Consider a wide $(50$ m $),$ deep $(5$ m $)$ rectangular cross$-$section channel. At noon on Thursday, there was a spill of $500$ litres of water containing the given concentrations of the contaminant. Assume the spill rapidly mixes vertically and across the channel, such that initially the entire mass of the contaminant can be considered as a line across the stream.

As a first approximation you estimate water velocity by timing how long it takes for oranges to float downstream along a $100$m section of channel to get the following times: $123,140,115,120,132,110,117,124,123($s$)$

Assume the velocity is uniform in the horizontal but varies linearly in the vertical, satisfying the non$-$slip condition at the bottom.

a$.$ Sketch the problem defining variables where appropriate. Include a sketch of the velocity distribution.

b$.$ Write an expression that describes the cross$-$stream average concentration of the contaminant. Define, and where appropriate, determine a value for each variable.

c$.$ At what time will dispersion become the dominant process spreading the pollutant? Create plots of the cross$-$stream averaged concentration versus distance of the contaminant at $\backslash ($ t$=12\backslash )$ hours, $1$ day, and $3$ days.

d$.$ How far downstream of the spill will the centroid of the concentration distribution be at $\backslash ($ t$=12\backslash )$ hours, $1$ day, and $3$ days? Is this consistent with your plot in part $\backslash ($ c $\backslash )?$

e$.$ How far downstream will the maximum concentration drop below the given standard?

Please provide Python codes for sketching graphs as well.