# Shear rate

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Shear rate is the rate at which a progressive shearing deformation is applied to some 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

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

where:

• $\dot\gamma$ 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:

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

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]

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

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

where:

• $\dot\gamma$ 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

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

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

$A = \pi r^2,$

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 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 Q and inner pipe radius r:

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

For a Newtonian fluid wall, shear stress ($\tau_w$) can be related to shear rate by $\tau_w = \dot\gamma_x \mu$, where $\mu$ is the 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". Retrieved 2007-06-10.
2. ^ Ron Darby, Chemical engineering fluid mechanics, 2nd ed. CRC Press, 2001, p. 64