Explanation of Equation 1 from Liquid Rise in a Capillary Tube by Wesley E. Brittin

 

 

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 water

= contact angle between the surface of the liquid and the wall of the tube

 

Surface Tension Force:

 

Graviational Force:

 

 

Poiseulle Viscous Force:

More information as to how the Poiseulle Viscous Force was derived can be found here: Poiseulle Viscous Force

 

End-Effect Drag:

 

By Newton's Second Law of Motion we know:

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:

 

 

 

 

 Now, an easy first check on your derivation is to look at the steady-state solution. That is, set the time derivatives

to zero in the last equation and solve for Z. This should agree with our purely steady-state analysis of the rise

of fluid in a capillary tube. From above, I find

 

 

A check for the dimensions of the steady state solution can be found on this page.

 

Notice that this does differ from the analysis that Anson presented because here, the angle that the capillary tube

makes with the horizontal surface of the fluid is not assumed to be ninety degrees. But, when this angle is ninety

degrees, the sine term in the denominator becomes one, this should then agree with our steady-analysis.

 

Anson - might be a good idea to redo your analysis taking the angle the tube meets the fluid into account.

 

Explanation of Initial Velocity

The initial velocity is assumed to be finite and is obtained from the differential equation when setting Z(0) = 0.

 

A check for the dimensions of the initial velocity can be found on this page.

 

Plotted Solution to Differential Equation

This a graph of the five-term approximation (Equ 36) given in Brittin's Paper in comparison to the steady state

solution that was obtained with using values given in the paper.

 

 

 

Plotting of Numerical Solution

This is a plot of the numerical solution given by MatLab for the Brittin Differential Equation that describes the rise of a liquid in a capillary tube.

The numerical solution is plotted against the experimental data and the steady state solution.

 

 Experiments Corresponding to Equations

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