In materials science and engineering, there is often interest in understanding the forces or stresses involved in the deformation of a material. For instance, if the material were a simple spring, the answer would be given by Hookes law, which says that the force experienced by a spring is proportional to the distance displaced from equilibrium. Stresses which can be attributed to the deformation of a material from some rest state are called elastic stresses. In other materials, stresses are present which can be attributed to the deformation rate over time. These are called viscous stresses. For instance, in a fluid such as water the stresses which arise from shearing the fluid do not depend on the distance the fluid has been sheared; rather, they depend on how quickly' the shearing occurs.
Viscosity is the material property which relates the viscous stresses in a material to the rate of change of a deformation (the strain rate). Although it applies to general flows, it is easy to visualize and define in a simple shearing flow, such as a planar Couette flow.
In the Couette flow, a fluid is trapped between two infinitely large plates, one fixed and one in parallel motion at constant speed  (see illustration to the right). If the speed of the top plate is low enough (to avoid turbulence), then in steady state the fluid particles move parallel to it, and their speed varies from  at the bottom to  at the top. Each layer of fluid moves faster than the one just below it, and friction between them gives rise to a force resisting their relative motion. In particular, the fluid applies on the top plate a force in the direction opposite to its motion, and an equal but opposite force on the bottom plate. An external force is therefore required in order to keep the top plate moving at constant speed.
In many fluids, the flow velocity is observed to vary linearly from zero at the bottom to  at the top. Moreover, the magnitude of the force, , acting on the top plate is found to be proportional to the speed  and the area  of each plate, and inversely proportional to their separation : 
The proportionality factor is the dynamic viscosity of the fluid, often simply referred to as the viscosity. It is denoted by the Greek letter mu (). The dynamic viscosity has the dimensions , therefore resulting in the SI units and the derived units:
: <math>  
  [\mu]
= \frac {\rm kg} {\rm m{\cdot}s} 
= \frac {\rm N} {\rm m^2}{\cdot}{\rm s}
= {\rm Pa{\cdot}s}
 energy per unit volume multiplied by time.
The aforementioned ratio  is called the rate of shear deformation or shear velocity, and is the derivative of the fluid speed in the direction parallel to the normal vector of the plates (see illustrations to the right). If the velocity does not vary linearly with , then the appropriate generalization is:
where , and  is the local shear velocity. This expression is referred to as Newtons law of viscosity. In shearing flows with planar symmetry, it is what defines' . It is a special case of the general definition of viscosity (see below), which can be expressed in coordinate-free form.
Use of the Greek letter mu () for the dynamic viscosity (sometimes also called the absolute viscosity) is common among mechanical and chemical engineers, as well as mathematicians and physicists. However, the Greek letter eta () is also used by chemists, physicists, and the IUPAC. The viscosity  is sometimes also called the shear viscosity. However, at least one author discourages the use of this terminology, noting that  can appear in non-shearing flows in addition to shearing flows.