CL: Viscous Stresses

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.

DEFINITION:

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.

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

DEFINITION:

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.

DEFINITION:

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:

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

NOTE:

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.

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

OUTCOME:

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

NOTE:

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