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 TubeWedgeSponge

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 = 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 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 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 Z o = 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 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 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 Further Information

(u, v, w) u - velocity in Z-dir v - velocity in r -dir w - 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: