In the study of partial differential equations, particularly in 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. Self-similar solutions appear whenever the problem lacks a characteristic length or time scale (for example, the Blasius boundary layer of an infinite plate, but not of a finite-length plate). These include, for example, the Blasius boundary layer or the Sedov–Taylor shell.^[1][2]

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 length or time scale, while the solution depends on space or time. It is then necessary to construct a scale using space or 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.

Classification

The normal self-similar solution is also referred to as a 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 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.^[10] 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

$${\displaystyle u{(y\!=\!0)}=U}$$

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

$${\displaystyle u{(y\!\to \!\infty )}=0.}$$

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

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

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

References