Shear rate

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

${\displaystyle {\dot {\gamma }}={\frac {v}{h}},}$

where:

• ${\displaystyle {\dot {\gamma }}}$ 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:

${\displaystyle {\dot {\gamma }}_{ij}={\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}.}$

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

${\displaystyle {\dot {\gamma }}={\sqrt {2\varepsilon$ :\varepsilon }}} .

The shear rate at the inner wall of a Newtonian fluid flowing within a pipe[2] is

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

where:

• ${\displaystyle {\dot {\gamma }}}$ 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

${\displaystyle v={\frac {Q}{A}},}$

where A is the cross-sectional area of the pipe, which for an inside pipe radius of r is given by

${\displaystyle A=\pi r^{2},}$

thus producing

${\displaystyle v={\frac {Q}{\pi r^{2}}}.}$

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:

${\displaystyle {\dot {\gamma }}={\frac {8v}{d}}={\frac {8\left({\frac {Q}{\pi r^{2}}}\right)}{2r}},}$

which simplifies to the following equivalent form for wall shear rate in terms of volumetric flow rate Q and inner pipe radius r:

${\displaystyle {\dot {\gamma }}={\frac {4Q}{\pi r^{3}}}.}$

For a Newtonian fluid wall, shear stress (τ_w) can be related to shear rate by ${\displaystyle \tau _{w}={\dot {\gamma }}_{x}\mu }$ 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.