Similarity solution

In study of partial differential equations, particularly 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. The self-similar solution appears whenever the problem lacks a characteristic length or time scale (for example, self-similar solution describes Blasius boundary layer of an infinite plate, but not the finite-length plate). These include, for example, the Blasius boundary layer or the Sedov-Taylor shell. In study of partial differential equations, particularly 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. The self-similar solution appears whenever the problem lacks a characteristic length or time scale (for example, self-similar solution describes Blasius boundary layer of an infinite plate, but not the finite-length plate). These include, for example, the Blasius boundary layer or the Sedov-Taylor shell. A powerful tool in physics is the concept of dimensional analysis and scaling laws; by looking at 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 (timescale) while the solution depends on space (time). It is then necessary to construct a lengthscale (timescale) using space (time) and the other dimensional quantities present - such as the viscosity ν {displaystyle u } . These constructs are not 'guessed' but are derived immediately from the scaling of the governing equations. 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 a different context in the 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 t = 0 {displaystyle t=0} the wall is made to move with constant speed U {displaystyle U} in a fixed direction (for definiteness, say the x {displaystyle x} direction and consider only the x − y {displaystyle x-y} 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 u = U {displaystyle u=U} on y = 0 {displaystyle y=0}