Fluid Boundary Layer Mathy Question

[member 428835]

Hi PF!

So after scaling Navier-Stokes for a flow over a flat plate we ultimately arrive at $f f'' + f''' = 0$ subject to $f(0)=0$, $f'(0)=0$, and $f'(\infty) = 1$ where independent variable is $\eta$. The source I was reading is trying to reduce this BVP to an IVP. Thus they suggest for some solution $F(\eta)$, $CF(C \eta)$ also is a solution. Then we have $$1 = \lim_{\eta \to \infty} f'(\eta) = C^2 \lim_{\eta \to \infty} F'(C \eta) \implies C = \left( \lim_{\eta \to \infty} F'(\eta) \right)^{-1/2}$$. But this is where it get's strange. They then say "if we specify $F''(0) = 1$... but how can they do this? We know $f''(0) = C^3 F''(0) \implies F''(0) = f''(0) C^{-3}$ but $C$ has already been specified.

The link to this is here: http://web.mit.edu/fluids-modules/www/highspeed_flows/ver2/bl_Chap2.pdf around eq. (3.48)

Any help at understanding this would be awesome!

 

[pasmith]

Let $f$ be the solution of the BVP $ff'' + f''' = 0$ with $f(0) = f'(0) = 0$ and $f'(\infty) = 1$.

Now for each $C > 0$ define a function $g_C(\eta) = Cf(C\eta)$. Then $g_C$ is the solution of the BVP $g_Cg_C'' + g_C''' = 0$ with $g_C(0) = g_C'(0) = 0$ and $g_C'(\infty) = C^2$. (This is not the original BVP unless $C = 1$, and by definition $g_1 = f$ anyway.)

Consider the IVP $hh'' + h''' = 0$ subject to $h(0) = h'(0) = 0$ and $h''(0) = 1$. Knowing $h$ we can find $f$, because $h = g_C$ where $C^2 = L = \lim_{x \to \infty} h'(x)$ (assuming, of course, that this limit exists). Hence by definition of $g_C$ we have $$f(\eta) = C^{-1}g_C(C^{-1}\eta) = L^{-1/2}h(L^{-1/2}\eta).$$

 

[member 428835]

Gotcha, I think this is making sense! I have a corollary question then, though I can post as a separate thread if that's more appropriate. I am trying to transform this equation to transform this BVP $y'' + 6 y^{2/3} = 0$ subject to $y'(0)=0$ and $y(1)=0$ into an IVP so I can numerically solve it. Right not when I use NDsolve in mathematica I get no output. Any insight on when this transformation is possible, and when it is, how to go about doing it?

I noticed the BVP is invariant when $y(x) = \lambda^n Y(\lambda^{-n/6} x)$. I'm just unsure how to proceed.