\documentclass[11pt]{article}
\usepackage{graphicx, epsf, rotate}
\usepackage{epsfig}
\usepackage{amsfonts}
\pagestyle{empty}
\parindent=0.3cm
\topmargin=0.in
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.in}
\setlength{\evensidemargin}{0.in}
\setlength{\oddsidemargin}{0.in}
\setlength{\topmargin}{0.0in}
\parskip=1.0ex
\begin{document}
%\begin{figure}
%\begin{center}
%\centerline{$~~~~~~~~~~~~~~~~~$}
%\vspace*{-6.5in}
%\includegraphics[width=1.4\textwidth]{~/STRIDER/forcebody}
%\centerline{$~~~~~~~~~~~~~~~~~$}
%\end{center}
%\caption{A schematic illustration of the experimental apparatus. Glycerol-water
%solutions were pumped through the source nozzle with a prescribed flux $Q$. The outer
%depth $H$ was controlled by an outer wall whose height was adjustable.}
%\end{figure}
%\clearpage
\begin{center}
{\bf LECTURE 6: Fluid Sheets}
\end{center}
\vspace*{.2in}
The dynamics of fluid sheets was first considered by Savart after his early
work on electromagnetism with Biot, and was subsequently examined in a series
of papers by Taylor (1959). They have recently received a great deal of attention
owing to their relevance in a number of spray atomization processes. Such sheets
may be generated from a variety of source conditions, for example, the
collision of jets on rigid impactors.
There is generally a curvature force acting on the
sheet edge which acts to contain the fluid sheet. For a 2D (planar) sheet, the
magnitude of this curvature force is given by
\begin{equation}
{\bf F_c} ~=~ \int_S ~\sigma \nabla \cdot {\bf n}~ d\ell~~~,
\end{equation}
Using the first Frenet-Serret equation,
\begin{equation}
(\nabla \cdot {\bf n})~ {\bf n} ~=~ \frac{d{\bf t}}{d\ell} ~~,
\end{equation}
thus yields
\begin{equation}
{\bf F_c} ~=~ \int_S ~\sigma ~ \frac{d{\bf t}}{d\ell}~d\ell~= ~ \sigma ({\bf t}_1 - {\bf t}_2) ~=~ 2 \sigma {\bf x}.
\end{equation}
There is thus an effective force per unit length $2 \sigma$ along the length of the sheet rim
acting to contain the rim.
We now consider how this result may be applied to compute sheet shapes for three
distinct cases: i) a circular sheet, ii) a lenticular sheet with unstable rims, and iii)
a lenticular sheet with stable rims.
\vspace*{.2in}
\noindent{\bf 6.1 Circular Sheet}
\vspace*{.1in}
\begin{figure}
\begin{center}
\centerline{$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$}
%\centerline{$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$}
\includegraphics[width = 0.7\textwidth]{circular}
\end{center}
\caption{A surface $S$ and bounding surface contour $C$. ${\bf n}$ represents
the unit normal to the surface; ${\bf m}$ the unit tangent to the contour $C$ and ${\bf s}$
the unit vector normal to $C$ but tangent to $S$.}
\label{fig2}
\end{figure}
We consider the geometry considered in Savart's (1833) original experiment.
A vertical fluid jet strikes a small horizontal impactor. If the flow rate is sufficiently high
that gravity does not influence the sheet shape, the fluid is ejected radially,
giving rise to a circular free fluid sheet. The sheet radius is prescribed by a balance
of radial forces; specifically, the inertial force must balance the curvature force:
\begin{equation}
\rho u^2 h ~ = ~ 2 \sigma
\end{equation}
\noindent Continuity requires that the sheet thickness $h$ depend on the
speed $u$, jet flux $Q$ and radius $r$ as
\begin{equation}
h ~=~ \frac{Q}{2 \pi r u} ~~~.
\end{equation}
\noindent Experiments (specifically, tracking of particles suspended within the sheet)
indicate that the sheet speed $u$ is independent of radius; consequently, the
sheet thickness decreases as $1/r$. Substituting the form (5) for $h$ into the
force balance (4) yields the sheet radius, or so-called Taylor radius:
\begin{equation}
R_T ~=~ \frac{\rho Q u}{4 \pi \sigma}
\end{equation}
\noindent The sheet radius increases with source flux and sheet speed, but
decreases with surface tension. We note that the fluid proceeds radially to the
sheet edge, where it accumulates until going unstable via a modified
Rayleigh-Plateau instability, often referred to as the Rayleigh-Plateau-Savart
instability, as it was first observed on a sheet edge by Savart.
