In solid bodies, $\tau_{xy}=\tau_{yx}$ makes sense to me because the volume elements "hold together" and can not spin against each other and therefore the resulting torque from the shear stresses has to be zero.
However, in fluids I imagine that the volume elements are able to spin against each other, and I was really surprised when I learned in my fluid mechanics lecture that $\tau_{xy}=\tau_{yx}$ holds for fluids too.
I realize that in this case a resulting torque would lead to an acceleration of the rotational speed of the volume elements, but I can't see anything which prevents that.
I unsuccessfully googled around a lot about this issue. I also asked my professor and several assistants, but neither of them was able to provide a satisfying explanation. Therefore I'm guessing my question doesn't really make sense in this way and/or is based on a very basic lack of understanding.
Best Answer
This follows from rotational invariance. If the stress tensor is not symmetric then the angular momentum of the fluid is not conserved. More explicitly, momentum conservation is the equation $$ \frac{\partial}{\partial t}\pi_i + \nabla_j\tau_{ij} = 0 $$ where $\pi_i=\rho v_i$ is the momentum density. The density of angular momentum (about the origin) is $l_i=\epsilon_{ijk}x_j\pi_k$ and $l_i$ is conserved if $\epsilon_{ijk}\tau_{jk}=0$ (that is, if $\tau_{ij}$ is symmetric). We get $$ \frac{\partial}{\partial t}l_i + \nabla_j m_{ij} = 0 $$ where $m_{ij}=\epsilon_{ikl}x_k\tau_{lj}$ is the angular momentum flux.
Of course, the angular momentum of the fluid can change because of external torques, and the angular momentum of a fluid cell can change because of surface stresses. (That is, I can integrate the conservation law over a volume inside the fluid, and the angular momentum of the fluid volume changes because of surface torques. Of course, the total angular momentum of the fluid is conserved.)
All of this applies to any fluid that is described by a rotationally invariant Hamiltonian, that means any fluid made of atoms, electrons, quarks, gluons, etc.
An interesting question arises if rotational invariance is broken spontaneously, for example in the case of a liquid crystal. In that case, angular momentum is still conserved, and the stress tensor is symmetric, but the hydrodynamic description of the fluid (and the stress tensor itself) depend on an extra vector field $n_i$, which arises from the order parameter.