Rate of change in the shear deformation of a material with respect to time

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".^{[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.