Shear rate

In physics, mechanics and other areas of science, shear rate is the temporal rate at which a progressive shear strain is applied to some material, causing shearing to the material. Shear rate has quantity dimension of velocity per distance, which simplifies to reciprocal time.

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 (τ_w) 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.

See also

• Shear strain
• Strain rate
• Non-Newtonian fluid