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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
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            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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