A simple example is a semi-infinite domain bounded by a rigid wall and filled with viscous fluid.  At time  the wall is made to move with constant speed  in a fixed direction (for definiteness, say the  direction and consider only the  plane), one can see that there is no distinguished length scale given in the problem. This is known as the Rayleigh problem. The boundary conditions of no-slip is
Also, the condition that the plate has no effect on the fluid at infinity is enforced as
Now, from the Navier-Stokes equations
one can observe that this flow will be rectilinear, with gradients in the  direction and flow in the  direction, and that the pressure term will have no tangential component so that
.  The  component of the Navier-Stokes equations then becomes
and the scaling arguments can be applied to show that
which gives the scaling of the  co-ordinate as
This allows one to pose a self-similar ansatz such that, with  and  dimensionless,
The above contains all the relevant physics and the next step is to solve the equations, which for many cases  will include numerical methods.  This equation is
with solution satisfying the boundary conditions that
which is a self-similar solution of the first kind.