Shear Rate

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

:$\dot\gamma = \frac,$

where:

*$\dot\gamma$ is the shear rate, measured in reciprocal seconds;
* is the velocity of the moving plate, measured in meters per second;
* is the distance between the two parallel plates, measured in meters.

Or:

:$\dot\gamma_ = \frac + \frac.$

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

:$\dot\gamma = \frac,$

where:

*$\dot\gamma$ is the shear rate, measured in reciprocal seconds;
* is the linear fluid velocity;
* is the inside diameter of the pipe.

The linear fluid velocity is related to the volumetric flow rate by

:$v = \frac,$

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

:$A = \pi r^2,$

thus producing

:$v = \frac.$

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 :

:$\dot\gamma = \frac = \frac,$

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

:$\dot\gamma = \frac.$

For a Newtonian fluid wall, shear stress () can be related to shear rate by $\tau_w = \dot\gamma_x \mu$ where 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.

References

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Continuum mechanics
Temporal rates