Material Derivative

Method
Symbol$D/Dt$
AliasesSubstantial Derivative, Total Derivative

Definition

The Material Derivative describes the time rate of change of a physical quantity experienced by a fluid element moving with the flow field.

Formula

DϕDt=∂ϕ∂t+v⋅∇ϕ\frac{D\phi}{Dt} = \frac{\partial\phi}{\partial t} + \mathbf{v} \cdot \nabla\phi where:

  • ∂ϕ/∂t\partial\phi/\partial t: Local rate of change (unsteady term).
  • v⋅∇ϕ\mathbf{v} \cdot \nabla\phi: Convective rate of change (due to motion).

Relationships