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

Couette flow

In fluid dynamics, Couette flow refers to the laminar flow of a viscous fluid in the space between two parallel plates, one of which is moving relative to the other. The flow is driven by virtue of viscous drag force acting on the fluid and the applied pressure gradient parallel to the plates...

), is defined by



where

• = The shear rate, measured in reciprocal seconds
• = The velocity of the moving plate, measured in meters per second
• = The distance between the two parallel plates, measured in meters

Or,


For the simple shear

Simple shear

In fluid mechanics, simple shear is a special case of deformation where only one component of velocity vectors has a non-zero value:\ V_x=f\ V_y=V_z=0And the gradient of velocity is constant and perpendicular to the velocity itself:...

 case, it is just a gradient

Gradient

In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....

 of velocity

Velocity

In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

 in a flowing material. The SI unit of measurement for shear rate is sec^-1, expressed as "reciprocal seconds" or "inverse seconds."

The shear rate at the inner wall of a Newtonian fluid

Newtonian fluid

A Newtonian fluid is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.-Definition:...

 flowing within a pipe is:


where:

• = The shear rate, measured in reciprocal seconds.
• = The linear fluid velocity.
• = The inside diameter of the pipe.

The linear fluid velocity v is related to the volumetric flow rate Q by:


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


thus producing:


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:


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

For a Newtonian fluid

Newtonian fluid

A Newtonian fluid is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.-Definition:...

 wall shear stress

Shear stress

A shear stress, denoted \tau\, , is defined as the component of stress coplanar with a material cross section. Shear stress arises from the force vector component parallel to the cross section...

 () can be related to shear rate by
, where is the viscosity

Viscosity

Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

 of the fluid. For Non-Newtonian fluids, there are different constitutive laws

Constitutive equation

In physics, a constitutive equation is a relation between two physical quantities that is specific to a material or substance, and approximates the response of that material to external forces...

 depending on the fluid, which relates the stress tensor

Stress tensor

Stress tensor may refer to:* Stress , in classical physics* Stress-energy tensor, in relativistic theories* Maxwell stress tensor, in electromagnetism...

to the shear rate tensor.