Page 583 - Modelling in Transport Phenomena A Conceptual Approach

B.3. SECOND-ORDER PARTIAL DIFFERENTIAL EQUATIONS 563

B .3.6.2 Similarity solution
This is also known as the method of combination of variables. Similarity solutions
are a special class of solutions used to solve parabolic second-order partial differ-
ential equations when there is no geometric length scale in the problem, i.e., the
domain must be either semi-infinite or infinite. Besides, the initial condition should
match the boundary condition at infinity.
The basis of this method is to combine the two independent variables in a single
variable so as to transform the second-order partial differential equation into an
ordinary differential equation.
Let us consider the following parabolic second-order partial differential equation
together with the initial and boundary conditions:

av, a2v,
-=u- (B.3-60)
at 6x2
at t < 0 v, = 0 for all x (B.3-61)
atx=O w,=V fort>O (B.3-62)
at x=oo wz=O fort>O (B.3-63)
Such a problem represents the velocity profile in a fluid adjacent to a wall suddenly
set in motion and is also known as Stokes’ first problem.
The solution is sought in the form

11-4. = f (rl) (B .3-64)
V
where
7) = /3tmxn (B.3-65)
The term 7 is called the similarity wariable. The proportionality constant p is
included in J3q. (B.3-65) so as to make q dimensionless.
The chain rule of differentiation gives

(B.3-66)

- p2 2 2m 2(,-1,!!Y + pn(n - 1)t 2 n-2df (B.3-67)
- nt x
dV2 d77
Substitution of Eqs. (B.3-66) and (B.3-67) into Eq. (B.3-60) gives