Shear rate
# Shear rate

In physics, 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". 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 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 $(τ_w)$ 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.

