# The Constancy of Couette Flow Shear Stress

### Key Takeaways

• The shear stress is a function of the rate of strain (d/dt) in fluids.

•  In Newtonian fluids, shear stress is constant and showcases a linear velocity profile.

• The constant shear stress is the notable property in the Couette laminar flow regime.

The flow of oil in a journal bearing can be idealized using Couette flow

Consider two infinite, parallel, flat walls that are in a straight line motion relative to each other. If a fluid is contained between the walls, there is a shear-induced motion called plane-Couette flow. Plane-Couette flow is the simplest shear flow known. The Couette flow shear stress is constant throughout the flow and is regarded as a unique property of flow.

In turbulent wall shear flows, the constant shear stress is observed only in a region close to the boundary of the flow. It is the Couette flow shear stress constancy that keeps it distinguished from other types of flows.

Let’s discuss Couette flow and its shear stress.

## Shear Stress in Fluids

A fluid at rest experiences shearing forces and continuous deformation. When the fluid is in motion, it resists the action of shearing forces. In terms of response to shearing forces, solids and fluids can be described in the following way:

Solids - In solids, the shear stress is a function of shear stress .

Fluids - The  shear stress is a function of the rate of strain (d/dt) in fluids.

## Newton’s Law of Viscosity

Generally, the property of viscosity is defined as the ability of the fluid to resist shear deformation growth. Most fluids obey Newton’s law of viscosity.

Shear stress is directly proportional to shear strain rate. The fluids obeying the above equation are collectively called Newtonian fluids.

## Newtonian Fluids and Viscous Flows

Consider a viscous fluid filled in the gap between two parallel horizontal plates. The upper plate is moving with velocity V. The distance between the plates is negligible compared to the dimensions of the plate. As the distance between the plates is small, the flow is parallel. There is no dependency between the velocity and horizontal coordinates. The pressure gradient is zero or the pressure is constant between two plates. The only force acting on the fluid element between the plates is due to shear stress on the boundaries.

In steady flows, the force due to shear stress is zero and shear stress remains constant. In Newtonian fluids, shear stress is constant and showcases a linear velocity profile. The corresponding shear stress is dependent on dynamic viscosity, the velocity profile, and the gap between the plates.

## Couette Flow

The flow of Newtonian fluids mentioned in the above section is shear-driven fluid motion, called Couette flow. The Couette flow is the laminar flow of viscous fluid in the gap between two parallel plates.

The Couette flow is due to a few things:

1. The viscous drag force is subjected to the fluid.
2. The applied pressure gradient parallel to the plates.

The constant velocity profile of the laminar Couette flow proves the unidirectionality of the flow. Only one of the velocity components will be non-trivial in Couette flow.

## Couette Flow Shear Stress

Laminar Couette flow is characterized by a linear velocity profile. However, as the value of the Reynolds number goes up, turbulence is introduced into the flow and the velocity profile exhibits nonlinearity. Even though the velocity distribution becomes non-linear, the shear stress distribution in the flow between the plates remains constant. At any point, the shear stress is equal to that at the wall, or wall shear stress.

The constant shear stress is the notable property in the Couette laminar flow regime. The Couette shear stress can be expressed as the product of (0h) and fluid viscosity according to Newton’s law of viscosity, where 0 is the velocity of the top plate and h is the distance between the plates.

## The Importance of Couette Flow

Couette flow is significant when analyzing the heat transfer in a fluid flow between two parallel plates or coaxial cylinders. The Couette flow is observed in applications such as turbines and viscosity pumps. In food processing, polymers, and lubrication, Couette flow is often used for modeling.

For example, the flow of oil in a journal bearing can be idealized using Couette flow. Particularly when the load applied on the bearing is small, the bearing and rotating shaft remain concentric. In this case, the lubricant flow characteristics can be modeled as Couette flow. The Couette flow shear stress study on the model can give an insight into shear distribution in the journal bearing.