Explain what is meant by a material time derivative, $D / D t$. Show that if the material velocity is $\mathbf{u}(\mathbf{x}, t)$ then

$$D / D t=\partial / \partial t+\mathbf{u} \cdot \nabla$$

When glass is processed in its liquid state, its temperature, $\theta(\mathbf{x}, t)$, satisfies the equation

$$D \theta / D t=-\theta \text {. }$$

The glass flows in a two-dimensional channel $-1<y<1, \quad x>0$ with steady velocity $\mathbf{u}=\left(1-y^{2}, 0\right)$. At $x=0$ the glass temperature is maintained at the constant value $\theta_{0}$. Find the steady temperature distribution throughout the channel.