Shear rate

 ( Continuum Mechanics ) 

In physics, mechanics and other areas of science, 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.

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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

$\backslash dot\backslash gamma\; =\; \backslash frac\{v\}\{h\},$

where:

• $\backslash dot\backslash gamma$ 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_{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

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

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

is

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

where:

• $\backslash dot\backslash gamma$ 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

$v\; =\; \backslash frac\{Q\}\{A\},$

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

$A\; =\; \backslash pi\; r^2,$

thus producing

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

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 :

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

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

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

For a Newtonian fluid wall, shear stress () can be related to shear rate by $\backslash tau\_w\; =\; \backslash dot\backslash gamma\_x\; \backslash mu$ 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.

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See also

• Shear strain
• Strain rate
• Non-Newtonian fluid

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