Incompressible fluid of constant viscosity $\mu$ is confined to the region $0<y<h$ between two parallel rigid plates. Consider two parallel viscous flows: flow A is driven by the motion of one plate in the $x$-direction with the other plate at rest; flow B is driven by a constant pressure gradient in the $x$-direction with both plates at rest. The velocity mid-way between the plates is the same for both flows.

The viscous friction in these flows is known to produce heat locally at a rate

$$Q=\mu\left(\frac{\partial u}{\partial y}\right)^{2}$$

per unit volume, where $u$ is the $x$-component of the velocity. Determine the ratio of the total rate of heat production in flow A to that in flow B.