# Unsteady-State Flow in a Tube by Orthogonal Collocation Method

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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.

Contributed by: Jorge Gamaliel Frade Chávez (March 2011)

Open content licensed under CC BY-NC-SA

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## Details

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.

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