Problem Capillary Viscometer

A capillary viscometer is a vertical, cylindrical capillary tube filled with a liquid. The top of the

tube is open to the atmosphere. At the start of an experiment, the bottom of the tube is opened to

the atmosphere, and the liquid starts to flow due to gravity. We can assume that flow becomes

steady $($i$.$e$.$ independent of time$)$ very rapidly.

We measure the time required for the liquid meniscusto fall a distance $L.$ This is equivalent

to drainage of a liquid volume $V=R^{2}L$ where $R$ is the inner radius of the tube. The viscosity of

the liquid can be determined $($in principle$)$ from the measured drainage time for this volume. $*$

$($a$)(12$ points$)$ Draw a diagram and develop a model for steady flow of a liquid through the tube,

far away from the meniscus and either end of the tube. Clearly state all other assumptions you

wish to make. Develop all required governing equations and boundary conditions, simplified in

accord with your assumptions.

$($b$)(24$ points$)$ What are the velocity and pressure profiles in the liquid far away from either end

of the tube?

$($c$)(24$ points$)$ What is the volumetric flow rate of liquid through the tube? How long does it take

for a volume $V=R^{2}L$ to drain out of the tube? How would you use this result to determine the

liquid viscosity?$*$

$($d$)(8$ points$)$ The "strain rate", a scalar measure of the rate of strain in the liquid, may be

calculated from

$=[\frac{1}{2}tr({}_{{?}_{?}}*{}_{{?}_{?}}){]}^{\frac{1}{2}}$

where $"$tr$"$ is the trace. What is the strain rate in this capillary viscometer flow? Is $$ a constant,

or does it vary with position in the liquid?

$($e$)(8$ points$)$ What is the stress vector exerted by the liquid on the inner surface of the capillary

tube?

$($f$)(24$ points$)$ In capillary flows of some particle suspensions $($like blood, for example$),$ a particle$-$

free "clear layer" of thickness $$ may develop near the inner surface of the capillary tube. We may

model this as a two phase flow with the liquid suspension $($viscosity ${}^s)$ located in the center of

the tube ${}^F$