# Why is the solution to the Blasius boundary layer problem self-similar?

In every course or textbook that I encountered so far, the authors transform the Navier-Stokes equations of the Blasius boundary layer problem into the Blasius ODE. The problem with many of those texts is that they claim that the solution is self-similar and give very weak justifications, or none at all and that really bothers me as I'm trying to understand rigorously why such a transformation is possible.

Does anyone know of references to look at for a rigorous proof of the above claim, or at least an experience showing that the plot of $\frac{u}{U_{\infty}}$ vs $\frac{y}{\sqrt{x}}$ doesn't change when changing $x$ ?

The wiki page on Blasius boundary layers is a useful and thorough resource in this case.

Blasius boundary layers arise in steady, laminar 2D flow over a semi-infinite plate oriented parallel to the flow. In this scenario, the Navier-Stokes equations are particularly simple and amount to a leading-order balance between inertia and viscous forces. The similarity solutions are derived from scaling arguments of these equations. It's not worth me repeating the derivation of the ode as the wiki page is very thorough.

• Actually the scaling argument only gives you an idea about the order of magnitude of the boundary layer thickness and has nothing to do with $\frac{u}{U_{\infty}}$ depending only on the Prandtl variable $\eta$. Wikipedia is among those sources that give no proof that the problem can in fact be expressed using only the reduced coordinate $\eta$. It seems to me that most just find it covenient, and also a little bit intuitive, and as such impose such a form. – BS. May 1 '15 at 13:14
• @B.S. Yes, the scaling arguments given you an order of magnitude for the boundary layer thickness, but then you use this as a length scale to construct the similarity variable, $\eta$. Once we find a similarity variable, we can construct a similarity solution for the velocity, $u(x,y) = U_{\infty} f(\eta)$ which, of course, depends only on $\eta \sim y/\sqrt{x}$. – Dai May 3 '15 at 19:04
• Yes I understand the intuition behind the construction of the similarity variable, but still, it is not a proof that $u$ can in fact be written as a function of $\eta$ only. At the moment one can only write $u (x,y)=F(\eta,x)$. – BS. May 4 '15 at 10:15

The self similarity condition is applied when deriving the Blasius equation, implying that the streamwise velocity $u(x, y)$ across a boundary layer is a function of the derivative of $f(\eta)$ scaled by the flow $U$ as follows $$u(x, y) = Uf'(\eta)$$ The self-similar streamwise profile derives from the following stream function. $$\psi(x,y) =(U\nu x)^{\frac{1}{2}}f(\eta)$$ To see this clearly, we differentiate the streamfunction with respect to $x$, finding the $x$ component of the velocity. $$u(x,y) = \frac{\partial \psi}{\partial y} = U(U\nu x)^{\frac{1}{2}}\,\,f'(\eta)\frac{d\eta}{dy} = U(U \nu x)^{\frac{1}{2}}\,\,f'(\eta)\bigg(\frac{U}{\nu x}\bigg)^\frac{1}{2}$$ Which simplifies to the desired equation.

## protected by ACuriousMind♦Apr 26 '17 at 10:21

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