Self-similar 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. 

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 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 .  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 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. 

Example - Rayleigh problem 

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 

 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. 

References

Category:Fluid dynamics