In this special case, the differential equation (12) in the formula sheet can be used to find the temperature profile of the ol inside the gap. a) Simplify equation (12)

as much as possible. Explain the physical meanings of the terms in the equation. (2 marks)b) Integrate the simplified differential equation you have found in a) andexpress the temperature profile of the oll Ty) as a function of y, k. . U.L and the two unknown constants of integration, C. and C. Do notsubstitute the numertcal values at this stage c) At y=L the surface is adiabatic and at yo the surface is at constant temperature Te In mathematical form, write down the two boundary conditions needed to solve the differential equation(2 marks) (2 marks]d) Using the results from b) and c), express the temperature profile of the oil, Toy), as a function of y k H.U, L and Te Do not substitute the numerical values at this stage e) By applying the Fourier's law at y-0, write down the mathematicalexpression for the rate of heat transfer per unit length of the cylinder as afuniction of D. p.U, L and calculate.its numerical value (Wim)A marks) ) It occurs that the heat-transfer rate per unit length (W/m) calculated in e) is equal to the power per unit length needed to rotate the cylinder (W/m). Explain why this is true