Shear stress is most commonly applied to solids. Shear forces acting tangentially to a surface of a solid body cause deformation. In contrast to solids that can resist deformation, liquids lack this ability, and flow under the action of the force. When the fluid is in motion, shear stresses are developed due to the particles in the fluid moving relative to one another.
For a fluid flowing in a pipe, fluid velocity will be zero at the pipe wall. Velocity will increase while moving towards the center of the pipe. Shear forces are normally present because adjacent layers of the fluid move with different velocities compared to each other.
By considering the velocity of this relative motion, shear rate, , can be calculated. Shear rate is defined as a measure of the extent or rate of relative motion between adjacent layers of the moving fluid. Shear rate for the fluid flowing between two parallel plates, one moving at a constant speed and one is stationary, is determined by:
is velocity gradient (can be written in differential form ).
Shear rate is normally expressed in units of reciprocal seconds (sec-1). For Newtonian fluids in laminar flow, shear stress is proportional to shear rate where viscosity is the proportionality coefficient. This is known as Newton’s law of viscosity.
µ is dynamic viscosity of the fluid.
The above example presents simplified laminar flow with idealized flow pattern for the ease of the model visualization. In reality, such model can be applied only to flows with very low Reynolds numbers and smooth wall surfaces.
In turbulent flow, there are no well-defined layers. Instead, unsteady vortices appear of many sizes that interact with each other in a rather chaotic manner. The characteristic feature of the turbulence is that the fluid velocity varies significantly and irregularly both in position and in time. As a result, the shearing of the fluid in turbulence is best described with statistical techniques. Kolmogorov’s theory on incompressible turbulence suggests that the turbulence can be seen isotropic on a small scale. The energy dissipation rate can be used to estimate the level of intensity of the turbulence. Read more in turbulence and multiphase flow.
Shear in different types of liquids
Fluids obeying Newton’s law where the value of µ is constant are known as Newtonian fluids. If µ is constant, the shear stress is linearly dependent on the velocity gradient. This is true for most common fluids.
Fluids in which the value of µ is not constant are known as non-Newtonian fluids. Shear stress for these fluids is a more complex function of the shear rate and can be expressed as:
A, B and n are constants. For Newtonian fluids, A=0, B=µ, and n=1.
More details on non-Newtonian fluids are in the article: Shear in non-Newtonian liquids