Convective derivative

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The convective derivative, also known as the Lagrangian derivative, total time derivative, substantive derivative, and by several other names, is a derivative taken with respect to a coordinate system moving with velocity u, and is often used in fluid mechanics and classical mechanics. It is defined for a scalar function $φ$ and vector v by:

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

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

where $\nabla$ is the gradient operator del and $\frac{\partial}{\partial t}$ denotes the partial derivative with respect to t.

The convective derivative expresses the Eulerian derivative (written $\partial/\partial t$ ) in Lagrangian coordinates.

Consider water undergoing steady flow through a hosepipe that has a gradually decreasing cross section. Because water is incompressible in practice, conservation of mass requires that the flow is faster at the end of the pipe than at the start. Because the flow is steady, the Eulerian derivative of velocity is everywhere zero, but the substantive derivative is nonzero because any individual parcel of fluid accelerates as it moves down the hose.

For tensor fields we usually want to take into account not only translation of the coordinate system due to the fluid movement but also its rotation and stretching. This is achieved by the upper convected time derivative.

[edit] Proof

Proof is via the chain rule for partial derivatives. In tensor notation (with the Einstein summation convention), the derivation may be written:

$\left[\frac{D\mathbf{B}}{Dt}\right]_j = \frac{D}{D t} \hat{B_j}(t, x_i(t)) = \frac{\partial B_j}{\partial t} + \frac{\partial B_j}{\partial x_i} \frac{\partial x_i}{\partial t} = \frac{\partial B_j}{\partial t} + \frac{\partial x_i}{\partial t} \frac{\partial}{\partial x_i} B_j = \frac{\partial B_j}{\partial t} + \left[(\mathbf{u}\cdot\nabla)\mathbf{B}\right]_j$