Ashu’s Shorthand definition

A material Derivative expresses the rate of change of a function of both time and space.

The substantial derivative $\frac{D}{D t}$ of a quantity $ϕ$ is the rate of change of $ϕ$ belonging to a certain particle moving within a material or substance.

Proof

Temperature of a moving particle $p$ .

T_{p} = T_{p} [x_{p} (t), y_{p} (t), y_{p} (t), z_{p} (t), t]

Applying multi-variable chain rule

\frac{d T _{p}}{d t} = \frac{\partial T _{p}}{\partial t} + \frac{\partial T _{p}}{\partial x _{p}} \frac{d x _{p}}{d t} + \frac{\partial T _{p}}{\partial y _{p}} \frac{d y _{p}}{d t} + \frac{\partial T _{p}}{\partial z _{p}} \frac{d z _{p}}{d t}

After simplification

\frac{d T _{p}}{d t} = \frac{\partial T _{p}}{\partial t} + u \frac{\partial T _{p}}{\partial x _{p}} + v \frac{\partial T _{p}}{\partial y _{p}} + w \frac{\partial T _{p}}{\partial z _{p}} = \frac{D T _{p}}{D t}

Using vector calculus notation

\frac{D T}{D t} = \frac{\partial T}{\partial t} + convective der (U \cdot \nabla T)

Note: The fluid particle in this context refers to the Fluid parcel in continuum mechanics.

References

Faculty of Khan - The Material Derivative | Fluid Mechanics

Ashuthosh Shridhar ~

Explorer

Material, Substantial or Lagrangian Derivative

Ashu’s Shorthand definition

Proof

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