Self-similar solution

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]

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.

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.

The early identification of self-similar solutions of the second kind can be found in problems of imploding shock waves (Guderley–Landau–Stanyukovich problem), analyzed by G. Guderley (1942) and Lev Landau and K. P. Stanyukovich (1944),^[3] and propagation of shock waves by a short impulse, analysed by Carl Friedrich von Weizsäcker^[4] and Yakov Borisovich Zel'dovich (1956), who also classified it as the second kind for the first time.^[5] A complete description was made in 1972 by Grigory Barenblatt and Yakov Borisovich Zel'dovich.^[6] The self-similar solution of the second kind also appears in different contexts such as in boundary-layer problems subjected to small perturbations,^[7] as was identified by Keith Stewartson,^[8] Paul A. Libby and Herbert Fox.^[9] 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.^[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.

In transient heat transfer applications, such as impingement heating on a ship deck during missile launches and the sizing of thermal protection systems, self-similar solutions can be found for semi-infinite solids.^[11][12] The governing equation when heat conduction is the primary heat transfer mechanism is the one-dimensional energy equation:$${\displaystyle \rho c_{p}{\frac {\partial T}{\partial t}}={\frac {\partial }{\partial x}}\left(k{\frac {\partial T}{\partial x}}\right)}$$where ${\displaystyle \rho }$ is the material's density, ${\displaystyle c_{p}}$ is the material's specific heat capacity, ${\displaystyle k}$ is the material's thermal conductivity. In the case when the material is assumed to be homogeneous and its properties constant, the energy equation is reduced to the heat equation:$${\displaystyle {\frac {\partial T}{\partial t}}=\alpha {\frac {\partial ^{2}T}{\partial x^{2}}},\quad \alpha ={\frac {k}{\rho c_{p}}}}$$with ${\displaystyle \alpha }$ being the thermal diffusivity. By introducing the similarity variable ${\displaystyle \eta =x/{\sqrt {t}}}$ and assuming that ${\displaystyle T(t,x)=f(\eta )}$, the PDE can be transformed into the ODE:$${\displaystyle f''(\eta )+{\frac {1}{2\alpha }}\eta f'(\eta )=0}$$If a simple model of thermal protection system sizing is assumed, where decomposition, pyrolysis gas flow, and surface recession are ignored, with the initial temperature ${\displaystyle T(0,x)=f(\infty )=T_{i}}$ and a constant surface temperature ${\displaystyle T(t,0)=f(0)=T_{s}}$, then the ODE can be solved for the temperature at a depth ${\displaystyle x}$ and time ${\displaystyle t}$:^[12]$${\displaystyle T(t,x)={\text{erf}}\left({\frac {x}{2{\sqrt {\alpha t}}}}\right)\left(T_{i}-T_{s}\right)+T_{s}}$$where ${\displaystyle {\text{erf}}(\cdot )}$ is the error function.