A powerful tool in physics is the concept of dimensional analysis and scaling laws. By examining the physical effects present in a system, we may estimate their size and hence which, for example, might be neglected. In some cases, the system may not have a fixed natural lengthscale while the solution depends on space. It is then necessary to construct a lengthscale using space and the other dimensional quantities present - such as the viscosity. These constructs are not 'guessed' but are derived immediately from the scaling of the governing equations.
Classification
The normal self-similar solution is also referred to as self-similar solution of the first kind since another type of self-similar exists for finite-sized problems, which cannot be derived from dimensional analysis, known as self-similar solution of the second kind. The discovery of solution of the second kind was due to Yakov Borisovich Zel'dovich, who also named it as second kind in 1956. A complete description was made in 1972 by Grigory Barenblatt and Yakov Borisovich Zel'dovich. The self-similar solution of the second kind also appears in different contexts such as in boundary-layer problems subjected to small perturbations, as was identified by Keith Stewartson, Paul A. Libby and Herbert Fox. Moffatt eddies are also a self-similar solution of the second kind.
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, 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 on Also, the condition that the plate has no effect on the fluid at infinity is enforced as 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 or which is a self-similar solution of the first kind.
