Shear rate

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{v}{h},

where:

• \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:

: \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".{{Cite web |url-status = dead

:\dot{\gamma}=\sqrt{2 \varepsilon:\varepsilon}.

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

:\dot\gamma = \frac{8v}{d},

where:

• \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 = \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 = \pi r^2,

thus producing

: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 :

:\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:

:\dot\gamma = \frac{4Q}{\pi r^3}.

For a Newtonian fluid wall, shear stress (τ) can be related to shear rate by \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.

References

References

1. Darby, Ron. (2001). "Chemical Engineering Fluid Mechanics". CRC Press.