Similarity solution

In study of partial differential equations, particularly fluid dynamics, a similarity solution is a form of solution in which at least one co-ordinate lacks a distinguished origin; more physically, it describes a flow which 'looks the same' either at all times, or at all length scales. 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. If we have cataloged these effects we will occasionally find that the system has not fixed a natural lengthscale (timescale), but that 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 ${\displaystyle \nu }$. These constructs are not 'guessed' but are derived immediately from the scaling of the governing equations.

Consider a semi-infinite domain bounded by a rigid wall and filled with viscous fluid. At time ${\displaystyle t=0}$ the wall is made to move with constant speed ${\displaystyle U}$ in a fixed direction (for definiteness, say the ${\displaystyle x}$ direction and consider only the ${\displaystyle x-y}$ plane). We can see that there is no distinguished length scale given in the problem, and we have the boundary conditions of no slip

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