Physics: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 $\nu$. 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.

Example - Rayleigh problem

A simple example is a semi-infinite domain bounded by a rigid wall and filled with viscous fluid. At time $t=0$ the wall is made to move with constant speed $U$ in a fixed direction (for definiteness, say the $x$ direction and consider only the $x-y$ 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

$u=U$ on $y=0$

Also, the condition that the plate has no effect on the fluid at infinity is enforced as

$u\to 0$ as $y\to \mathrm{\infty }$.

Now, from the Navier-Stokes equations

$\rho (\frac{\partial \overrightarrow{u}}{\partial t}+\overrightarrow{u}\cdot \mathrm{\nabla }\overrightarrow{u})=-\mathrm{\nabla }p+\mu {\mathrm{\nabla }}^2\overrightarrow{u}$

one can observe that this flow will be rectilinear, with gradients in the $y$ direction and flow in the $x$ direction, and that the pressure term will have no tangential component so that $\frac{\partial p}{\partial y}=0$. The $x$ component of the Navier-Stokes equations then becomes

$\frac{\partial \overrightarrow{u}}{\partial t}=\nu \underset{y}{\overset{2}{\partial }}\overrightarrow{u}$

and the scaling arguments can be applied to show that

$\frac{U}{t}\sim \nu \frac{U}{y^2}$

which gives the scaling of the $y$ co-ordinate as

$y\sim (\nu t{)}^{1/2}$.

This allows one to pose a self-similar ansatz such that, with $f$ and $\eta$ dimensionless,

$u=Uf(\eta \equiv \frac{y}{(\nu t{)}^{1/2}})$

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

$-\eta f^{\prime }/2=f^{″}$

with solution satisfying the boundary conditions that

$f=1-\mathrm{erf}(\eta /2)$ or $u=U(1-\mathrm{erf}(-y/(4\nu t{)}^{1/2}))$

which is a self-similar solution of the first kind.