André Lévêque - His Doctoral Thesis, Page 285
2006
Where Lévêque describes convective heat transfer close to a surface.
On p285, Lévêque writes that neglecting 
, and substituting the simplified velocity profile (
) into Eqn. 85, results in Eqn. 95:
With the boundary conditions described, it is a fact that Eqn. 95 has a solution of the form:
Or, simply:
where 
is some, as yet undefined, function. The only constraint on 
is that it satisfies the solution to Eqn. 95 and it is a function of 
and 
.
Putting this solution for 
into Eqn. 85 (on p277), you end up with:
Rewriting:
Lévêque observed that if we set 
, and we define 
, then 
. By defining 
and 
in this way, this equation reduces to:
This is something we can solve. The important point to note is this: 
is some function of 
that satisfies Eqn. 95 - it translates from 
to 
. 
depends on 
raised to any power 
, so we are free to define 
. Lévêque chose some 
that makes the equation easy to solve. Once 
in defined, then 
and a 
that satisfies Eqn. 95 are automatically defined.
This is how and why Lévêque chose 
.
The rest of p286 and p287 is pretty straightforward, leading to a solution for 
and 
.
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