In physics, shear rate is the rate at which a progressive shear strain is applied to some material, causing shearing to the 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

${\displaystyle {\dot {\gamma ))={\frac {v}{h)),}$

where:

• ${\displaystyle {\dot {\gamma ))}$ 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:

${\displaystyle {\dot {\gamma ))_{ij}={\frac {\partial v_{i)){\partial x_{j))}+{\frac {\partial v_{j)){\partial x_{i))}.}$

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

${\displaystyle {\dot {\gamma ))={\sqrt {2\varepsilon :\varepsilon ))}$.

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

${\displaystyle {\dot {\gamma ))={\frac {8v}{d)),}$

where:

• ${\displaystyle {\dot {\gamma ))}$ 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

${\displaystyle v={\frac {Q}{A)),}$

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

${\displaystyle A=\pi r^{2},}$

thus producing

${\displaystyle v={\frac {Q}{\pi r^{2))}.}$

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 d = 2r:

${\displaystyle {\dot {\gamma ))={\frac {8v}{d))={\frac {8\left({\frac {Q}{\pi r^{2))}\right)}{2r)),}$

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

${\displaystyle {\dot {\gamma ))={\frac {4Q}{\pi r^{3))}.}$

For a Newtonian fluid wall, shear stress (τw) can be related to shear rate by ${\displaystyle \tau _{w}={\dot {\gamma ))_{x}\mu }$ 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". Archived from the original on 2007-06-09. Retrieved 2007-06-10.
2. ^ Darby, Ron (2001). Chemical Engineering Fluid Mechanics (2nd ed.). CRC Press. p. 64. ISBN 9780824704445.