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The Relationship Between Wall Shear Stress and the Maximum Shear Stress Formula

Key Takeaways

  • In fluids, shear stress is a function of the rate of shear strain, which is related to the velocity gradient of the fluid flow.

  • The property of the material to resist the development of shear deformation is called  viscosity.

  • The shear stress is maximum at the walls of the pipe. 

  Fluid flow

The shear stress is different at different points in a fluid flow

The most common classifications for materials are solids and liquids; however, there are other properties that can specify materials, including elasticity and viscosity. There are even materials that are partially elastic or viscous or neither solid nor liquid. Such materials are called viscoelastic liquids or viscoelastic solids.

To better understand this concept, consider materials that are viscous fluids. The two-plates model helps mathematically describe these materials in terms of viscosity. In the two-plates model, the shear stress and shear rate experienced by the viscous fluids are explained. The shear stress is different at different points in a fluid flow. We will discuss finding the shear stress and shear rate with the two-plates model as well as solving the maximum shear stress formula for fluid flow through a pipe. 

Defining Deformation Using Shear Stress and Viscosity 

When in use, solids and liquids are both subjected to forces. When shearing forces are applied to these materials, they deform. The action of the shearing forces on the materials is called shear stress.

Shear Strain

In solids, shear stress is a function of shear strain. Shear strain represents the deformations that are applied parallel to the cross-section of material. In fluids, shear stress is a function of the rate of shear strain, which is related to the velocity gradient of the fluid flow.

Viscosity

The ability of a material to resist the development of shear deformation is called viscosity. Viscosity is an important rheological parameter that describes the deformation of materials.  The resistance to deformation is low in low-viscosity materials. High viscosity fluid resists deformation and does not flow easily.

Let’s see how viscosity and shear stress are related in fluids using the two-plates model. 

Two-Plates Model

Viscosity is an important parameter expressing the resistance to shearing flows. In shearing flow, layers of fluid flow parallel to each other at different speeds. The relationship between viscosity and shear stress can be explained using the two-plates model. The two-plates model represents the idealized condition of Couette flow, where the fluid is trapped between two horizontal plates of area A and separated by a distance of y. The upper plate moves at a constant speed, u, while the lower plate is fixed.

There are two criteria to apply to the two-plates model:

  1. There is an adhesive force between the fluid and plates so that the fluid is in contact with the plates, but without any wall-slip effect.
  2. The fluid flow is laminar with the infinitesimally thin fluid layers and there is no turbulence in the flow.

Consider the fluid particles moving parallel to the upper plate at speeds ranging from zero at the bottom to u at the top. Each layer moves faster than the layer beneath it. The friction between the layers gives rise to a force opposing the relative motion. The fluid imparts a force in the opposite direction to the fluid motion on the upper plate and equal but opposite force on the lower plate. An external force is required to keep the upper plate moving at a constant speed and can be expressed as:

Shear stress and force relationship

It can be concluded that the ratio of speed to the height between the plates is proportional to shear stress, and new equality can be obtained by applying the proportionality constant as follows:

Shear stress and viscosity relationship

Note that μ is the dynamic viscosity or absolute viscosity.

In differential form, the equation can be written as:

 Shear stress and  shear rate relationship

According to Newton’s law of viscosity, the shear stress in Newtonian fluids is proportional to the velocity gradient in the direction perpendicular to the flow, as given in equation (5) where du/dy is the shear rate.

Wall Stress in a Pipe Flow and the Maximum Shear Stress Formula 

The flow types–laminar or turbulent–influence the nature of pipe flow. The shear stress in laminar and turbulent flow is different and plays a significant role in defining the pipe flow. The transfer of mometum among the randomly moving molecules produces shear stress in laminar flow. In turbulent flow, shear stress is largely a result of a momentum transfer among the randomly moving, finite-sized fluid particles. The physical properties of shear stress are different for laminar and turbulent flows.

In a fully developed laminar flow, there is constant axial velocity and velocity profile in the direction of flow. The radial component of velocity  and acceleration are zero in a steady, fully developed laminar flow. In a pipe, the shear stress distribution is a linear function of the radial coordinate. The shear stress is maximum at the wall and is defined as wall shear stress. The shear stress can be defined using wall shear stress in a fully developed laminar flow as follows:

Shear stress and  wall shear stress relationship

Note that r is the radial coordinate and D is the diameter of the pipe. The shear stress is maximum at the walls of the pipe and minimum at the center of the pipe. The maximum shear stress formula can be derived from equation (6) as:

 Maximum shear stress formula

The maximum shear stress formula reveals that the stress is maximum at the walls in a laminar flow. The shear stress distribution and velocity profile in a pipe can be studied using CFD models. You can rely on Cadence’s CFD tools to help you model fluid behavior in piping systems.

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