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]

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.

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, 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.

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

${\displaystyle u=U}$ 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.

References

1. Gratton, J. (1991). Similarity and self similarity in fluid dynamics. Fundamentals of Cosmic Physics. 15. New York: Gordon and Breach. pp. 1–106. OCLC 35504041.
2. Barenblatt, Grigory Isaakovich (1996). Scaling, self-similarity, and intermediate asymptotics: dimensional analysis and intermediate asymptotics. Vol. 14. Cambridge University Press. ISBN 0-521-43522-6.
3. Stanyukovich, K. P. (2016). Unsteady motion of continuous media. Elsevier. Page 521
4. Weizsäcker, CF (1954). Approximate representation of strong unsteady shock waves through homology solutions. Zeitschrift für Naturforschung A , 9 (4), 269-275.
5. Zeldovich, Y. B. (1956). "The motion of a gas under the action of a short term pressure shock". Akust. zh. 2 (1): 28–38.
6. Barenblatt, G. I.; Zel'dovich, Y. B. (1972). "Self-similar solutions as intermediate asymptotics". Annual Review of Fluid Mechanics. 4 (1): 285–312. doi:10.1146/annurev.fl.04.010172.001441.
7. Coenen, W.; Rajamanickam, P.; Weiss, A. D.; Sánchez, A. L.; Williams, F. A. (2019). "Swirling flow induced by jets and plumes". Acta Mechanica. 230 (6): 2221–2231. doi:10.1007/s00707-019-02382-2.
8. Stewartson, K. (1957). "On asymptotic expansions in the theory of boundary layers". Journal of Mathematics and Physics. 36 (1–4): 173–191. doi:10.1002/sapm1957361173.
9. Libby, P. A.; Fox, H. (1963). "Some perturbation solutions in laminar boundary-layer theory". Journal of Fluid Mechanics. 17 (3): 433–449. doi:10.1017/S0022112063001439.
10. Batchelor (2000) [1967]. An Introduction to Fluid Dynamics. p. 189.