Shear rate

Shear rate is the rate at which a shear is applied.

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

• $\dot\gamma$ = The shear rate, measured in reciprocal seconds
• $v \$ = The velocity of the moving plate, measured in meters per second
• $h \$ = 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 sec^-1, expressed as "reciprocal seconds" or "inverse seconds."

The shear rate at the inner wall of a Newtonian fluid flowing within a pipe is:

$\dot \gamma = \frac {8v} {d}$

where:

• $\dot\gamma$ = The shear rate, measured in reciprocal seconds.
• $v \$ = The linear fluid velocity.
• $d \$ = 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 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.