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Stressing Over Stress Tensor Symmetry in Navier-Stokes
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[QUOTE="spin2he2, post: 6084806, member: 650016"] It looks like the method of control volumes can be used to generate additional physical equations from the Navier-Stokes equation (in conservation form). If you take the cross product of both sides with the radial vector and pull the cross product through the divergence operators, additional terms must be subtracted out as per the chain rule. Taking integrals of both sides over some control volume and using the divergence theorem yields that the time rate of change of the angular momentum within the control volume is equal to (negative of) the rate of transported loss of angular momentum through the surface plus the torque generated on the surface - which must from an equation that holds on its own by physical grounds (and is also equivalent to a continuity diffusion equation for angular momentum with a torque source) - PLUS the volume integral of the term shed off by the chain rule (now integration by parts) which must therefore be zero. This vanishing latter term is just the volume integral of the stress tensor contracted with the Levi-Civita symbol, and so the stress tensor is arguably symmetric. If one takes the dot product of both sides of the Navier-Stokes equations with the velocity vector and follows the analogous steps, one gets the enthalpy- continuity kinetic energy-diffusion equation with a pressure source plus a second equation for diffusion of internal kinetic energy with mechanical and dissipative sources. As far as deriving the value of the shear stress in the y-direction perpendicular to the shear flow above (which has no momentum in the y-direction) directly from molecular dynamic considerations (as opposed to deriving it as a consequence of the conservation of angular momentum), it is still not clear to me how to visualize the causal mechanism. If one puts a set of blocks in a row on (the x-axis of) a table and exerts a shear force on them in the x-direction, they might rotate except that a shear force would be generated in the y-direction on each of their front faces as they collide with the back face of the block in front of them... Is this the right way to visualize the molecular dynamics that cause the stress tensor to be symmetric? Regarding the fluid beginning in rigid body rotation with a radial temperature gradient, it looks like the viscous torque term from the gradient of the anti-symmetric/rotational viscosity times the vorticity (twice the initial uniform angular rotation rate) does lead to an increase in angular speed everywhere (except at the boundary) generating angular momentum "out of nowhere" during the first time step dt. Physically, it still seems to me that the (temperature enhanced) rate of inward diffusion of tangential velocity should induce angular acceleration of the inner bands during the first time step away from rigid body rotation - but it is not clear to me how to model this. [/QUOTE]
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