PX285 - I2b - advective derivative

in fluid mechanics, capital $D$ is used to denote total derivatives:

\frac{D \vec{u}}{D t} = \frac{d \vec{u}}{d t} = lim_{Δ t \to 0} [\frac{\vec{u} (\vec{r} (t + Δ t), t + Δ t) - \vec{u} (\vec{r} (t), t)}{Δ t}]

using the taylor expansion for $\vec{u} (\vec{r} (t + Δ t), t + Δ t);$

\begin{matrix} \frac{D \vec{u}}{D t} \approx lim_{Δ t \to 0} {. \frac{1}{Δ t} [\vec{u} (\vec{r}, t) + \frac{\partial \vec{u}}{\partial x} Δ x + \frac{\partial \vec{u}}{\partial y} Δ y + \frac{\partial \vec{u}}{\partial z} Δ z + \frac{\partial \vec{u}}{\partial t} Δ t - \vec{u} (\vec{r}, t)]} \\ = \frac{\partial \vec{u}}{\partial t} + u_{x} \frac{\partial \vec{u}}{\partial x} + u_{y} \frac{\partial \vec{u}}{\partial y} + u_{z} \frac{\partial \vec{u}}{\partial z} \\ = \frac{\partial \vec{u}}{\partial t} + (\vec{u} \cdot \vec{\nabla}) \vec{u} \end{matrix}

where, $lim_{Δ \to 0} [Δ x (t) / Δ t] = u_{x}$

therefore, the advective derivative:

\frac{d}{d t} = \frac{D}{D t} = \frac{\partial}{\partial t} + \vec{u} \cdot \vec{\nabla}

the first term emphasises the rate of change of fluid property at a fixed point in space
the second term is for the additional rate of change due to the movement of the fluid element