Newton's Law of Viscosity and Newtonian Fluid
Definitions
$$ \mathbf{u} = \mathbf{u} \left( t ; \mathbf{x} \right) = \left( u_{1} \left( t ; \mathbf{x} \right) , u_{2} \left( t ; \mathbf{x} \right) , u_{3} \left( t ; \mathbf{x} \right) \right) $$ In particular, in three-dimensional space, let the velocity field at the point of observation $t$ and spatial coordinates $\mathbf{x} = \left( x_{1} , x_{2} , x_{3} \right)$, i.e. the velocity field, be represented by the velocity vector as above.
Newton’s law of viscosity
In fluid mechanics, the stress applied to an incompressible, isotropic fluid is proportional to the symmetrized gradient of the velocity; this relation is called Newton’s law of viscosity. In formula form, for the stress tensor $\tau \in \mathbb{R}^{3 \times 3}$ and the Jacobian $\nabla \mathbf{u}$ of the velocity field $\mathbf{u}$, it is expressed as follows. $$ \tau = \mu \left( \nabla \mathbf{u} + \left( \nabla \mathbf{u} \right)^{T} \right) $$ Written component-wise, it is as follows. $$ \left( \tau \right)_{ij} = \mu \left( {\frac{ \partial u_{i} }{ \partial x_{j} }} + {\frac{ \partial u_{j} }{ \partial x_{i} }} \right) $$ Here, $\mu$ is called the coefficient of dynamic viscosity. $\mu$ can also be expressed in terms of the density $\rho$ as follows, in which case $\nu$ is called the kinematic viscosity. $$ \mu = \rho \nu $$
Newtonian fluid
A fluid that obeys Newton’s law of viscosity is called a Newtonian fluid, and a fluid that does not obey Newton’s law of viscosity is called a non-Newtonian fluid.
Explanation
Often Newton’s law of viscosity is summarized starting from one-dimensional flow together with an ordinary differential equation like $\tau = \mu du /dy$ as “a linear relation between shear stress and velocity gradient”; in my personal view, that explanation is considerably more difficult and extending it to higher dimensions is awkward. Frequently the notation changes — in 1D it is $u$, in 2D it is $u, v$, and in 3D it is $\mathbf{u}$ — which often becomes an unnecessary and confusing digression. For that reason, I think it is better to start from a three-dimensional vector function from the outset.
An intuitive interpretation of Newton’s law of viscosity is that the fluid is a “reasonable” fluid. By reasonable, we mean it produces a proportionate amount of resistance in response to the forces applied to it.
Non-Newtonian fluids
However, for fluids that do not follow Newton’s law of viscosity, they can respond counterintuitively; for example, they may behave like a liquid under weak forces but suddenly behave like a solid under strong forces.