How do we know that the stress tensor must be symmetric in the Navier-Stokes equation? Here are some papers that discuss this issue beyond the usual derivations:(adsbygoogle = window.adsbygoogle || []).push({});

Behavior of a Vorticity Influenced Asymmetric Stress Tensor In Fluid Flow http://www.dtic.mil/dtic/tr/fulltext/u2/a181244.pdf

Vorticity and Stress Tensor https://www.rgnpublications.com/journals/index.php/jims/article/view/1061

Diffusion Stratification in a Rotating Linearly Stratified Fluid https://link.springer.com/article/10.1007/BF02698703

First, consider a shear flow v = (Cy,0,0) where C is a constant with the dimensions of 1/time. Diffusion of momentum at the molecular dynamic level implies that there is a shear stress in the x-direction along planes that have their normals in the y-direction. If the stress-tensor is symmetric, then there must be exactly the same magnitude of shear-stress in the y-direction along planes that have their normals in the x-direction, even though there is no momentum in the y-direction to diffuse to account for such a stress in the y-direction. If such an additional stress exists, by what molecular dynamic process is it brought into existence and how do we model it mathematically?

Note that, at the level of the Navier-Stokes equations, the term that is used to symmetrize the stress tensor only makes a non-vanishing contribution to the equations of motion if the flow either has density gradients and/or viscosity gradients – e.g. the flow is (roughly) either compressible and/or has temperature gradients. Thus a simple flow that might illustrate the presence of a non-zero antisymmetric (aka rotational) viscosity might be: an incompressible viscous flow of uniform density in a cylinder of radius R with full slip boundary conditions with a large and constantly maintained temperature gradient in the radial direction that is initially rotating uniformly (i.e. same angular speed everywhere) as fast as possible without inducing turbulence.

Physically, one would expect the higher temperature of the outer bands (of constant radius) would initially cause a net diffusion of angular momentum radially inward, create a viscous torque density, and induce angular acceleration of the inner bands – even at time t = 0 without an initial rate of deformation of the volume elements. One would expect angular momentum to be conserved globally since there are no external torques acting on the system – not because there are not equal and opposite torques between interacting fluid elements within the system - (i.e. the stress tensor does not necessarily need to be symmetric for angular momentum to be conserved globally).

Mathematically, in cylindrical polar coordinates, the theta component of the Navier-Stokes equation for the covariant velocity produces a viscous torque density term that is in fact proportional to the gradient of the rotational viscosity times the vorticity. Thus, if the rotational viscosity does not vanish, the initial uniform rotation of the fluid would induce a viscous torque on the inner bands that would cause them to undergo angular acceleration – consistent with what one might expect physically.

I’m not sure how to think about the final steady state that this system evolves to other than to suspect that: it conserves angular momentum globally, it dissipates as much angular kinetic energy as possible under this constraint, and all accelerations and force densities vanish. If one plugs in an ansatz such that the final distribution of angular speed is proportional to the radius raised to some power, one can show that the angular speed vanishes everywhere except at the origin (where it is infinite), that angular momentum is still conserved globally, that all the angular kinetic energy dissipates into heat, and that all force densities vanish.

There is some talk in the literature of angular speeds possibly blowing up at the origin for some Navier-Stokes flows. Needless to say, taking the limit as r goes to zero doesn’t make much sense for a physical fluid as it would be less than the inter-particle spacing (which implies a rotational inertia of the fluid that does not vanish and an angular speed that is consequently not infinite). There is also some talk of Coriolis forces and increases of angular speeds being quite noticeable for rotating flows with density (and presumably temperature) gradients.

So how do we know with certainty that the rotational viscosity is identically zero? And if it is identically zero, what molecular dynamic mechanism accounts for the presence of orthogonal shear forces in a pure shear flow?

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# A Stressing Over Stress Tensor Symmetry in Navier-Stokes

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