Novikov's compact leaf theorem

In 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.

Concept

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 catalogued 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 $\nu$. These constructs are not 'guessed' but are derived immediately from the scaling of the governing equations.

Example - The impulsively started plate

Consider 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). We can see that there is no distinguished length scale given in the problem, and we have the boundary conditions of no slip

$u=U$ on $y=0$

and that the plate have no effect on the fluid at infinity

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

Now, if we examine the Navier-Stokes equations

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

we 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

$\frac{\partial \overrightarrow{u}}{\partial t}=\nu {\partial }_y^2\overrightarrow{u}$

and we may apply scaling arguments to show that

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

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

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

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

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

We have now extracted all of the relevant physics and need only solve the equations; for many cases this will need to be done numerically. This equation is

$-\eta f^{\prime }/2=f^{″}$

with solution satisfying the boundary conditions that

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

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