Download Dynamics of a Capillary Tube

Dynamics of Capillary Surfaces
Lucero Carmona
Professor John Pelesko and Anson Carter
Department of Mathematics
University of Delaware
Explanation

When a rigid container is inserted into a fluid,
the fluid will rise in the container to a height
higher than the surrounding liquid
Tube
Wedge
Sponge
Goals
Map mathematically how high the liquid
rises with respect to time
 Experiment with capillary surfaces to
see if theory is in agreement with data
 If the preparation of the tube effects
how high the liquid will rise

Initial Set-up and Free Body Diagram
List of Variables:
volume =
g = gravity
r = radius of capillary tube
Z = extent of rise of the surface of the liquid,
measured to the bottom of the meniscus, at time t ≥ 0
= density of the surface of the liquid = surface tension
= the angle that the axis of the tube makes with the horizontal of
the stable immobile pool of fluid
= contact angle between the surface of the liquid and the wall of the tube
Explanation of the Forces

Surface Tension Force

Gravitational Force

Poiseulle Viscous Force
Explanation of the Forces

End-Effect Drag
Newton's
Second Law of Motion
Explanation of Differential Equation
From
our free body diagram and by Newton's Second Law of Motion:
Net Force = Surface Tension Force - End-Effect Drag - Poiseuitte Viscous Force - Gravitational Force
Net Force + End-Effect Drag + Poiseuitte Viscous Force + Gravitational Force - Surface Tension Force = 0
After
By
Subbing back in our terms we get:
Dividing everything by
where
Zo = Z(0) = 0
we get our differential equation:
Steady State

By setting the time derivatives to zero in the
differential equation and solving for Z, we are
able to determine to steady state of the rise
Set - Up
 Experiments were performed using
silicon oil and water
 Several preparations were used on the
set-up to see if altered techniques would
produce different results
 The preparations included:
• Using a non-tampered tube
• Extending the run time and aligning
the camera
• Aligning the camera and using an
non-tampered tube
•Disinfecting the Tube and aligning
the camera
• Pre-wetting the Tube and aligning
the camera
Set - Up
 The experiments were recorded with the high speed camera.
 The movies were recorded with 250 fps for Silicon Oil
and 1000 fps for water.
 Stills were extracted from the videos and used to process in MatLab.
 1 frame out of every 100 were extracted from the Silicon Oil experiments
so that 0.4 of a second passed between each frame.
 1 frame out of every 25 were extracted from the Water experiments
so that 0.025 of a second passed between each frame.
Set - Up
 MatLab was then used to measure the
rise of the liquid in pixels
 Excel and a C-program were used to
convert the pixel distances into MM and
to print out quick alterations to the data
Z
Capillary Tubes with Silicon Oil
Silicon Oil Data:
Steady State Solution
Initial Velocity
Eigenvalues
Capillary Tube with Water
Water Data:
Steady State Solution
Initial Velocity
Eigenvalues
Previous Experimental Data (Britten 1945)
Water Rising at Angle Data:
Steady State Solution
Initial Velocity
Eigenvalues
Results




There is still something missing from the
theory that prevents the experimental data to
be more accurate
The steady – state is not in agreement with
the theory
There is qualitative agreement but not
quantitative agreement
Eliminated contamination
Explanation of Wedges

When a capillary wedge is inserted into a
fluid, the fluid will rise in the wedge to a
height higher than the surrounding liquid
Goals
Map
mathematically how high the liquid
rises with respect to time
Wedge Set - Up
 Experiments were performed using
silicon oil
Two runs were performed with different
angles
 Experiments were recorded with the
high speed camera at 250 fps and 60 fps
Wedge Set - Up
 For first experiment, one still out of every
100 were extracted so that 0.4 sec passed
between each slide
 For second experiment, one still out of
every 50 were extracted so that 0.83 sec
passed between each slide
 MatLab was then used to measure the
rise of the liquid in pixels
 Excel and a C-program were used to
convert the pixel distances into MM and
to print out quick alterations to the data
Z
Wedge Data
Explanation of Sponges

Capillary action can be seen in porous
sponges
Goals
To
see if porous sponges relate to the
capillary tube theory by calculating what
the mean radius would be for the pores
Sponge Set - Up
 Experiments were performed using
water
Three runs were preformed with varying
lengths
 Experiments were recorded with the
high speed camera at 250 fps and 60 fps
Sponge Set - Up
 For first and second experiments, one still
out of every 100 were extracted so that
0.4 sec passed between each slide
 For third experiment, one still out of
every 50 were extracted so that 0.83 sec
passed between each slide
 MatLab was then used to measure the
rise of the liquid in pixels
 Excel and a C-program were used to
convert the pixel distances into MM and
to print out quick alterations to the data
Z
Sponge Data
The effects of widths and swelling
Future Work




Refining experiments to prevent undesirable
influences
Constructing a theory for wedges and
sponges
Producing agreement between theory and
experimentation for the capillary tubes
Allowing for sponges to soak overnight with
observation
References



Liquid Rise in a Capillary Tube by W. Britten
(1945). Dynamics of liquid in a circular capillary.
The Science of Soap Films and Soap Bubbles by C.
Isenberg, Dover (1992).
R. Von Mises and K. O. Fredricks, Fluid Dynamics
(Brown University, Providence, Rhode Island, 1941), pp
137-140.
Further Information

http://capillaryteam.pbwiki.com/here
Explanation of the Forces

Poiseulle Viscous Force:
Since we are only considering the liquid movement in the Z-dir:
u = u(r)
v=w=0
(u, v, w)
u - velocity in Z-dir
v - velocity in r -dir
w - velocity in θ-dir
The shearing stress,τ, will be proportional to the rate of change of velocity across the surface.
Due to the variation of u in the r-direction, where μ is the viscosity coefficient:
Since we are dealing with cylindrical coordinates
From the Product Rule we can say that:
Solving for u:
Explanation of the Forces

Poiseulle Viscous Force:
If
From this we can solve for c:
then:
Sub back into the equation for u:
Sub back into the
original equation for u:
Average Velocity:
So then for
:
Explanation of the Forces

Poiseulle Viscous Force:
Equation, u, in terms of Average Velocity
Further Anaylsis on shearing stress, τ:
for
,
The drag, D, per unit breadth exerted on the wall
of the tube for a segment l can be found as: