A fluid is sheared between two parallel plates as we saw in Chapter 4.

If the gap between the plates is small and the shearing rate is high, friction between the fluid molecules will generate heat. Energy will flow by conduction in the \(y\) direction from the hot plate toward the cold plate. As we will see later in Chapter 10, the heat generation rate will be given by:

\[\dot{q}\left(\frac{W}{m^{3}}\right)=\mu\left(\frac{d v_{x}}{d y}\right)^{2}\]

a. For a fluid of viscosity, \(\mu\), a gap width between the plates of \(\delta\), and an upper plate velocity of \(v_{0}\), what is the heat generation rate within the fluid?

b. If the upper surface of the plate is held at a temperature of \(T_{0}\) and the lower surface at a temperature of \(T_{1}\), what is the temperature profile between the plates?

c. For the parameters:

\[\begin{aligned}& \mu=0.25 \mathrm{Ns} / \mathrm{m}^{2} \quad \delta=0.001 \mathrm{~m} \\& v_{0}=1 \mathrm{~m} / \mathrm{s} \quad T_{0}=325 \mathrm{~K} \quad T_{1}=300 \mathrm{~K}\end{aligned}\]

evaluate the heat generation rate and the temperature profile.