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
- [math]\displaystyle{ \dot\gamma = \frac{v}{h}, }[/math]
where:
- [math]\displaystyle{ \dot\gamma }[/math] 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:
- [math]\displaystyle{ \dot\gamma_{ij} = \frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}. }[/math]
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
- [math]\displaystyle{ \dot{\gamma}=\sqrt{2 \varepsilon:\varepsilon} }[/math].
The shear rate at the inner wall of a Newtonian fluid flowing within a pipe[2] is
- [math]\displaystyle{ \dot\gamma = \frac{8v}{d}, }[/math]
where:
- [math]\displaystyle{ \dot\gamma }[/math] 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
- [math]\displaystyle{ v = \frac{Q}{A}, }[/math]
where A is the cross-sectional area of the pipe, which for an inside pipe radius of r is given by
- [math]\displaystyle{ A = \pi r^2, }[/math]
thus producing
- [math]\displaystyle{ v = \frac{Q}{\pi r^2}. }[/math]
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:
- [math]\displaystyle{ \dot\gamma = \frac{8v}{d} = \frac{8\left(\frac{Q}{\pi r^2}\right)}{2r}, }[/math]
which simplifies to the following equivalent form for wall shear rate in terms of volumetric flow rate Q and inner pipe radius r:
- [math]\displaystyle{ \dot\gamma = \frac{4Q}{\pi r^3}. }[/math]
For a Newtonian fluid wall, shear stress (τw) can be related to shear rate by [math]\displaystyle{ \tau_w = \dot\gamma_x \mu }[/math] 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.
References
- ↑ "Brookfield Engineering - Glossary section on Viscosity Terms". Archived from the original on 2007-06-09. https://web.archive.org/web/20070609171914/http://www.brookfieldengineering.com/education/viscosity_glossary.asp. Retrieved 2007-06-10.
- ↑ Darby, Ron (2001). Chemical Engineering Fluid Mechanics (2nd ed.). CRC Press. p. 64. ISBN 9780824704445. https://books.google.com/books?id=hoXH5qWVnpMC&pg=PA64.
![]() | Original source: https://en.wikipedia.org/wiki/Shear rate.
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