Convective derivative

The convective derivative, also known as the Lagrangian derivative, is a derivative taken with a respect to a coordinate system moving with velocity u, and is often used in fluid mechanics. It is defined for a scalar function <math>\phi<math> and vector v by:

<math>\frac{D\phi}{Dt} = \frac{\partial \phi}{\partial t} + (\mathbf{u}\cdot\nabla)\phi<math>

<math>\frac{D\mathbf{v}}{Dt} = \frac{\partial \mathbf{v}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{v}<math>

where <math>\nabla<math> is the gradient operator del and <math>\frac{\partial}{\partial t}<math> denotes the partial derivative with respect to t. Proof is via the chain rule for partial derivatives.

Note the following identities when taking the convective derivative of an integral:

<math>\frac{D}{Dt}\int_{V(t)} f(\mathbf{x})\, dV

= \int_{V(t)} \left( \frac{\partial f}{\partial t} + \nabla\cdot(f\mathbf{u}) \right) \, dV = \int_{V(t)} \left( \frac{Df}{Dt} + f ( \nabla\cdot\mathbf{u} ) \right) \, dV<math>