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Convective DerivativeThe 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 and vector v by: -
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where is the gradient operator del and 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: -
= \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
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