Shear rate

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 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

↑ "Brookfield Engineering - Glossary section on Viscosity Terms". Retrieved 2007-06-10.
↑ Ron Darby, Chemical engineering fluid mechanics, 2nd ed. CRC Press, 2001, p. 64