Shear rate: Definition from Answers.com

Shear rate is the rate at which a shear is applied.

Simple Shear

Shear rate for a fluid flowing between two fixed parallel plates is defined using the following equation:

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

Where:

$\dot\gamma$ = The shear rate, measured in reciprocal seconds
$v \$ = The velocity, measured in meters per second
$h \$ = The distance between the two parallel faces that are experiencing the shear, 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 sec^-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:

$\dot\gamma$ = The shear rate, measured in reciprocal seconds.
$v \$ = The linear fluid velocity.
$d \$ = 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 ( $τ w$ ) can be related to shear rate by $\tau_{w} = {\dot \gamma_{x} \mu}$ , where $μ$ 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

^ "Brookfield Engineering - Glossary section on Viscosity Terms". http://www.brookfieldengineering.com/education/viscosity_glossary.asp. Retrieved on 2007-06-10.
^ Ron Darby, Chemical engineering fluid mechanics, 2nd ed. CRC Press, 2001, p. 64

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