Self similarity for slowpokes

I am struggling with the concept of self similarity.

If I google search, I either find examples that don't explain all too much or detailed explanations that go waaaaay over my head.

I've found a lot of worked examples of how you are to use the technique and once you've made the ansatz $u = U(\eta)$ where $\eta$ is dimensionless, it's all mechanical calculations from there. Although I understand the "mathematical" steps in getting the answer, I have no idea what's going on in the sense of "why am I doing this" and "when does this technique apply".

So what I am looking for is a concise (if this means loss of generality, then so be it) explanation of what the deal with self-similarity is when applied to fluid mechanics and hopefully some physical intuition of the technique.

• You do it to make the equations dimensionless. – Kyle Kanos Dec 4 '15 at 15:28

Imagine we look at the case of transient Couette flow where initially the fluid is stationary and the top wall starts moving with speed $U$. This is described by the following equations: $$\partial_t u_x = \nu \partial_y^2 u_x \quad u_x(0,x) = 0$$ with boundary conditions: $$u_x(t,0) = 0 \quad u_x(t,H) = U$$
In steady state, this is easily solved to give a linear profile: $$u_x(y) = U\frac{y}{H}$$ but as you should notice this steady-state does not include the effects of viscosity $\nu$. Those effects only are important for the speed at which the steady state is reached. Imagine qualitatively the profiles as as they move towards the steady-state: 