Shear rate

In physics, 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". 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 ($&tau;w$) 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.