Problem $3.($total $40$ points$)$ Subject: flow rate $($symmetric surface$)$

Point$-$of$-$care testing using nano biosensors represents a novel paradigm in modern medical engineering,

allowing real$-$time, continuous monitoring of various biomarkers in a patient's bloodstream. A deployed

biosensor testing for metabolite concentrations should be capable of pumping microscopic amounts of

blood through a test chamber where the measurements are performed.

Assume that such a device is being designed by a team of engineers, including yourself, and some

specifications must be determined before a working prototype can be built. For proper metabolite

detection, an optimal flow rate through the chamber must be achieved. During the process of

identifying design requirements, a non$-$dimensional constant $($alpha$)$ was identified as a parameter of

the flow that can be altered by changing the magneto$-$electrical specifications of the device. To link the

required flow rate with the device specifications, the total flow rate dependence on $($alpha$)$ must be

established. The chamber's cross$-$section through which blood flows is a semicircle. To calculate the flow rate

through this cross$-$section and its dependence on $($alpha$),$ use the following input. Figure A shows the

geometric parameters of the semicircle and the origin of the $(x,y)$ coordinate system. The flow

$($schematically represented by a red arrow$)$ crosses the cross$-$section at a varying angle $,$ changing

linearly from $0$ at $y=0$ to the value shown in the figure at $y=h.$ The magnitude $v_0$ of the velocity

vector field depends on the coordinate $y$ as shown in the figure.

Using the final formula for the dependence of flow on alpha, $Q(),$ determine the value of the flow if

$=0((h)=45)$ and if $>>1((h)$~~$0).$ Hint: use Taylor's expansion series to approximate the flow

when $>>1.$

B

$\{\begin{array}{c}\text{:}[{]}_y=0=0^0\text{;}[{]}_y=h=\frac{{45}^0}{(1+)}\end{array}$

$v_0(y)=\frac{V}{h}\sqrt[\phantom{2}]{2hy-y^2}$