How can I break down the Navier Stokes equation? (momentum equation)

[Carbon273]

Homework Statement

So I am trying to break down the complexity of my homework assignment where I need to perform the operation: (velocity vector)*(momentum equation) as a step to define the transport of kinetic energy equation. For the sake of academic integrity I wish to fully understand the concept by breaking it down. So my first question is, how do I gain understanding of the momentum equation in this context? How can I understand to unpack this equation. I have a feeling this is the navier stokes equation in a very condensed manner. If it is, how do I break it down, especially with the tensor embedded in there.

Relevant Equations

Equation shown below:

 

[BvU]

of my homework assignment

Please provide a complete problem statement, explain what the variables used represent, etc.

Do you have access to a copy of Bird, Stewart, Lightfoot ? It spells out the eqn in your picture (over several pages !)

 

[jack476]

Sorry if this comes too close to the line for self-promotion, but a few months ago I wrote a Medium article about the derivation of the Navier-Stokes equation that mostly followed the approach in Grainger's textbook:



(Yes, I know that it says "part 1" in the subtitle, implying the existence of a "part 2". I'll get to it eventually.)

 

[hunt_mat]

What do you mean by "break down" exactly?

 

[Chestermiller]

The momentum equation (aka, the equation of motion) is a differential version of Newton's 2nd law of motion applied to a fluid. Is that what you were asking? You are supposed to take the dot product of the velocity vector with the momentum equation. Are you asking how one mathematically takes the dot product of the velocity vector with the gradient of the stress tensor?