Page 497 - Modelling in Transport Phenomena A Conceptual Approach

11.1. UNSTEADY LAMINAR FLOWIN A TUBE 477

(1 1.1-33)

at <=1 &=O (1 1 .l-34)
which can be solved by the method of separation of variables.
Representing the solution as a product of two functions of the form
(11.1-35)

1dF
1 d
Fdr - Gt@('Z) (1 1.1-36)
While the left side of Eq. (11.1-36) is a function of r only, the right side is dependent
only on 5. This is possible if both sides of Eq. (11.1-36) are equal to a constant,
__- -&f(tg)=-XZ (11.1-37)
2
say - X , i.e.,
1 dF
d7
Equation (11.1-37) results in two ordinary differential equations. The equation for
F is given by
dF
+
- X2F = 0 (11.1-38)
dr
The solution of Eq. (11.1-38) is
=
e-
~(7) X2~ (1 1.1-39)
On the other hand, the equation for G is

f (t$)+X2tG=0 (1 1.1-40)

and it is subject to the boundary conditions

(11.1-41)

at <=l G=O (11.1-42)
Note that Eq. (11.1-40) is a Sturm-Liouville equation with a weight function of (.
The solution of Eq. (11.1-40) is given in terms of the hsel functions as

G(t) = A JO(M + €3 Yo(%) (11.1-43)
where A and B are constants. Since Y,(O) = - 00, B = 0. Application of the Eq.
( 11.1-42) gives
&(A) = 0 =+ X = XI,&, ... (1 1.1-44)