

\noindent{\bf LECTURE 1: The definition and scaling of surface tension}

\vspace*{.1in}

\noindent{\bf 1.1 Surface tension: a working definition}

\vspace*{.1in}

Discussions of the molecular origins of surface or interfacial tension may be found 
elsewhere (e.g. Israelachvili 1995, Rowlinson \& Widom 1982).
For our purposes, it is sufficient to say that interfacial tension is a material 
property of a fluid-fluid interface
who origins lie in the differing intermolecular forces that act in the two 
fluid phases. For example, water
molecules have a mutual attraction owing to their electrostatic dipole moment. 
Consequently, a water molecule adjacent to a water surface
prefers to be in the water than the air. Increasing the surface area requires that
we pull molecules from the bulk to the surface, thereby doing work. 
The result is a surface energy per area or that acts to resist the creation of new surface,
and that is equivalent to a line tension acting in all directions parallel to the surface.

Fluids between which no interfacial tension arises are said to be immiscible.
For example, salt molecules will diffuse freely across a boundary between fresh
and saltwater; consequently, these fluids are immiscible, and there is no interfacial
tension between them. Our discussion will be confined to miscible fluid-fluid
interfaces (or fluid surfaces), at which an effective interfacial (or surface) tension
must act.

Surface tension $\sigma$ has the units of force/length or equivalently energy/area, and
so may be thought of as a surface pressure. Pressure is generally an
isotropic force per area that acts throughout the bulk of a fluid: a small surface
element $dS$ will feel a total force $p({\bf x})dS$  owing to the local pressure field
$p({\bf x})$. If the surface $S$ is closed, and the pressure uniform, the net pressure
force acting on $S$ is zero and the fluid remains static. Pressure gradients
correspond to body forces (with units of force per unit volume) within a fluid, 
and so appear explicitly in the Navier-Stokes equations.

Surface tension has the units of force per length, and its action is confined to
the free surface. Consider for the sake of simplicity a perfectly flat interface. 
A surface line element $d\ell$ will feel a total force $\sigma d\ell$
owing to the local surface tension $\sigma({\bf x})$. If the surface line element is
a closed loop $C$, and the surface tension uniform, the net surface tension force
acting on $C$ is zero, and the fluid remains static. If surface tension gradients
arise, there may be a net force on the surface element that acts to distort it
through driving flow. 

\vspace*{.2in}

\noindent{\bf 1.2 Governing Equations}

\vspace*{.1in}

The motion of a fluid of uniform density $\rho$ and viscosity $\mu$ is governed 
by the Navier-Stokes equations, which represent a continuum statement of 
Newton's laws.

\begin{equation}
\rho \left(\frac{\partial {\bf u}}{\partial t}~+~{\bf u} \cdot \nabla {\bf u} \right) ~=~ - \nabla p ~+~ {\bf F}
~+ ~ \mu \nabla^2 {\bf u}
\end{equation}

\begin{equation}
\nabla \cdot {\bf u} ~=~0.
\end{equation}

This represents a system of 4 equations in 4 unknowns (the fluid pressure $p$ and
the three components of the velocity field ${\bf u}$). Here ${\bf F}$ represents any
body force acting on the fluid; for example, in the presence of a gravitational field,
${\bf F} = \rho {\bf g}$, where ${\bf g}$ is the acceleration due to gravity.

Surface tension acts only at the free surface; consequently, it does not appear in the 
Navier-Stokes equations, but rather enters through the boundary conditions.
The boundary conditions appropriate at a fluid-fluid interface 
are formally developed in Lecture 2. We here simply state them for the simple
case of a free surface (such as air-water, in which one of the fluids is not
dynamically significant) in order to get a feeling for the scaling of surface tension.

The normal stress balance at a free surface must be balanced by the curvature
force associated with the surface tension:
\begin{equation}
 {\bf n} \cdot {\bf T} \cdot {\bf n} 
=~ \sigma ~(\nabla \cdot {\bf n}) 
\end{equation}
where $T = -p {\bf I} + \mu [\nabla {\bf u} + (\nabla {\bf u})^T] = -p {\bf I} + 2 \mu {\bf E}$ is the stress
tensor, ${\bf E}= 1/2 ~[\nabla {\bf u} + (\nabla {\bf u})^T]$ is the deviatoric stress tensor,
and ${\bf n}$ is the unit normal to the surface. The tangential stress
at a free surface must balance the local surface tension gradient:
\begin{equation}
 {\bf n} \cdot {\bf T} \cdot {\bf t} 
=~ \nabla \sigma \cdot {\bf t} 
\end{equation}
where ${\bf t}$ is the unit tangent to the interface.


