Table of Contents
For this experiment we want to investigate how the shape of the neck of a drop changes through out the pinch off process. Fluids are complex systems which can be difficult to characterize in terms of fundamental physics principles. But with a little bit of knowledge of some of the basic parameters of fluids, we can use the technique of dimensional analysis to predict possible relationships among the processes which are most likely to be involved in driving the behavior which we observe.
We will consider the following physical properties of fluids:
- Surface Tension $[\gamma]$. This can be thought of as a force which tries to minimize the surface area of the fluid. For example, surface tension is responsible for causing water to bead up in small droplets.
- Viscosity $[\eta]$. A measure of how easily a fluid flows. You can think of this parameter as being related to an internal friction that resists the internal motion of the fluid.
- Density $[\rho]$. A measure of inertia. The more dense a fluid is the more inertia (i.e. mass) a given volume element will possess.
Dimensional analysis begins with identifying a relevant set of physical parameters, and then looking for combinations which produce dimensionless quantities. The dimensions of the above parameters are:
- Surface Tension $[\gamma] = \frac{M}{T^2}$.
- Viscosity $[\eta] = \frac{M}{LT}$.
- Density $[\rho] = \frac{M}{L^3}$.
- Radius $[R] = L$.
- Time $[\tau] = T$.
Where $M$ is units of mass, $L$ is units of length, and $T$ is units of time.
We have assembled a set of 5 parameters ($\gamma, \eta, \rho, R, \tau$) which are made up of 3 dimensions ($M, L, T$). The Buckingham Pi theorem from dimensional analysis says that the number of dimensionless quantities which can be assembled from this set of parameters is 5 - 3 = 2. It does not take long to find that the two dimensionless arrangements of these parameters are:
$\frac{R^{3}\rho}{\tau^{2}\gamma}$ which we will denote as $\Pi_{0}$,
and
$\frac{\rho r^{2}}{\tau \eta}$ which we will denote as $\Pi_{1}$.
The technique of dimensional analysis does not yield exact formulas which describe the phenomena of interest. Rather it produces combinations of parameters which might describe the functional form of a more complete solution, but it does not tell you what the constants of proportionality are. Because they are dimensionless quantities, each of the above arrangements $\Pi_{0-}$ and $\Pi_{1}$ can be set equal to an arbitrary constant which we will choose to be 1. Using $\Pi_{0}$ and $\Pi_{1}$ we can build three different functional forms which may describe the pinch-off process.
First Form
We set $\Pi_{0} \sim 1$ and rearrange to get a function of the form $R = f(\tau)$.
$\frac{R^{3}\rho}{\tau^{2}\gamma} \sim 1$
$R \sim (\frac{\gamma}{\rho})^3 \tau ^\frac{2}{3}$
This form predicts a dependence on the ratio of surface tension to density. You can think of this as the competition between surface tension which tries to pull the surface of the fluid inwards, but this is being opposed by the inertia of the fluid.
Second Form
We set $\Pi_{1} \sim 1$ and rearrange to get a function of the form $R = f(\tau)$.
$\frac{R^{3}\rho}{\tau^{2}\gamma} \sim 1$
$R \sim \sqrt{\frac{\eta}{\rho}} \tau ^\frac{1}{2}$
This form predicts a dependence on the ratio of viscosity to density. This form is more difficult to interpret without a deeper understanding of fluid mechanics. This ratio is related to something called the vorticity constant. Vorticity is rotational motion that drives instabilities within the fluid.
Third Form
Finally we can set the ratio $\frac{\Pi_{0}}{\Pi_{1}} \sim 1$ and rearrange to get a function of $R = f(\tau)$.
$\frac{R^{3}\rho}{\tau^{2}\gamma} \sim 1$
$R \sim \frac{\gamma}{\eta} \tau $
This form predicts a dependence on the ratio of surface tension to viscosity. You can think of this as the competition between surface tension which tries to pull the surface of the fluid inwards, but this is being opposed by the effect of viscosity which opposes the motion of the fluid.
What we have done is to use a little bit of knowledge about fluids to identify a small set of physical parameters which might play important roles in the drop pinch-off process. We then used dimensional analysis to construct three different arrangements of these parameters where the units work out. We are not pretending that this is a rigorous development of a theory to describe the pinch-off of a fluid. What we can draw from this exercise is that it is reasonable to expect that the pinch-off process could follow one of these power law models. It is also possible that the process may exhibit power law behavior that follows one of the above forms and then transitions into one of the other as different physical parameters become more and less important.
The only way to know if any of these models describe the phenomena is to do the experiment.