Self-similar solution

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 ${\displaystyle \nu }$. 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.[3] A complete description was made in 1972 by Grigory Barenblatt and Yakov Borisovich Zel'dovich.[4] 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[5], Paul A. Libby and Herbert Fox[6]. 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.[7] 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), 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

${\displaystyle u=U}$ on ${\displaystyle y=0}$

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

${\displaystyle u\rightarrow 0}$ as ${\displaystyle y\rightarrow \infty }$.

Now, from the Navier-Stokes equations

${\displaystyle \rho \left({\dfrac {\partial {\vec {u}}}{\partial t}}+{\vec {u}}\cdot \nabla {\vec {u}}\right)=-\nabla p+\mu \nabla ^{2}{\vec {u}}}$

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

${\displaystyle {\dfrac {\partial {\vec {u}}}{\partial t}}=\nu \partial _{y}^{2}{\vec {u}}}$

and the scaling arguments can be applied to show that

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

which gives the scaling of the ${\displaystyle y}$ co-ordinate as

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

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

${\displaystyle u=Uf\left(\eta \equiv {\dfrac {y}{(\nu t)^{1/2}}}\right)}$

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

${\displaystyle -\eta f'/2=f''}$

with solution satisfying the boundary conditions that

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

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