91Ó°ÊÓ

The expression of the volume flow rate is,

\(Q = \frac{{\pi {R^4}\left( {{P_1} - {P_2}} \right)}}{{8\eta l}}\)

Here\(R\)is the inside radius,\(\eta \)is the coefficient of viscosity,\(Q\)is the volume flow rate,\(l\)is the length, and\(\left( {{P_1} - {P_2}} \right)\)is the pressure difference.

From the above equation, we can conclude thatthe volume flow rate is directly proportional to the fourth power of the inside radius. That is,

\(Q \propto {R^4}\)

The expression of the radius in terms of diameter is,

\(R = \frac{d}{2}\)

Here\(d\)is the diameter.

Substitute\(\frac{d}{2}\)in the expression volume flow rate, we get

\(Q \propto {\left( {\frac{d}{2}} \right)^4}\)

Assume the diameter of the artery is\(d\). The cholesterol reduces the diameter by25%. The new diameter\({d^\prime }\)is,

\(\begin{array}{c}{d^\prime } = d - \left( {\frac{{25}}{{100}}} \right)d\\ = \;0.75\;d\end{array}\)

The initial flow rate is directly proportional to the initial diameter.

\({Q_{{\rm{in }}}} \propto {\left( {\frac{d}{2}} \right)^4}\)

Here\({Q_{{\rm{in }}}}\)is the initial flow rate.

The final flow rate is directly proportional to the final diameter.

\({Q_{{\rm{final }}}} \propto \;{\left( {\frac{{{d^\prime }}}{2}} \right)^4}\)

Here \({Q_{{\rm{final }}}}\) is the final flow rate.

The ratio of the final volume flow rate to the initial volume flow rate is,

\(\begin{array}{c}\frac{{{{\rm{Q}}_{{\rm{in}}}}}}{{{{\rm{Q}}_{{\rm{final}}}}}} = \frac{{{{\left( {\frac{d}{2}} \right)}^4}}}{{{{\left( {\frac{{{d^\prime }}}{2}} \right)}^4}}}\\{Q_{{\rm{final}}}} = {Q_{{\rm{in}}}}\left( {\frac{{{d^4}}}{{{d^4}}}} \right)\end{array}\)

Substitute\(0.75d\)for\({d^\prime }\), we get

\(\begin{array}{c}{Q_{{\rm{final}}}} = \;{Q_{{\rm{in}}}}\left( {\frac{{{{(0.75d)}^4}}}{{{d^4}}}} \right)\\{Q_{{\rm{final}}}} = \;{(0.75)^4} \times \;{Q_{{\rm{in}}}}\left( {\frac{{{d^4}}}{{{d^4}}}} \right)\\{Q_{{\rm{final}}}}\; = \;(0.316){Q_{{\rm{in}}}}\\{Q_{{\rm{final}}}}\; = \;(31.6\% ){Q_{{\rm{in }}}}\end{array}\)

Therefore, the flow rate is \(31.6\% \) of the initial flow rate.