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

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

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