# 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 = 2r:

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 (τw) can be related to shear rate by τw = xμ, 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

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