Transient momentum transport in a bounded system: For the problem of transient velocity profiles in a bounded system, oil is the fluid between the plates. The oil properties are: $=800\frac{kg}{m^3},=0\mathrm{.}5\frac{kg}{m*s}.$ The distance between plates is $)=(0\mathrm{.}5m.$ The upper plate is green and the lower plate is a particularly vibrant shade of electric orange.

The equation for the velocity profile $($in dimensionless terms$)$ is given below $($to save you from having to flip through your notes$).$

$(,)=(1-)-{\displaystyle \underset{n=1}{\overset{}{}}}(\frac{2}{n})exp(-n^2{}^2)sin(n)$

a$)$ Note the summation term in the solution. When simulating velocity profiles $($at different times$)$ you will need to determine how many terms in the summation are required $($meaning at what point does the contribution of the velocity for a given value of $n$ not significantly change the velocity profile$).$ Calculate the value of each term to determine how many terms are required in the summation. Justify your answer.

b$)$ Generate a set of velocity profiles through the oil $($at various times$)$ and plot them on a single plot. Three profiles are sufficient. Please go all "Goldilocks and The Three Bears" for this $-$ meaning one profile is a short time from the plate moving, one is a medium time and the last is at a time that approaches steady state. Please use time $(t$ not $$ on the plot $($but it will be better for all involved if you use $$ and $$ instead of $x$ and velocity $(v).$

c$)$ At what time does steady$-$state occur?