Outer product of flow velocities in Navier-Stokes equation

[snoopies622]

Reading the Wikipedia entry about the Navier–Stokes equation, and I don't understand this second term, the one with the outer product of the flow velocities. I mean, I understand the literal mathematical meaning, but I don't have an intuitive idea of what it physically represents. When I make up velocity fields and compute its value, I get something similar to the flow velocity but not exactly, and something related to divergence, but not exactly that either.

 

[Chestermiller]

Reading the Wikipedia entry about the Navier–Stokes equation, and I don't understand this second term, the one with the outer product of the flow velocities. I mean, I understand the literal mathematical meaning, but I don't have an intuitive idea of what it physically represents. When I make up velocity fields and compute its value, I get something similar to the flow velocity but not exactly, and something related to divergence, but not exactly that either.

The usual interpretation of this term is the rate of flow of momentum into and out of a differential control volume.

 

[snoopies622]

Thanks Chestermiller, will give this some more thought and get back to you here.

 

[Chestermiller]

Thanks Chestermiller, will give this some more thought and get back to you here.

I really like your avatar. Is that your pet?

 

[snoopies622]

Nah, just a cute doggie pick I found on line years ago. In other places I use a picture of a mug of hot cocoa.

So about the momentum flow . . Would you say that this second term is basically a kind of vector version of the divergence? Never thought about it before, but I suppose one could find not only how fast fluid is rushing away from a certain location but its overall net direction as well.

 

[Chestermiller]

Nah, just a cute doggie pick I found on line years ago. In other places I use a picture of a mug of hot cocoa.

So about the momentum flow . . Would you say that this second term is basically a kind of vector version of the divergence?

It's basically the divergence of the "momentum flux tensor." Physically, it represents the difference between the rate of momentum flowing out of the control volume and the rate of momentum flowing into the control volume. If we move it to the other side of the equation it will have a minus sign and, including the minus sign, it would represent the net rate of momentum flowing into the control volume. The term on the left side represents physically the rate of increase of momentum within the control volume. So the rate of increase of momentum within the control volume is equal to the net rate of momentum flowing into the control volume plus the various forces on the right hand side. The overall equation is basically a differential momentum balance (i.e., force balance/Newton's 2nd law) on the flow.

 

[snoopies622]

Thank you very much!