Shear rate

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;
• Template:Mvar is the velocity of the moving plate, measured in meters per second;
• Template:Mvar 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". 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

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

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

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

where:

• $\dot{\gamma }$ is the shear rate, measured in reciprocal seconds;
• Template:Mvar is the linear fluid velocity;
• Template:Mvar is the inside diameter of the pipe.

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

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

where Template:Mvar is the cross-sectional area of the pipe, which for an inside pipe radius of Template:Mvar 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 Template:Math:

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

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

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

For a Newtonian fluid wall, shear stress (Template:Mvar) can be related to shear rate by ${\tau }_w={\dot{\gamma }}_x\mu$ where Template:Mvar 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