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
.
This page was last rendered on June 29, 2023.