# Shear rate

**Shear rate** is the rate at which a progressive shearing deformation is applied to some material.

## Simple shear

The shear rate for a fluid flowing between two parallel plates, one moving at a constant speed and the other one stationary (Couette flow), is defined by

where:

*is the shear rate, measured in reciprocal seconds;**v*is the velocity of the moving plate, measured in meters per second;*h*is the distance between the two parallel plates, measured in meters.

Or:

For the simple shear case, it is just a gradient of velocity in a flowing material. The SI unit of measurement for shear rate is s^{−1}, expressed as "reciprocal seconds" or "inverse seconds".[1]

The shear rate at the inner wall of a Newtonian fluid flowing within a pipe[2] is

where:

*is the shear rate, measured in reciprocal seconds;**v*is the linear fluid velocity;*d*is the inside diameter of the pipe.

The linear fluid velocity *v* is related to the volumetric flow rate *Q* by

where *A* is the cross-sectional area of the pipe, which for an inside pipe radius of *r* is given by

thus producing

Substituting the above into the earlier equation for the shear rate of a Newtonian fluid flowing within a pipe, and noting (in the denominator) that *d* = 2*r*:

which simplifies to the following equivalent form for wall shear rate in terms of volumetric flow rate *Q* and inner pipe radius *r*:

For a Newtonian fluid wall, shear stress (τ_{w}) can be related to shear rate by τ_{w} = * _{x}μ*, where

*μ*is the dynamic viscosity of the fluid. For non-Newtonian fluids, there are different constitutive laws depending on the fluid, which relates the stress tensor to the shear rate tensor.

## References

- "Brookfield Engineering - Glossary section on Viscosity Terms". Archived from the original on 2007-06-09. Retrieved 2007-06-10.
- Darby, Ron (2001).
*Chemical Engineering Fluid Mechanics*(2nd ed.). CRC Press. p. 64.