In Aris' book, non-polar fluids have a symmetric stress tensor ( named T ), and are NOT subject to body torque forces, though they may still have so-called "external" angular momentum ( and "rigid" body rotation ), in the form of a moment of linear momentum, as shown in equation (5.11.4), p.100. Non-polar fluids may or may not be irrotational ( the two terms are NOT synonymous ): meaning their vorticity may or may not be zero. Note: In the derivation of the rate of change of kinetic energy for non-polar fluids in section 6.14: "Dissipation of Energy by Viscous Forces", p.117, the reason the velocity gradient tensor is replaced at one point with the deformation tensor, is because the double dot product ( aka. dyadic product ) of a symmetric tensor ( here: T ) and an anti-symmetric one ( here: Omega, the anti-symmetric part of the velocity gradient tensor ) is always zero ( which does not necessarily mean that Omega was null, which would otherwise be interpreted as a condition of irrotationality, since vorticity vector w = curl( velocity v ) = -2*vec( Omega ) = 2 * angular velocity ). In the book, non-polar fluids refer to Stokesian as well as Newtonian fluids. Note: Newtonian fluids are basically a type of Stokesian fluids in which the eigenvalues of the viscosity tensor are related to those of the deformation tensor via a linear equation rather than via a quadratic, as evidenced p.110.
Polar fluids, on the other hand, have a non-symmetric stress tensor, and ARE subject to body torque, as stated p. 102. They may or may not have rotationality ( vorticity ), depending on the case. Polar fluids are only briefly mentioned in the book. He mentions two kinds in passing: "poly-atomic fluids and certain non-Newtonian fluids", p.102. Note: non-polar fluids can be thought of as a special case of polar fluids, in which the vector of the stress tensor ( Tx = 2 * vec(T) ) is null ( which is another way of saying that the stress tensor of non-polar fluids is symmetric, since its anti-symmetric part would be null ). The "general" equation for the conservation ( aka. "balance" ) of TOTAL ( internal + external ) angular momentum for polar fluids is given in (5.13.10), p.104. Just set Tx = 0 in equation (5.13.10), and you'll get back equation (5.11.4), p.100, which is the version of conservation of -- external only -- angular momentum for non-polar fluids. Aris thus explicitly states, p.123: "For certain class of fluids however ( here called polar fluids ), the stress tensor is NOT symmetric and there may be an internal [ aka. "intrinsic" ] angular momentum [ caused by body torque + stress couple ], as well as an external moment of [ linear ] momentum [ aka. external angular momentum, caused by body force + normal stress ]." Together, these form the "total angular momentum", whose rate of change is given in "general" equation (5.13.6), p.103.