physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

mechanics Mechanics () is the area of physics concerned with the relationships between force, matter, and motion among Physical object, physical objects. Forces applied to objects may result in Displacement (vector), displacements, which are changes of ...

and other areas of science, shear rate is the rate at which a progressive shear strain is applied to some material, causing

shearing Sheep shearing is the process by which the woollen fleece of a sheep is cut off. The person who removes the sheep's wool is called a '' shearer''. Typically each adult sheep is shorn once each year (depending upon dialect, a sheep may be sai ...

to the material. Shear rate is a measure of how the velocity changes with distance.

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,

where: *

\dot\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_ = \frac + \frac.

For the simple shear case, it is just a

gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...

velocity Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...

in a flowing material. The

SI unit The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of units of measurement, system of measurement. It is the only system ...

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 :

\dot=\sqrt

. The shear rate at the inner wall of a

Newtonian fluid A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of cha ...

flowing within a pipe is :

\dot\gamma = \frac,

where: *

\dot\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 = \frac,

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

A = \pi r^2,

thus producing :

v = \frac.

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 = \frac,

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

\dot\gamma = \frac.

For a

wall,

shear stress Shear stress (often denoted by , Greek alphabet, Greek: tau) is the component of stress (physics), stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross secti ...

() can be related to shear rate by

\tau_w = \dot\gamma_x \mu

where is the dynamic

viscosity Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...

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

{{reflist

Simple shear

References

See also