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```					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   Subbing back in our terms we get:

By   Dividing everything by                 we get our differential equation:

where
Zo = Z(0) = 0
   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:

Initial Velocity

Eigenvalues
Capillary Tube with Water

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            Z
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
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                     Z

 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
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:                                                               (u, v, w)
Since we are only considering the liquid movement in the Z-dir:                            u - velocity in Z-dir
u = u(r)                                                                                   v - velocity in r -dir
w - velocity in θ-dir
v=w=0
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        then:           From this we can solve for c:

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:

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