\vspace*{.2in}

\noindent{\bf 1.3 The scaling of surface tension}

\vspace*{.1in}

We consider a fluid of density $\rho$ and viscosity $\mu = \rho \nu$ with a free surface characterized
by a surface tension $\sigma$. The flow is marked by characteristic length- and velocity-scales of,
respectively, $a$ and $U$ and evolves in the presence of a gravitational field ${\bf g} = -g \hat{z}$.
We thus have a physical system defined in terms of six physical variables $(\rho, \nu, \sigma, a, U, g)$
that may be expressed in terms of three fundamental units: mass, length and time. Buckingham's
Theorem thus indicates that the system may be uniquely prescribed in terms of three dimensionless
groups. We choose

\begin{equation}
Re ~=~ \frac{Ua}{\nu} ~=~ \frac{INERTIA}{VISCOSITY} ~=~ \hbox{Reynolds number}
\end{equation}

\begin{equation}
Fr ~=~ \frac{U^2}{ga} ~=~ \frac{INERTIA}{GRAVITY} ~=~ \hbox{Froude number}
\end{equation}

\begin{equation}
Bo ~=~ \frac{\rho g a^2}{\sigma} ~=~ \frac{GRAVITY}{CURVATURE} ~=~ \hbox{Bond number}
\end{equation}

The Reynolds number prescribes the relative magnitudes of inertial and viscous forces in the
system, while the Froude number those of inertial and gravity forces. The Bond number 
indicates the relative importance of forces induced by gravity and surface tension. Note
that these two forces are comparable when $Bo = 1$, which arises on a lengthscale
corresponding to the capillary length: $\ell_c = (\sigma/(\rho g))^{1/2}$. For an air-water
surface, for example, $\sigma \approx 70$ dynes/cm, $\rho= 1$ g/cc and $g= 980$ cm/s$^2$,
so that $\ell_c \approx 2$mm. Bodies of water in air are dominated by the influence of surface
tension provided they are smaller than the capillary length. Roughly speaking, the  
capillary length prescribes the maximum size of pendant drops that may hang inverted 
from a ceiling, water-walking insects, and raindrops. Note that as a fluid system becomes
progressively smaller, the relative importance of surface tension and gravity increases;
it is thus that surface tension effects are dominant in microscale engineering processes.

Finally, we note that other frequently arising dimensionless group may be formed 
from the product of $B$, $Re$ and $Fr$:
\begin{equation}
We ~=~ \frac{\rho U^2 a}{\sigma} ~=~ \frac{INERTIA}{CURVATURE} ~=~ \hbox{Weber number} ~~.
\end{equation}
\begin{equation}
Ca ~=~ \frac{\rho \nu U}{\sigma} ~=~ \frac{VISCOUS}{CURVATURE} ~=~ \hbox{Capillary number} ~~.
\end{equation}
The Weber number indicates the relative magnitudes of inertial and curvature forces within
a fluid, and the capillary number those of viscous and curvature forces. Finally, we note that
if the flow is marked by a Marangoni stress of characteristic magnitude $\Delta \sigma/L$,
then an additional dimensionless group arises that characterizes the relative magnitude
of Marangoni and curvature stresses:  $a \Delta \sigma/(L \sigma)$.

We now demonstrate how these dimensionless groups arising naturally from
the nondimensionalization of Navier-Stokes equations and the surface boundary conditions.
We first introduce a dynamic pressure: $p_d = p - \rho {\bf g} \cdot {\bf x}$, so that
gravity appears only in the boundary conditions. 
We consider the special case of high Reynolds number flow, for which the
characteristic dynamic pressure is $\rho U^2$. We define dimensionless primed variables
according to:
\begin{equation}
{\bf u} = U {\bf u}'~~,~~p_d = \rho U^2 p_d'~~, ~~{\bf x} = a{\bf x}'~~, ~~t = \frac{a}{U}t'~~.
\end{equation}
Nondimensionalizing the Navier-Stokes
equations and appropriate boundary conditions yields the following system:

\begin{equation}
\left(\frac{\partial {\bf u}'}{\partial t'}~+~{\bf u}' \cdot \nabla' {\bf u}' \right) ~=~ - \nabla p'_d ~+ ~ \frac{1}{Re}\nabla'^2 {\bf u}' ~~~~~~,~~~~~~\nabla' \cdot {\bf u}' ~=~0.
\end{equation}
The normal stress condition assumes the dimensionless form:
\begin{equation}
-p'_d~+~\frac{1}{Fr} z'~+~\frac{2}{Re} ~{\bf n} \cdot {\bf E}' \cdot {\bf n} ~=~ \frac{1}{We}~
\nabla \cdot {\bf n}
\end{equation}
The importance of surface tension relative to gravity and viscous stresses is prescribed by the relative magnitudes of the Weber, Froude and Reynolds numbers. In the high $Re$ limit of
interest, the normal force balance requires that the dynamic pressure be balanced by either
gravitational or curvature stresses, the relative magnitudes of which are prescibed by the
Bond number.

The nondimensionalization scheme will depend on the physical system of interest.
Our purpose here was simply to illustrate the manner in which the dimensionless
groups arise in the theoretical formulation of the problem. Moreover, we see that 
those involving surface tension enter exclusively through the boundary conditions.


\vspace*{.2in}

\noindent{\bf References}

While this list of relevant textbooks is far from complete, we include it as a source of 
additional reading for the interested student.

Rowlinson, J.S. and Widom, B., 1982, Molecular theory of capillarity, Dover.

Israelachvili, J., 1995. Intermolecular and surface forces, 2nd Edition, Academic 
Press.

DeGennes, P., Brochard-Wyart, M. and Quere, D., 2002. Perles, Gouttes, bulles, perles et
ondes, Belin.


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