capillary presentation
TRANSCRIPT
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Dynamics of Capillary SurfacesLucero CarmonaProfessor John Pelesko and Anson Carter
Department of Mathematics
University of Delaware
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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
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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
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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
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Explanation of the Forces
Surface Tension Force
Gravitational Force
Poiseulle Viscous Force
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Explanation of the Forces End-Effect Drag
Newton's Second Law of Motion
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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
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Steady State
By setting the time derivatives to zero in thedifferential equation and solving for Z, we are
able to determine to steady state of the rise
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Set - Up Experiments were performed usingsilicon oil and waterSeveral preparations were used on the
set-up to see if altered techniques wouldproduce different results
The preparations included:
Using a non-tampered tube
Extending the run time and aligningthe camera
Aligning the camera and using an
non-tampered tube
Disinfecting the Tube and aligningthe camera
Pre-wetting the Tube and aligning
the camera
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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.
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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 andto print out quick alterations to the data
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Capillary Tubes with Silicon Oil
Silicon Oil Data:
Steady State Solution
Initial Velocity
Eigenvalues
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Capillary Tube with Water
Water Data: Steady State Solution
Initial Velocity
Eigenvalues
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Previous Experimental Data (Britten 1945)
Water Rising at Angle Data:Steady State Solution
Initial Velocity
Eigenvalues
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Results
There is still something missing from the
theory that prevents the experimental data to
be more accurate
The steadystate is not in agreement with
the theory There is qualitative agreement but not
quantitative agreement
Eliminated contamination
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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
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Wedge Set - Up Experiments were performed usingsilicon oilTwo runs were performed with different
angles
Experiments were recorded with the
high speed camera at 250 fps and 60 fps
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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 secpassed between each slide
MatLab was then used to measure the
rise of the liquid in pixels
Excel and a C-program were used toconvert the pixel distances into MM and
to print out quick alterations to the data
Z
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Wedge Data
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Explanation of Sponges
Capillary action can be seen in porous
sponges
Goals
To see if porous sponges relate to thecapillary tube theory by calculating what
the mean radius would be for the pores
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Sponge Set - Up Experiments were performed usingwaterThree runs were preformed with varying
lengths
Experiments were recorded with the
high speed camera at 250 fps and 60 fps
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Sponge Data
The effects of widths and swelling
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Future Work Refining experiments to prevent undesirable
influences Constructing a theory for wedges and
sponges
Producing agreement between theory andexperimentation for the capillary tubes
Allowing for sponges to soak overnight with
observation
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References Liquid Rise in a Capillary Tubeby W. Britten
(1945). Dynamics of liquid in a circular capillary. The Science of Soap Films and Soap Bubblesby C.
Isenberg, Dover (1992).
R. Von Mises and K. O. Fredricks, Fluid Dynamics
(Brown University, Providence, Rhode Island, 1941), pp137-140.
Further Information http://capillaryteam.pbwiki.com/here
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(u, v, w)
u - velocity in Z-dirv - velocity in r -dir
w - velocity in -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,, will be proportional to the rate of change of velocity across the surface.
Due to the variation of u in the r-direction, where is the viscosity coefficient:
Since we are dealing with cylindrical coordinates
From the Product Rule we can say that:
Solving for u:
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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:
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Explanation of the Forces Poiseulle Viscous Force:Equation, u, in terms of Average Velocity
Further Anaylsis on shearing stress, :
for,
The drag, D, per unit breadth exerted on the wall
of the tube for a segment l can be found as: