Shear rate explained

In physics, shear rate is the rate at which a progressive shear strain is applied to some material, causing shearing to the material. Shear rate is a measure of how the velocity changes with distance.

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

• \gamma = (v)/(h) ,

where:

• \gamma is the shear rate, measured in reciprocal seconds;

• is the velocity of the moving plate, measured in meters per second;
• is the distance between the two parallel plates, measured in meters.

Or:

• \gamma _{ij} = (\partial v _{i})/(\partial x _{j}) + (\partial v _{j})/(\partial x _{i}) .

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] However, when modelling fluids in 3D, it is common to consider a scalar value for the shear rate by calculating the second invariant of the strain-rate tensor

• \gamma = \sqrt{ 2 \varepsilon:\varepsilon} .

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

• \gamma = (8 v)/(d) ,

where:

• \gamma is the shear rate, measured in reciprocal seconds;

• is the linear fluid velocity;
• is the inside diameter of the pipe.

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

v = (Q)/(A) ,

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

A = \pi r ^{2,}

thus producing

v = (Q)/(\pi r ^{2}) .

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 :

• \gamma = (8 v)/(d) = (8 \left( (Q)/(\right) \pi r ^{2}))/(2 r) ,

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

• \gamma = (4 Q)/(\pi r ^{3}) .

For a Newtonian fluid wall, shear stress can be related to shear rate by

\tau _{w} = • \gamma _{x} \mu where 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.

See also

• Shear strain
• Strain rate
• Non-Newtonian fluid

Notes and References

1. Web site: Brookfield Engineering - Glossary section on Viscosity Terms . 2007-06-10 . dead . https://web.archive.org/web/20070609171914/http://www.brookfieldengineering.com/education/viscosity_glossary.asp . 2007-06-09.
2. Book: Darby, Ron. Chemical Engineering Fluid Mechanics. 2nd. CRC Press. 2001. 64. 9780824704445.