Recall that viscous stresses (sometimes called "frictional
stresses") arise from adjacent fluid elements *moving relative
to one another*. As the pieces of fluid "slide" past each other
they exert a friction-like force from layer to layer.

The viscous stresses therefore are the forces per unit are acting on fluid elements, due to friction.

It should not be surprising, therefore, that the
*velocities* of these fluid elements must actually be
different from each other (or else they would not be moving
relative to each other). The simplest case of this are shear
stresses arising from fluid deformation.

The **rate of strain (or shear rate)** is the rate at which a
fluid element is deformed. this can be shown to be related to the
slope (gradient) of the velocity.

$\displaystyle{\dot \gamma = \frac{d\gamma}{dt} = \frac{dv_x}{dy}}$

We can actually define a **fluid** as a material whose
*rate* of strain (or shear rate) is proportional to the
imposed shear stress, whereas a solid is a material whose
*total* strain is proportional to the shear stress.

A special case of a fluid (but the only thype of fluid that we will spend any real time studying) is one whose relationship between shear rate and stress is linear.

A **Newtonian fluid** is one whose rate of strain (or shear
rate) is *linearly* proportional to the imposed shear
stress, according to the following expression:

$\displaystyle{\tau = \mu \dot \gamma = \mu \frac{dv_x}{dy}}$

This expression is often called **Newton's law of
viscosity** and the constant of proportionality is called the
*viscosity*.

As we will see in future fluid studies, the ratio of the viscosity to the fluid density often appears in equation and is thus given a name: the kinematic viscosity.

$\displaystyle{\nu = \frac{\mu}{\rho}}$

Calculate viscous shear stresses (momentum fluxes!) directly from velocity profiles

We do this in the **exact** same fashion as getting a heat
flow/flux from a temperature profile!