In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material derivative can serve as a link between Eulerian and Lagrangian descriptions of continuum deformation.
For example, in fluid dynamics, the velocity field is the flow velocity, and the quantity of interest might be the temperature of the fluid. In this case, the material derivative then describes the temperature change of a certain fluid parcel with time, as it flows along its pathline (trajectory).
Other names
<!-- in bold, since the redirects lead to here -->
There are many other names for the material derivative, including:
- advective derivative
- convective derivative
- derivative following the motion
- particle derivative
- substantial derivative
- substantive derivative
- Stokes derivative although the material derivative is actually a special case of the total derivative
Only this spatial term containing the flow velocity describes the transport of the field in the flow, while the other describes the intrinsic variation of the field, independent of the presence of any flow. Confusingly, sometimes the name "convective derivative" is used for the whole material derivative <math>\mathrm{D}/\mathrm{D}t</math>, instead for only the spatial term <math>\mathbf u \cdot \nabla</math>.
<math display="block">[\left(\mathbf{u} \cdot \nabla \right)\mathbf{A}]_j =
\sum_i \frac{u_i}{h_i} \frac{\partial A_j}{\partial q^i} + \frac{A_i}{h_i h_j}\left(u_j \frac{\partial h_j}{\partial q^i} - u_i \frac{\partial h_i}{\partial q^j}\right),
</math>
where the are related to the metric tensors by <math>h_i = \sqrt{g_{ii.</math>
In the special case of a three-dimensional Cartesian coordinate system (x, y, z), and being a 1-tensor (a vector with three components), this is just:
<math display="block">(\mathbf{u}\cdot\nabla) \mathbf{A} =
\begin{pmatrix}
\displaystyle
u_x \frac{\partial A_x}{\partial x} + u_y \frac{\partial A_x}{\partial y}+u_z \frac{\partial A_x}{\partial z}
\\
\displaystyle
u_x \frac{\partial A_y}{\partial x} + u_y \frac{\partial A_y}{\partial y}+u_z \frac{\partial A_y}{\partial z}
\\
\displaystyle
u_x \frac{\partial A_z}{\partial x} + u_y \frac{\partial A_z}{\partial y}+u_z \frac{\partial A_z}{\partial z}
\end{pmatrix} =
\frac{\partial (A_x, A_y, A_z)}{\partial (x, y, z)}\mathbf{u}
</math>
where <math>\frac{\partial(A_x, A_y, A_z)}{\partial(x, y, z)}</math> is a Jacobian matrix.
There is also a vector-dot-del identity for the case <math> \mathbf{u} = \mathbf{A} </math>, for which the material derivative for a vector field <math>\mathbf A</math> can be expressed as:
:<math> {\displaystyle (\mathbf {A} \cdot \nabla )\mathbf {A} = {\frac {1}{2\nabla |\mathbf {A} |^{2}-\mathbf {A} \times (\nabla \times \mathbf {A} )={\frac {1}{2\nabla |\mathbf {A} |^{2}+(\nabla \times \mathbf {A} )\times \mathbf {A} }.</math>
See also
- Navier–Stokes equations
- Euler equations (fluid dynamics)
- Derivative (generalizations)
- Lagrangian and Eulerian specification of the flow field
- Lie derivative
- Levi-Civita connection
- Spatial acceleration
- Spatial gradient