Figure B.1: Gravity driven laminar flow between two "plates". The velocity profile is a second-order polynomium.
The velocity variations through the soap film should be small, if we want the soap film to resemble a two-dimensional fluid. I will present an estimate of the influence the air that surrounds the film has on velocity variations through the film.
If we assume that the air at the interfaces of the film is moving at the same constant velocity along the direction of the channel, it is possible to reduce the problem to that of a pressure or gravity driven flow between two plates. What the velocity of the surrounding air actually is, and the viscous forces in the air can then be ignored in this argument.
Let x denote the axis along the direction of the channel, let z denote the
axis perpendicular to the channel, and let z=0 be the centre of the film. If
we assume that the system is in a stationary state, and that there only are
variations along the z-axis, the Navier-Stokes equation
(1.1) reduces to
where u denotes the velocity along the x-axis, and denotes the x-component of the acceleration of gravity (). The solution to this equation is a second-order polynomium of the form:
where and are constants of integration which are to be determined from the boundary conditions.
The two interfaces are located respectively at and at
, where t is the thickness of the film, .
It is our assumption that the air is moving at the same speed at the two
interfaces and thus also that the water at the two interfaces is moving at the
The equation of motion for the water in the film can thus be reduced to
The largest velocity difference through the film is the difference between the
velocity at one of the surfaces and the velocity in the middle of the film.
This value should be compared to the typical velocity uncertainties shown in table 4.1. You will see that the uncertainties in the measurements are a factor ten larger than the calculated velocity difference through the film. - I am thus not able to measure exact enough to let velocity differences through the film be discernible in my data.
Please notice that we have only looked at the velocity profile far from the boundaries of the channel, where the drag from the channel "walls" can be ignored.