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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

γ ˙ = v h , {\displaystyle {\dot {\gamma }}={\frac {v}{h}},}

where:

• γ ˙ {\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:

γ ˙ i j = ∂ v i ∂ x j + ∂ v j ∂ x i . {\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

γ ˙ = 2 ε : ε {\displaystyle {\dot {\gamma }}={\sqrt {2\varepsilon :\varepsilon }}} .

The shear rate at the inner wall of a Newtonian fluid flowing within a pipe² is

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

where:

• γ ˙ {\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

v = Q A , {\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

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

thus producing

v = Q π r 2 . {\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:

γ ˙ = 8 v d = 8 ( Q π r 2 ) 2 r , {\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:

γ ˙ = 4 Q π r 3 . {\displaystyle {\dot {\gamma }}={\frac {4Q}{\pi r^{3}}}.}

For a Newtonian fluid wall, shear stress (τw) can be related to shear rate by τ w = γ ˙ x μ {\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.

See also

• Shear strain
• Strain rate
• Non-Newtonian fluid

References

1. "Brookfield Engineering - Glossary section on Viscosity Terms". Archived from the original on 2007-06-09. Retrieved 2007-06-10. https://web.archive.org/web/20070609171914/http://www.brookfieldengineering.com/education/viscosity_glossary.asp ↩

2. Darby, Ron (2001). Chemical Engineering Fluid Mechanics (2nd ed.). CRC Press. p. 64. ISBN 9780824704445. 9780824704445 ↩