Newtonian fluid mechanics: Navier-Stokes equation

[Feodalherren]

Homework Statement

Homework Equations

Navier-Stokes

The Attempt at a Solution

Not really trying to solve a problem, trying to understand what is going on in my textbook. So look at the stuff in red first. I see where all that is coming from, it's clear to me. However, the stuff in green indicates that in the example case the right side of the equation should be zero, instead they throw in the stuff in blue out of the blue. What exactly happened here that they totally missed to explain?!

 

[Chestermiller]

Are you familiar with the material derivative, d/dt? It is defined as:
$$\frac{d}{dt}=\frac{\partial}{\partial t}+u\frac{\partial}{\partial x}+v\frac{\partial}{\partial y}+w\frac{\partial}{\partial z}$$
The material derivative of u is equal to the x-component of the fluid acceleration.
$$\frac{du}{dt}=a_x=\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}$$

Chet

 

[Feodalherren]

Okay yeah I remember that from class. Thanks.

I'm still slightly confused though. How am I supposed to know if it's a material derivative of just a regular derivative? What exactly makes it a material derivative? If I take the expression "at face value" then u does NOT depend on t and it equal zero.
Thanks again!

 

[Chestermiller]

Okay yeah I remember that from class. Thanks.

I'm still slightly confused though. How am I supposed to know if it's a material derivative of just a regular derivative? What exactly makes it a material derivative? If I take the expression "at face value" then u does NOT depend on t and it equal zero.
Thanks again!

It's all a matter of the terminology your textbook or professor uses. Some fluids textbooks use d/dt and others use D/Dt. Of course, if you are familiar with the NS equations, you know to look for that.

Chet