$1.$ The left atrial appendage in the heart is a region in the atrium where blood clots $($thrombi$)$ can form in patients with atrial fibrillation. One strategy to prevent this is to seal off the appendage such that blood cannot enter the appendage. This can be done surgically using a staple or suture. However, cardiothoracic surgery is quite invasive and carries additional risks to the patient$$s wellbeing as well as being costly$.$ One possible solution to this challenge is to use a cardiac catheterization procedure to fill the appendage with a substance that is liquid during delivery, but as it fills the appendage and is contacted by blood it solidifies, resulting in a permanent $$plug$$ in the appendage. The solution to be deployed has a viscosity of $10,000$ cP$,$ a density of $1\mathrm{.}26$ g$/$cm$3$ and the maximum diameter of catheter that can be used is $8$ French. A typical cardiac catheter is $110$ cm in length. The pressure is the atrium is referred to as the $$preload$$ of the heart and this must be overcome for the fluid to be able to fill the appendage. Normal preload is about $10$ mm Hg$.$ However, in patients with atrial fibrillation, it can often be double that value. As a design engineer, you must determine the amount of force that must be applied to the plunger of a $20$ ml syringe hooked up to the catheter. The left atrial appendage is usually determined by computed tomography on a patient$-$by$-$patient basis, but your design must accommodate a volume of up to $15$ ml$.$ One the syringe pump is turned on; you can assume that the system is operating at steady state until the appendage is full $($e$.$g$.$ all $15$ ml has been dispensed$).$ The time desired for dispensing the solution is not yet known, but is estimated not to exceed $5$ min. You need to determine the relationship between filling time and force required to deliver the fluid. Your results should be reported in graphical form using Excel $($plunger force $($lb$)$ vs filling time $($min$).$ Also discuss what you think the range of reasonable filling times would constitute good design with respect to the ability to control the procedure $(10$ points$)$