Absorption in a Falling Thin Liquid Film at Low Reynolds Numbers

Consider the absorption without chemical reaction of a gaseous species in a thin liquid film of a nonvolatile compound flowing vertically at low Reynolds numbers.
The governing dimensionless equation is:
,
with the initial and boundary conditions
, ,
, ,
, ,
where is the concentration of compound , and and are the positions.
The Demonstration plots the solution for any value of in the interval . You can vary the values of as well as the number of Chebyshev collocation points, .

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DETAILS

In the discrete Chebyshev–Gauss–Lobatto case, the interior points are given by . These points are the extrema of the Chebyshev polynomials of the first kind, .
The Chebyshev derivative matrix at the quadrature points is an matrix given by
, , for , and for , , and ,
where for and .
The matrix is then used as follows: and , where is a vector formed by evaluating at , , and and are the approximations of and at the .
References
[1] P. Moin, Fundamentals of Engineering Numerical Analysis, Cambridge, UK: Cambridge University Press, 2001.
[2] L. N. Trefethen, Spectral Methods in MATLAB, Philadelphia: SIAM, 2000.
[3] R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd ed., New York: John Wiley & Sons, 2002.
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