Hello,

There are important situations where swirl of particle/parcel is encountered, such as in combustion. For these cases there is an important question: is stress tensor assumed to be symmetric?

We know that in NS equation only linear momentum is considered, and the general form of NS equation does not assume that stress tensor is symmetric. Physically, if the tensor is asymmetric then there is torque on the microscopic volume *m*, and within its streamline in general it is subjected to the influence of:

- gravity
- normal stress
- shear force ⇒ torque ⇒ angular acceleration
*mra*(*r*assumed to be mean radius of*m*)

I think this understand is basically correct regarding torque: visualize a crowed square in which thousands of people, shoulder-to-shoulder, are moving toward a same direction (for example pilgrims in Mecca), then each body is subjected to torque from the neighbor, and in general he will not always orient toward the same direction during his course along the streamline. The orientation change is therefore due to the angular *a* above, and his movement can always be decomposed into

- linear translation ⇐ normal stress
- rotation about his own axis ⇐ torque

therefore the assumption seems perfectly valid, especially when liquid swirl is encountered.

And although in general is *a* present, the accumulated *mr⋅v(rotational)* is small because temporarily (time) microscopically, and individual “rotation” not in-phase/aligned with its neighbors increases stress which soon acts to decelerate that.

However, if many places I read that the symmetry of stress is assumed, such as with Rutherford Aris’s *Vectors, Tensors, and the Basic Equations of Fluid Mechanics*, in section 6.41 and he also used a somewhat questionable term “polar fluid” to refer to fluid with torque/asymmetric stress.

I would like to know that in popular CFD packages *Solidworks Flow Simulation, Fluent, CFX or FloEFD*, is stress tensor assumed to be symmetric or not? I wonder if they do assume symmetric, then for viscous fluid symmetric shear stress implies a symmetric velocity field which is unrealistic for example for swirls, so the software will fail.

I just started learning this subject for a few days so the question might seem very basic to experienced users. Sincerely hope someone could resolve my puzzle!

Andy

I believe you're referring to the Reynolds stress tensor, which is symmetric (the diagonal elements are the normal stresses, the off-diagonal elements are the shear stresses; the shear stresses are symmetric so R_ij = R_ji)