The derivative, \( D/Dt \), defined by the previous slide, is commonly called the substantial derivative to remind us that it is computed for a particle of "substance." It is often also called the material derivative or particle derivative.

The physical significance of the terms is: \( \underbrace{\frac{D \vec{V}}{D t} = \vec{a}_p}_{\text{total acceleration of a particle}} = \underbrace{ v_x \frac{\partial \vec{V}}{\partial x} + v_y \frac{\partial \vec{V}}{\partial y} + v_z \frac{\partial \vec{V}}{\partial z} }_{\text{convective acceleration}} + \underbrace{ \frac{\partial \vec{V}}{\partial t} }_{\text{local acceleration}} \)

We recognize that a fluid particle moving in a flow field may undergo acceleration for either of two reasons:

• A steady flow convects particles toward a low-velocity region and then away to a high-velocity region.
• If a flow field is unsteady, a fluid particle will undergo an additional local acceleration because the velocity field is a function of time.

The convective acceleration can be rewritten using the gradient operator \( \nabla \): \( v_x \frac{\partial \vec{V}}{\partial x} + v_y \frac{\partial \vec{V}}{\partial y} + v_z \frac{\partial \vec{V}}{\partial z} = (\vec{V} \cdot \nabla) \vec{V} \)

where \( \cdot \) is the dot product. Thus, the material derivative may be rewritten as: \( \frac{D \vec{V}}{D t} = \vec{a}_p = (\vec{V} \cdot \nabla) \vec{V} + \frac{\partial \vec{V}}{\partial t} \)