Shear rate

Short description: Rate of change in the shear deformation of a material with respect to time

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". 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 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 (τ_w) 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.