Page 585 - Modelling in Transport Phenomena A Conceptual Approach

B.3. SECONDORDER PARTIAL DIFFERENTIAL EQUATIONS 565

Integration of Eq. (B.3-78) gives

f = CI (B.3-79)
Ja
where u is a dummy variable of integration. Application of the boundary condition
defined by Eq. (B.3-75) gives C2 = 1. On the other hand, the use of the boundary
condition defined by Q. (B.3-76) gives

(B.3-80)

Therefore, the solution becomes

f=1-- 1" e-"' du. = 1 - erf(7) (B.3-81)

where erf(x) is the emr function defined by

(B.3-82)

Finally, the velocity distribution as a function oft and z is given by

(B.3-83)

mFERENCES

Bird, R.B., R.C. Armstrong and 0. Hassager, 1987, Dynamics of Polymeric Liquids,
Volume 1: Fluid Dynamics, 2nd Ed., Wiley, New York.

Hildebrand, F.B., 1976, Advanced Calculus for Applications, 2nd Ed., Prentice-
Hall, Englewood Cliffs, New Jersey.

Murray, D.A., 1924, Introductory Course in Differential Equations, Longmans,
Green and Co., London.