Shear rate
Shear rate is the rate at which a progressive shearing deformation is applied to some material.
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
where:
Or:
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".
The shear rate at the inner wall of a Newtonian fluid flowing within a pipe is
where:
The linear fluid velocity v is related to the volumetric flow rate Q by
where A is the cross-sectional area of the pipe, which for an inside pipe radius of r is given by
thus producing
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:
which simplifies to the following equivalent form for wall shear rate in terms of volumetric flow rate Q and inner pipe radius r:
For a Newtonian fluid wall, shear stress (τw) can be related to shear rate by τw = xμ, 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.
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