I was wondering what is the difference between the convective/material derivative and the total derivative. We were introduced to the notion of material derivative

$$ \frac{D\vec{u}}{Dt}=\frac{\partial \vec{u}}{\partial t}+(\vec u.\vec{\triangledown})\vec u $$

in fluid mechanics but I can not see how this is any different to the time derivative of a vector field

$$ \vec{u}({\vec{x(t)}},t) $$ and just applying the chain rule?

If there is no difference, why introduce new notation of $ \frac{ D\vec{u}}{D t} $ instead of just using the standard notation $\frac{d\vec{u}}{dt} $?

  • 3
    $\begingroup$ Different authors use different notation. $\endgroup$ – Qmechanic Apr 19 at 12:27

Take a look at the definitions below

Total derivative: $$\frac{d}{dt} = \frac{\partial}{\partial t} + \frac{dx_i}{dt}\frac{\partial}{\partial x_i}$$

Material derivative: $$\frac{D}{Dt} = \frac{\partial}{\partial t} + u_i\frac{\partial}{\partial x_i}$$

Would you agree they are the same if $u_i=\frac{dx_i}{dt}$?

Why people use inconsistent notation for both? I do not know.

In my opinion the total derivative is used most often in mathematics whereas the material derivative is used most often in physics. In physics, $\frac{dx_i}{dt}$ has a clear physical interpretation as the instantaneous velocity. In mathematics, there is not necessarily a physical interpretation and the variable notation may be arbitrary.

The next question is, do you understand what the physical interpretation of the material derivative is?

  • $\begingroup$ I believe physically it describes how a quantity (scalar or vector) of an element changes as the element moves in time and space? $\endgroup$ – Dan Apr 25 at 11:00

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