Chapter 10, Problem 88GP

To determine

To find: The effect on blood flow due to reduction of the diameter.

Answer to Problem 88GP

The flow becomes $52\%$ of the initial value.

Explanation of Solution

Given:

Diameter of the artery reduces by $15\%$ .

Formula used:

Write the expression for the coefficient of viscosity.

  $\eta =\frac{\pi \times \Delta P\times r^4}{8\times L\times F}$

Here,

  $\eta$ is the coefficient of viscosity,

  $\Delta P$ is the change of pressure,

  $L$ is the length,

  $r$ is the radius,

  $F$ is the flow of fluid.

Calculation:

From the above equation:

  $F=\frac{\pi \times \Delta P\times r^4}{8\times L\times \eta }$

As other quantities except $F$ and $r$ are constants, So

  $F\propto r^4$

Therefore , $\frac{F_{\text{1}}}{F_{\text{2}}}=\frac{r_1{}^{\text{4}}}{r_2{}^{\text{4}}}$

As diameter reduces by $15\%$

  $\frac{F_1}{F_2}=\frac{r^4{}_1}{{(r_1-0.15r_1)}^4}$

  $\frac{F_1}{F_2}=\frac{r^4{}_1}{{(0.85r_1)}^4}$

  $\begin{array}{l}\frac{F_2}{F_1}=\frac{{0.85}^4}{1}\\ =0.5222\end{array}$

  $F_2$ is $52\%$ of the initial value.

Conclusion:

Thus, the flow becomes $52\%$ of the initial value.