Past exam of the mathematics course of the University of Cambridge / 2006 / ib / Paper 2 / 8A / Solution

A material derivative measures the rate experienced by a moving fluid particle. If its position is $\mathbf{X}(t)$ with $\dot{\mathbf{X}}=\mathbf{u}(\mathbf{X},t)$, the chain rule for any scalar field $F$ givesHence the operator is $D/Dt={\partial }_t+\mathbf{u}\cdot \nabla$, and the material derivative differs from the time derivative at a fixed spatial point.

In the steady temperature problem ${\partial }_t\theta =0$ and the transverse velocity vanishes. Thus at each fixed $y\in (-1,1)$,Integrating in $x$ and applying the inlet value givesA particle's travel time from the inlet is $x/(1-y^2)$, so the formula is also the exponential decay along its characteristic curve. For fixed $x>0$ the temperature tends to zero near either wall, because the travel time becomes large. No value at $y=\pm 1$ is imposed by an inlet condition there, since the wall velocity is zero and the stated channel is open in $y$.