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, .
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 , 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 . [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.
