Problem $2$

"When the relevant dimensionless groups are maintained constant, dimensional analysis can tell us the proper way to interpret the results of scaled$-$up or scaled$-$down experiments."

Consider that a sophomore biomedical engineering student is interested in designing an artificial kidney but was told to figure out first the type of drag experienced by flowing blood in an artery. The student has decided to evaluate the drag coefficient for blood flowing through an artery using a $\frac{1}{5}$ scale model he$/$she built in his$/$her home garage. The real artery $($call it prototype$)$ would maintain the same dimensionless quantity called Reynold's number $($used to characterize flows including flows through arteries$)$ as the model built by the student. Based on the last statement, the Reynolds number for the student artery model $($m$)$ will be related to the real prototype $($p$)$ artery as:

$\frac{{}_m}{{}_m{V}_m{L}_m}=\frac{{}_p}{{}_p{V}_p{L}_p}$

where $$ is the fluid viscosity, $$ is the density of the fluid, $V$ is the fluid velocity and $L$ is the characteristic length of the artery. Note subscript $m$ stands for model while $p$ is for the real or prototype artery.

$($a$)$ If $L_m=\frac{1}{5}{L}_p,$ what are the choices the student has for the model?

$($b$)$ Which $($if any$)$ of those choices above would not make practical sense to implement? Explain