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;
$v$ 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] 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\varepsilon :\varepsilon }}

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 $d = 2 r$ :

{\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 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". Archived from the original on 2007-06-09. Retrieved 2007-06-10.
^ Darby, Ron (2001). Chemical Engineering Fluid Mechanics (2nd ed.). CRC Press. p. 64. ISBN 9780824704445.

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