1
$\begingroup$

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} $?

$\endgroup$
  • 3
    $\begingroup$ Different authors use different notation. $\endgroup$ – Qmechanic Apr 19 at 12:27
0
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.