For most general cases, stress tensor need not be symmetric in fluid mechanics. Unfortunately, in many books the development follows the treatment used in solid mechanics for systems which do not have distributed couple. In fluid mechanics, such situation will be reached, if the vorticity time rate is zero. This can be rigorously shown. It is good to remember that symmetry of stress is related to rotation rate of an element, and convection is related to translational motion.

Symmetry is not telling anything about zero or non-zero components of the stress tensor. A matrix with all zero's is symmetric as well.

Regards,

Dr. Th. Frank.

First, the symmetry is a property of the stress tensor T that is hystorically a consequence of the angular momentum, then the Newtonian model prescribes a linear relation between the stress tensor and the velocity gradient.

Note that the antisymmetric part of the velocity gradient is linked to a rigid rotation of the volume of fluid while the symmetric part describes the dilatation/contrction of the volume (the trace) and the deformation of the angles (the zero-trace part)

For most general cases, stress tensor need not be symmetric in fluid mechanics. Unfortunately, in many books the development follows the treatment used in solid mechanics for systems which do not have distributed couple. In fluid mechanics, such situation will be reached, if the vorticity time rate is zero. This can be rigorously shown. It is good to remember that symmetry of stress is related to rotation rate of an element, and convection is related to translational motion.

Tapan K. Sengupta addressed an issue that is, indeed, someway a hystorical lack also in old textbooks and exists yet at present. We see the equation for the angular moment is not considered and the stress tensor is assumed symmetric.

If I remember well, some hints about that are in the Landau's textbook.

What I understand, for Newtonian fluids angular momentum conservation directly leads to the symmetry of the stress tensor. One may consider some non Newtonian flows (for example Micro polar fluids) where this is not the case. Also, I suspect, under extreme confinement (water in a carbon Nanotube) one can observe breakage of symmetry.

Anirudh Rana : It is not correct to presume symmetry because it is ensured by conservation of angular momentum. Work out the exercises given in books where symmetry is proven. It is premised on the basis of the control volume not experiencing any angular acceleration, i.e. time rate of vorticity is zero. That is not generally true for any fluid motion, irrespective of whether it is Newtonian/ Nen-Newtonian or micro/nano-fluids.

Dear @Tapan K. Sengupta:

I agree what you say: want I meant was that in many Newtonian fluids (gases, for examples), symmetry is a consequence of conservation of the angular momentum. Some non- Newtonian fluid (for example micro-polar fluid) does not have symmetry in the stress tensor. Even, the Newtonian fluids (for example Water) may even show non symmetric stress when exposed to ultra confined geometries, they even show Weissenberg like effects, atypical for the non- Newtonian fluids. Is it not correct?

Anirudh: You misunderstand the fact that in fluid mechanics we do not satisfy angular momentum conservation - period! That is a shortcoming not very easy to overcome. Taking stress as symmetric is not the same as applying conservation principle for angular momentum. I have written about it in one of my book, without providing any solution. It is a difficult problem to tackle for fluid flow. In solid mechanics it is much easier as the inertia properties are time invariant.

Santosh Kumar Singh. Your answer appears logical to me but is there literature or data to back up your answer?

I agree with answers. Developments of this point are given in Rutherford Aris, Vectors, Tensors, and the Basic Equations of Fluid Mechanics., Dover Publications, 1962 (ISBN 0-486-66110-5)