"A thin layer of oil $(\backslash (\backslash$left$.\backslash$mu$=0\mathrm{.}104\backslash$mathrm$\{$~N$\}.\backslash$mathrm$\{$s$\}/\backslash$mathrm$\{$m$\}^\{2\},\backslash$rho$=917\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash$right$)\backslash )$ is to be removed from the inside of a pipe by blowing air through the pipe. The inside diameter of the pipe is $2\mathrm{.}0$ cm and the oil layer is $0\mathrm{.}2$ mm thick. If the pressure gradient in the pipe is $\backslash (1\mathrm{.}5\backslash$mathrm$\{$kPa$\}/\backslash$mathrm$\{$m$\}\backslash )$ and the flow of air and oil is laminar and fully developed, determine the shear stress at the oil$-$air interface."

a$)$ With the shear stress at the oil$-$air interface determined through this previous problem analysis, determine the rate of mechanical energy dissipation into thermal energy in the oil over a $1\mathrm{.}0$m$-$long segment of the pipe.

Hint: Determine the rate of work done by the air on the oil.

b$)$ If we neglect the heat transfer between the oil and the air, and the heat transfer between the oil and the pipe, determine the change in the bulk temperature of the oil over $1$ m $-$long segment of pipe.

c$)$ With the shear stress at the oil$-$air interface determined through this previous problem analysis, determine the rate of mechanical energy dissipation into thermal energy in the oil over a $1\mathrm{.}0$m$-$long segment of the pipe.