Unsteady-State Flow in a Tube by Orthogonal Collocation Method

In transport phenomena it is very common to encounter unsteady-state problems. This Demonstration describes the solution of a partial differential equation that describes unsteady-state flow in a tube, using an orthogonal collocation method.

Unsteady flow in a tube is described by the following dimensionless partial differential equation:

,

with these boundary conditions:

at , ,

at , ,

and the initial condition:

at , .

In the partial differential equation is nondimensional velocity, is nondimensional time, and is nondimensional position.

References: J. V. Villadsen and W. E. Stewart, "Solution of Boundary-Value Problems by Orthogonal Collocation," Chemical Enginering Science,22, 1967 pp. 3981–3996.

R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd ed., New York: John Wiley and Sons.