Convective derivative

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The convective derivative, also known as the Lagrangian derivative, total time derivative, and by several other names, is a derivative taken with a 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 <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.

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>

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Proof

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

<math>\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</math>
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References

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