This demonstrates laminar flow of a viscous fluid between two plates. The plot shows fluid velocity as a function of vertical distance. Velocity is proportional to the length of the arrows. The lower plate is stationary and the upper plate is either stationary or moves to the right; its velocity is set using the slider. The pressure gradient is also set with a slider.

For steady-state, laminar flow of an incompressible viscous fluid between parallel plates, the ordinary differential equation (ODE) describing fluid velocity is

\( \frac{ d^{ 2 } u }{ d y^{ 2 } } = \frac{ 1 }{ \mu } \frac{ dp }{ dx } \)

where \( u \) is fluid velocity, \( y \) is distance from the center line, \( \mu \) is dynamic viscosity of the fluid, and \( \frac{ dp }{ dx } \) is the pressure gradient along the x-axis. The boundary conditions are

\( u \left( \mathop{\mbox{-$h$}} \right) = 0 \; \; \) (lower plate velocity)

\( u \left( h \right) = U_{ 0 } \; \; \) (upper plate velocity)

where \( 2h \) is the distance between the plates and \( U_{ 0 } \) is the velocity of the top plate. An analytical solution for this ODE is:

\( u \left( y \right) = \frac{ U_{ 0 } }{ 2 } \left( 1 + \frac{ y }{ h } \right) - \frac{ dp }{ dx } \frac{ h^{ 2 } }{ 2 \mu } \left( 1 - \left( \frac{ y }{ h } \right)^{ 2 } \right) \)

This simulation was created in the Department of Chemical and Biological Engineering, at University of Colorado Boulder for LearnChemE.com by Neil Hendren under the direction of Professor John L. Falconer. It is an HTML5/JavaScript implementation of the Wolfram Demonstration created by W.C. Guttner¹. This simulation was prepared with financial support from the National Science Foundation. Address any questions or comments to learncheme@gmail.com. All of our simulations are open source, and are available on our LearnChemE Github repository.

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References:

1. Guttner, W.C. (2011). Couette Flow. A Wolfram Demonstrations Project.