By Niall McMahon
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:
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:
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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