Understanding Momentum Transfer as a Tensor in Newton's Law of Viscosity

[asdf1]

What does it mean by momentum transfer is not a vector (3 components) but rather a tensor (9 components)?

 

[siddharth]

What does it mean by momentum transfer is not a vector (3 components) but rather a tensor (9 components)?

"momentum transfer"? Can you elaborate?

 

[FredGarvin]

Newton says that viscosity is due to molecular diffusion between layers in the fluid. A molecule leaves one layer and transfers it's momentum to the adjoining layer. That transfer creates an acceleration and that acceleration creates shear forces which is related to the viscosity.

Stresses in the most basic forms, i.e. no simplifying assumptions, are tensors. There are nine components 3 normal stresses and 6 shear stresses (note that for equillibrium, the 6 shear stresses are 2 groups of 3 that are equal, \tau_{xy} = \tau_{yx}).

You might like to read up on tensors by taking a look at this thread that Astronuc created in the tutorial section:
https://www.physicsforums.com/showthread.php?t=101414

 

[berkeman]

Thanks for the link to the tutorial, Fred. Cool stuff.

Tensor analysis is the type of subject that can make even the best of students shudder. My own post-graduate instructor in the subject took away much of the fear by speaking of an implicit rhythm in the peculiar notation traditionally used, and helped us to see how this rhythm plays its way throughout the various formalisms. Prior to taking that class, I had spent many years “playing” on my own with tensors. I found the going to be tremendously difficult but was able, over time, to back out some physical and geometrical considerations that helped to make the subject a little more transparent. Today, it is sometimes hard not to think in terms of tensors and their associated concepts. This article, prompted and greatly enhanced by Marlos Jacob, whom I’ve met only by e-mail, is an attempt to record those early notions concerning tensors. It is intended to serve as a bridge from the point where most undergraduate students “leave off” in their studies of mathematics to the place where most texts on tensor analysis begin. A basic knowledge of vectors, matrices, and physics is assumed. A semi-intuitive approach to those notions underlying tensor analysis is given via scalars, vectors, dyads, triads, and higher vector products. The reader must be prepared to do some mathematics and to think. For those students who wish to go beyond this humble start, I can only recommend my professor’s wisdom: find the rhythm in the mathematics and you will fare pretty well.

 

[asdf1]

thank you very much!