= Solution
{c}
At a fixed point beside a river, the <Eulerian coordinate> description measures $\mathbf u(\mathbf x,t)$. Its local <acceleration> $\partial_t\mathbf u$ measures change at that fixed point. Following a floating parcel gives a <Lagrangian trajectory> $\mathbf X(t)$ satisfying $\dot{\mathbf X}=\mathbf u(\mathbf X,t)$. The <chain rule> gives its physical <material acceleration>:
$$
\boxed{\ddot{\mathbf X}=\frac{D\mathbf u}{Dt}
=\partial_t\mathbf u+(\mathbf u\cdot\nabla)\mathbf u.}
$$
The extra term is the <convective acceleration>. For a river that narrows steadily, an observer at one fixed position measures a constant <velocity>, yet a parcel speeds up as it reaches the narrowing. A parcel following a steady bend also experiences a <centripetal acceleration> even when its speed is constant. An Eulerian calculation of the physical <acceleration> must include the same <convective acceleration>; the two descriptions do not predict different forces.
Back to article page
