Question $2(20$ Marks in Total$)$

A general expression for Poisseuille number for microchannels with arbitrary cross$-$sectional shape is given by

$($a$)$ Describe briefly in words how the form of this equation is obtained

$(5$ Marks$)$

$($b$)$ Starting with this equation, and noting that when the flow becomes developed the contribution of the developing and developed terms are equal, derive the following expression for hydrodynamic entrance length

$L_n^{**}=0\mathrm{.}0822(1+{)}^2[1-\frac{192}{{}^5}tanh(\frac{}{2}){]}^2$

$(7$ Marks$)$

$($c$)$ Determine the entrance length for a flow of deionised water in a rectangular channel $($height $150$ microns, width $200$ microns$)$ at Reynolds number of $10.$

$(3$ Marks$)$

$($d$)$ If the Reynolds number is increased, what trend does the entrance length follow? Elaborate on the physical effects in your answer.

$(3$ Marks$)$

$($e$)$ How long should a channel be relative to the entrance length to treat the channel pressure drop as fully developed with a small error?

$(2$ Marks$)$