\vspace*{.2in}
\noindent{\bf 6.2 Lenticular sheets with unstable rims}
\vspace*{.1in}
\begin{figure}
\begin{center}
\centerline{$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$}
%\centerline{$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$}
\includegraphics[width = 0.7\textwidth]{Taylorscan}
\end{center}
\caption{A water sheet generated by the collision of water jets at left.
The fluid streams radially outward in a thinning sheet; once the fluid reaches
the sheet rim, it is ejected radially in the form of droplets. From G.I. Taylor (1960).}
\label{fig2}
\end{figure}
We now consider the non-axisymmetric fluid sheet, such as may be formed by
the oblique collision of water jets, a geometry originally considered by
Taylor (1960). Fluid is ejected radially from the origin into a sheet
with flux distribution given by $Q(\theta)$, so that the volume flux flowing into the
sector between $\theta$ and $\theta + d\theta$ is $Q\left(\theta\right) d\theta$.
As in the previous case of the circular sheet, the sheet rims are unstable, and
fluid drops continuously ejected therefrom.
The sheet shape is computed in a similar manner, but now depends explicitly
on the flux distribution within the sheet, $Q(\theta)$. The normal force balance on the
sheet edge now depends on the normal component of the sheet speed, $u_n$:
\begin{equation}
\rho u_n^2 h ~ = ~ 2 \sigma ~~.
\end{equation}
The sheet thickness is again given by (4), but now $Q~=~Q(\theta)$, and the sheet
radius $R(\theta)$ is now given by
\begin{equation}
R(\theta) ~=~ \frac{\rho u_n^2 Q(\theta)}{4 \pi \sigma u}
\end{equation}
Computing sheet shapes thus relies on measurement of the flux distribution
$Q(\theta)$ within the sheet.
\vspace*{.2in}
\noindent{\bf 6.3 Lenticular sheets with stable rims}
\vspace*{.1in}
\begin{figure}[t]
\begin{center}
\centerline{$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$}
%\centerline{$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$}
\includegraphics[width = 0.7\textwidth]{fluidchain.pdf}
\end{center}
\caption{A schematic illustration of a fluid sheet bound by stable rims. }
\end{figure}
In a certain region of parameter space, specifically, with fluids more viscous than
water, one may encounter fluid sheets with stable rims
(see www-math.mit.edu/~bush/chains.html). The force balance describing
the sheet shape must change accordingly. When rims are stable, fluid entering the
rim proceeds along the rim. As a result, there is a centripetal force normal to the fluid
rim associated with flow along the curved rim that must be considered in order to
correctly predict the sheet shapes.
The relevant geometry is presented in Figure 3.
$r(\theta)$ is defined to be the distance from the origin to the rim centreline,
and $u_{n}(\theta)$ and $u_{t}(\theta)$ the normal and tangential components of
the fluid velocity in the sheet where it contacts the rim.
$v(\theta)$ is defined to be the velocity of flow in the rim,
$R(\theta)$ the rim radius, and $\psi(\theta)$ the angle between the
position vector $\mathbf{r}$ and the local tangent to the rim centreline.
Finally, $r_{c}(\theta)$ is defined to be the radius of curvature of the rim centreline,
and $s$ the arc length along the rim centreline.
The differential equations governing the shape of a stable fluid rim bounding
a fluid sheet may be deduced by consideration of conservation of mass in the
rim and the local normal and tangential force balances at the rim.
For a steady sheet, continuity requires that the volume flux from the sheet
balance the tangential gradient in volume flux along the rim:
\begin{equation}
0 = u_{n}h-\frac{
\partial}{\partial s}\left( v \pi R^{2}\right ) ~~~.
\label{continuity}
\end{equation}
The normal force balance requires that the curvature force associated with the
rim's surface tension balance the force resulting from the normal flow
into the rim from the fluid sheet and the centripetal force resulting from the
flow along the curved rim:
\begin{equation}
\rho u_{n}^{2}h+\frac{\rho\pi R^{2}v^{2}}{r_{c}}= 2\sigma ~~. \label{normalforcebalance}
\end{equation}
The tangential force balance at the rim requires a balance between
tangential gradients in tangential
momentum flux, tangential gradients in curvature pressure, viscous resistance to
stretching of the rim, and the tangential momentum flux arriving from the sheet.
