Self-similar solution
In the study of partial differential equations, particularly in fluid dynamics, a self-similar solution is a form of solution which is similar to itself if the independent and dependent variables are appropriately scaled. Self-similar solutions appear whenever the problem lacks a characteristic length or time scale (for example, the Blasius boundary layer of an infinite plate, but not of a finite-length plate). These include, for example, the Blasius boundary layer or the Sedov–Taylor shell. Concept. 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 length or time scale, while the solution depends on space or time. It is then necessary to construct a scale using space or time and the other dimensional quantities present—such as the viscosity formula_1. 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 a 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 a self-similar solution of the second kind. Self-similar solution of the second kind. The early identification of self-similar solutions of the second kind can be found in problems of imploding shock waves (Guderley–Landau–Stanyukovich problem), analyzed by G. Guderley (1942) and Lev Landau and K. P. Stanyukovich (1944), and propagation of shock waves by a short impulse, analysed by Carl Friedrich von Weizsäcker and Yakov Borisovich Zel'dovich (1956), who also classified it as the second kind  for the first time. 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. Example - Rayleigh problem. A simple example is a semi-infinite domain bounded by a rigid wall and filled with viscous fluid. At time formula_2 the wall is made to move with constant speed formula_3 in  a fixed direction (for definiteness, say the formula_4 direction and consider only the formula_5 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 formula_6 on formula_7 Also, the condition that the plate has no effect on the fluid at infinity is enforced as formula_8 as formula_9. Now, from the Navier-Stokes equations formula_10 one can observe that this flow will be rectilinear, with gradients in the formula_11 direction and flow in the formula_4 direction, and that the pressure term will  have no tangential component so that formula_13. The formula_4 component of the Navier-Stokes equations then becomes formula_15 and the scaling arguments can be applied to show that formula_16 which gives the scaling of the formula_11 co-ordinate as formula_18. This allows one to pose a self-similar ansatz such that, with formula_19 and formula_20 dimensionless, formula_21 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 formula_22 with solution satisfying the boundary conditions that formula_23 or formula_24 which is a self-similar  solution of the first kind.