Why is viscous stress a tensorial quantity? In an incompressible fluid, the viscous stress (in Cartesian) is defined by
\begin{align}
\tau_{ij} = \eta(\partial_i v_j + \partial_j v_i)
\end{align}
for dynamic viscosity $\eta$ and velocity field $v_i$. If $\tau_{ij}$ was a tensor (strictly, a tensor field), it must obey the tensor transformation law
\begin{align}
\tau_{ij}(x) \stackrel{?}{=} \frac{\partial \bar{x}^a}{\partial x^i}\frac{\partial \bar{x}^b}{\partial x^j} \bar{\tau}_{ab}(\bar{x})
\end{align}
However, the viscous stress obeys the transformation
\begin{align}
\tau_{ij} (x) = \eta\left(\frac{\partial \bar{x}^a}{\partial x^i}\frac{\partial\bar{x}^b}{\partial x^j} \frac{\partial \bar{v}_a(\bar{x})}{\partial \bar{x}^b} + \frac{\partial^2 \bar{x}^b}{\partial x^i \partial x^j} \bar{v}_b \right) + (i \leftrightarrow j)
\end{align}
The second term spoils the transformation law. This is as opposed to the EM field tensor $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\nu$, where the antisymmetry cancels out the second term and it does obey the tensor transformation law.
While $\tau_{ij}$ behaves tensorially under global transformations (e.g. global rotations and translations), a transformation to polar coordinates, for example, would spoil the tensor transformation law. Why then do people refer to $\tau_{ij}$ as a tensor (tensor field)?
 A: In general, you have to specify under what kind of transformations a given tensor transforms under. Normally, in physics, people aren't explicit about this when it should be clear from context, but that can sometimes lead to confusion.
In this case, saying $\tau_{ij}$ is a tensor means that it is a tensor under global rotations (the way you've defined it with partial derivatives). It is only a tensor in Cartesian coordinates.
You can easily promote it to be a tensor under general coordinate transformations, by replacing the partial derivatives with covariant derivatives.
The fact that whether or not an object is a tensor depends on the notation we use to express it and the specific kind of transformations being considered might bother you, but it actually is something you may have come across before if you have studied special relativity from a geometric, 4-dimensional point of view. The electric and magnetic fields $\vec{E}$ and $\vec{B}$ are vectors under three dimensional rotations, but do not transform as vectors under Lorentz transformations. However, we can combine them into a covariant electromagnetic field-strength tensor $F_{\mu\nu}$, which does transform as a tensor under Lorentz transformations.
