A thin film of a silicon oil, \(3 \mathrm{~mm}\) thick, sits on a long plate that is partially heated. The plate has a linear temperature distribution imposed on it. At \(x=0\), on the left-hand side of the plate, the temperature is \(0^{\circ} \mathrm{C}\). At \(x=0.005 \mathrm{~m}\), on the right-hand side, the temperature is \(100^{\circ} \mathrm{C}\). The silicone oil has a viscosity of \(0.1 \mathrm{~Pa} \cdot \mathrm{s}\) and its surface tension decreases with temperature at a rate, \(-6 \times 10^{-5} \mathrm{~N} / \mathrm{mK}\).

a. What is the shear stress on the upper surface of the oil assuming that at any \(x\)-position on the plate, the oil is at a uniform temperature throughout its thickness?

b. What is the velocity of the free surface of the silicone oil assuming steady-state, 1-D flow, i.e., \(v_{x}=f(y)\) ?