Shear rate

{{short description|Rate of change in the shear deformation of a material with respect to time}}

In physics, mechanics and other areas of science, 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

: $\dot\gamma = \frac{v}{h},$

where:

$\dot\gamma$ is the shear rate, measured in reciprocal seconds;
{{mvar|v}} is the velocity of the moving plate, measured in meters per second;
{{mvar|h}} is the distance between the two parallel plates, measured in meters.

Or:

: $\dot\gamma_{ij} = \frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}.$

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".{{Cite web

|title = Brookfield Engineering - Glossary section on Viscosity Terms

|accessdate = 2007-06-10

|url = http://www.brookfieldengineering.com/education/viscosity_glossary.asp

|url-status = dead

|archiveurl = https://web.archive.org/web/20070609171914/http://www.brookfieldengineering.com/education/viscosity_glossary.asp

|archivedate = 2007-06-09

}} 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

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

The shear rate at the inner wall of a Newtonian fluid flowing within a pipe{{cite book|first=Ron|last=Darby|title=Chemical Engineering Fluid Mechanics|edition=2nd|publisher=CRC Press|date=2001|url=https://books.google.com/books?id=hoXH5qWVnpMC&pg=PA64|page=64|isbn=9780824704445}} is

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

where:

$\dot\gamma$ is the shear rate, measured in reciprocal seconds;
{{mvar|v}} is the linear fluid velocity;
{{mvar|d}} is the inside diameter of the pipe.

The linear fluid velocity {{mvar|v}} is related to the volumetric flow rate {{mvar|Q}} by

: $v = \frac{Q}{A},$

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

: $A = \pi r^2,$

thus producing

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

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 {{math|1=d = 2r}}:

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

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

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

For a Newtonian fluid wall, shear stress ({{mvar|τ{{sub|w}}}}) can be related to shear rate by $\tau_w = \dot\gamma_x \mu$ where {{mvar|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.

Shear rate

Simple shear

References

See also