19.8 : Couette Flow

Couette flow represents the flow of fluid between two parallel plates, with one plate fixed and the other moving with a constant velocity. This configuration allows for a simplified analysis using the Navier-Stokes equations, which govern fluid motion under conditions of viscosity and incompressibility. For Couette flow, the assumptions include a steady, laminar, incompressible flow with a zero-pressure gradient in the flow direction. This flow type is beneficial for understanding shear-driven flows and is relevant to applications in lubrication and civil engineering scenarios such as sediment transport and erosion.

In a Couette flow setup, let the x-axis align with the moving direction of the upper plate while the y-axis is perpendicular to both plates. The flow is steady, laminar, and fully developed, with no variation in the

For Couette flow, the velocity distribution equation is given by:

Where:

1. U is the constant velocity of the upper plate,
2. b is the distance between the plates,
3. μ is the dynamic viscosity of the fluid.

In dimensionless form, dividing both sides by U, we get:

Where u/U represents the dimensionless velocity, and y/b represents the normalized distance from the stationary plate. To analyze specific conditions, we define the dimensionless parameter P as:

Under these conditions, P becomes zero when there is no pressure gradient (∂p/∂x=0) in the x-direction, reducing the velocity equation to:

This linear profile indicates that the fluid velocity increases linearly from zero at the stationary plate to U at the moving plate, resulting in a uniform shear rate across the fluid layer.