capillary presentation

8/11/2019 Capillary Presentation

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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:

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