Dynamics of Capillary SurfacesLucero Carmona

Professor John Pelesko and Anson CarterDepartment of MathematicsUniversity 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

List of Variables:volume =       g = gravityr = radius of capillary tubeZ = 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

Initial Set-up and Free Body Diagram

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 EquationFrom our free body diagram and by Newton's Second Law of Motion:

Net Force = Surface Tension Force - End-Effect Drag - Poiseuitte Viscous Force - Gravitational ForceNet 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:  

                                                                 whereZo = 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 usingsilicon 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

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

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

GoalsMap mathematically how high the liquid rises with respect to time

Wedge Set - Up Experiments were performed usingsilicon 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

GoalsTo 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 usingwater

Three runs were preformed with varyinglengths

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 DataThe 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

(u, v, w)u - velocity in Z-dirv - velocity in r -dirw - velocity in θ-dir

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 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: 

                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:

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: