Dynamics of Capillary Surfaces

1
Dynamics of Capillary Surfaces

• Lucero Carmona
• Professor John Pelesko and Anson Carter
• Department of Mathematics
• University of Delaware

2
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
3
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

4
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
5
Explanation of the Forces

• Surface Tension Force

• Gravitational Force

• Poiseulle Viscous Force

6
Explanation of the Forces

• End-Effect Drag

• Newton's Second Law of Motion

7
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
8
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

9
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

10
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.

11
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
12
Capillary Tubes with Silicon Oil
Silicon Oil Data                               
                                        
Steady State Solution                           
                 Initial Velocity
                                          
Eigenvalues                              
13
Capillary Tube with Water
Water Data                                     
                                   
Steady State Solution                           
                 Initial Velocity
                                          
Eigenvalues
14
Previous Experimental Data (Britten 1945)
Water Rising at Angle Data
Steady State Solution                           
               Initial Velocity
                                          
Eigenvalues
15
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

16
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

17
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

18
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
19
Wedge Data
20
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

21
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

22
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
23
Sponge Data
The effects of widths and swelling
24
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

25
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

26
Explanation of the Forces

• Poiseulle Viscous Force

(u, v, w) u - velocity in Z-dir v - velocity in r
-dir w - velocity in ?-dir
Since we are only considering the liquid movement
in the Z-dir u u(r) v w 0   The shearing
stress,t, 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                    Si
nce we are dealing with cylindrical
coordinates   From the Product Rule we can say
that                                      
Solving for u                                  
                             
27
Explanation of the Forces

• Poiseulle Viscous Force

 If             then                    Sub
back into the original equation for u
                     So  then  for      
                      
From this we can solve for c               Sub
back into the equation for u
                       
Average Velocity
28
Explanation of the Forces

• Poiseulle Viscous Force

Equation, u, in terms of Average Velocity  
                    
Further Anaylsis on  shearing stress, t  
                                          
for
,
The drag, D, per unit breadth exerted on the wall
of the tube for a segment l can be found as