For most applications, the Reynolds number characterizing the rim dynamics is
large, so viscous resistance may be safely neglected. Moreover, the curvature
term $\nabla\cdot \hat{n}$ generally depends on $\theta$; however, accurate
to $O(R/r_c)$, we may use $\nabla \cdot \hat{n}=1/R$. One thus obtains:
\begin{equation}
\frac{\partial}{\partial s}(\pi R^{2}v^{2})=hu_{t}^{s}u_{n}+3\nu \frac{
\partial}{\partial s}\left( R^{2}\frac{\partial v}{\partial s}\right) -
\frac{\pi R^{2}\sigma}{\rho}\frac{\partial}{\partial s}\left( \frac{1}{R}
\right). \label{tangentialforcebalance}
\end{equation}
Equations (9)-(11) must be supplemented by the continuity relation,
\begin{equation}
h(r, \theta) = \frac{Q(\theta)}{u_0r}~~,
\end{equation}
in addition to a number of relations that follow directly from the system geometry:
\begin{equation}
u_n = u_0 \hbox{sin}\Psi ~~~,~~~u_t = u_0 \hbox{cos}\Psi
\end{equation}
\begin{equation}
\frac{1}{r_c} ~=~\frac{\hbox{sin}\Psi}{r}~\left(\frac{\partial \Psi}{\partial \theta} + 1 \right)~~.
\end{equation}
This resulting system of equations may be nondimensionalized, and reduce to a set of coupled
ordinary equations in the four variables $r(\theta)$, $v(\theta)$, $R(\theta)$ and $\Psi(\theta)$.
Given a flux distribution, $Q(\theta)$, the system may be integrated to deduce the sheet shape.
\vspace*{.2in}
\noindent{\bf 6.4 Water bells}
\vspace*{.1in}
All of the fluid sheets considered thus far have been confined to a plane.
In \S 1, we considered circular sheets generated from a vertical jet striking
a horizontal impactor. The sheet remains planar only if the flow is sufficiently
fast that the fluid reaches its Taylor radius before sagging substantially under
the influence of gravity. Decreasing the flow rate will make this sagging more
pronounced, and the sheet will no longer be planar. Moreover, the sheet may
actually close upon itself, giving rise to a water bell, as illustrated in
Figure 5. We proceed by outlining the theory required to compute the
shapes of water bells.
\begin{figure}[t]
\begin{center}
\centerline{$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$}
%\centerline{$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$}
\includegraphics[width = 0.7\textwidth]{taylor}
\end{center}
\caption{A water bell produced by the impact of a descending water jet and a
solid impactor. The impactor radius is 1 cm. Fluid is splayed radially by the impact,
then sags under the influence of gravity. The sheet may close on itself owing to
the azimuthal curvature of the bell.}
\end{figure}
We consider a fluid sheet extruded radially at a speed $u_0$ and subsequently
sagging under the influence of a
gravitational field ${\bf g} = -g \hat{z}$. The inner and outer sheet surfaces are
characterized by a constant surface tension $\sigma$. The sheet has constant
density $\rho$, thickness $t$ and is assumed to be inviscid. $Q$ is the total
volume flux in the sheet.
We define the origin to be the center of the impact plate; $r$ and $z$ are,
respectively, the radial and vertical distances from the origin. $u$ is the sheet speed,
and $\phi$ the angle made between the sheet and the vertical. $r_c$ is the local
radius of curvature of a meridional line, and $s$ the arc length along a meridional
line measured from the origin. Finally, $\Delta P$ is the pressure difference between
the outside and inside of the bell.
Flux conservation requires that
\begin{equation}
Q ~=~ 2 \pi rtu ~~~,
\end{equation}
while Bernoulli's Theorem indicates that
\begin{equation}
u^2~=~u_0^2 ~+~2gz
\end{equation}
The total curvature force acting on the bell is given by $2 \sigma \nabla \cdot {\bf n} =2\sigma(1/r_c +$cos$\phi/y)$.
Therefore, the force balance normal to the sheet takes the form:
\begin{equation}
\frac{2\sigma}{r_c} ~+~\frac{2T\hbox{cos}\phi}{x} ~-~P~+~\rho g t ~\hbox{sin}\phi ~-~\frac{\rho t u^2}{r_c} ~=~0
\end{equation}
Equations (15)-(17) may be appropriately nondimensionalized, and integrated to determine the shape of the bell.
\end{document}