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Unsteady state Couette flow Calculator

This equation describes the time-dependent velocity distribution of a viscous fluid confined between two infinite parallel plates where one plate is suddenly set into motion.

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Overview

The equation is a specific application of the Navier-Stokes equations, simplifying to a diffusion-type partial differential equation for the velocity component parallel to the plates. It accounts for the momentum diffusion process driven by kinematic viscosity as the velocity profile develops over time from an initial state toward a steady-state linear profile. Understanding this evolution is critical for determining the transient behavior of fluid systems subject to sudden changes in boundary conditions.

Apply it well

When To Use

When to use: Use this equation when analyzing the transient velocity profile of an incompressible Newtonian fluid between parallel boundaries immediately following a sudden start-up or change in plate velocity.

Why it matters: It models the fundamental mechanism of momentum transport via viscous diffusion, which governs how shear effects propagate through a fluid over time.

Avoid these traps

Common Mistakes

  • Assuming the velocity profile is linear at all times during the transient phase.
  • Neglecting the impact of the kinematic viscosity on the time required to reach a steady state.

One free problem

Practice Problem

If the kinematic viscosity of a fluid increases, how does the time required for the flow to reach a steady-state Couette profile change?

parameterkinematic viscosity

Solve for:

Hint: Consider the relationship between viscosity and the diffusion rate of momentum.

The full worked solution stays in the interactive walkthrough.

References

Sources

  1. Bird, R. B., Stewart, W. E., & Lightfoot, E. N., Transport Phenomena, 2nd Edition, Wiley.
  2. White, F. M., Viscous Fluid Flow, McGraw-Hill Education.
  3. NIST CODATA
  4. IUPAC Gold Book
  5. White, Frank M. Fluid Mechanics. 8th ed., McGraw-Hill Education, 2016.
  6. Bird, R. Byron; Stewart, Warren E.; Lightfoot, Edwin N. (2007). Transport Phenomena (2nd ed.). John Wiley & Sons.
  7. White, Frank M. (2011). Fluid Mechanics (7th ed.). McGraw-Hill.
  8. NPTEL (National Programme on Technology Enhanced Learning) - Fluid Mechanics Course