Question $2(50$ pts$).($Collapse of a Constricted Vessel Segment$).$ Figure $6\mathrm{.}15$ shows the blood flow through a constricted circular vessel, where $d_i$ and $d_c$ denote the diameters at the inlet and constriction, respectively. The blood has the density $=1060kg{m}^{-3}$ and may be regarded as an ideal $($in$-$viscous$)$ fluid with the velocity $v_i=23\mathrm{.}2cm{s}^{-1}$ and the pressure $p_i=75mmHg$ at the vessel's inlet. The vessel wall may be regarded as a membrane that has no bending stiffness, and $p_0=12\mathrm{.}0mmHg$ determines the ambient pressure outside the vessel.

Fig. $6\mathrm{.}15$ Blood flow through a vessel constriction

a$)$ Consider incompressible blood and express the flow velocity $v_G$ in the constriction.

b$)$ Use Bernoulli's equation

$\frac{v^2}{2}+gh+\frac{p}{}=C$

and express the pressure $p_i$ in the constriction.

c$)$ Specify the relative diameter stenosis $S_{\text{d}\text{i}\text{a}\text{m}\text{e}\text{t}\text{e}\text{r}}=\frac{d_i-d_c}{d_i}$ as well as the relative area stenosis $S_{\text{a}\text{r}\text{e}\text{a}}=\frac{A_i-A_c}{A_i},$ at which the vessel would collapse $($as percentages$).$ Here, $A_i$ and $A_c$ denote the cross$-$sectional areas at the inlet and constriction, respectively.