Self-similar solution

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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 . 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.

Self-similar solution of the second kind[edit]

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.

Example: Rayleigh problem[edit]

A simple example is a semi-infinite domain bounded by a rigid wall and filled with viscous fluid.[10] At time the wall is made to move with constant speed in a fixed direction (for definiteness, say the direction and consider only the 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

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

Now, from the Navier-Stokes equations

one can observe that this flow will be rectilinear, with gradients in the direction and flow in the direction, and that the pressure term will have no tangential component so that . The component of the Navier-Stokes equations then becomes
and the scaling arguments can be applied to show that
which gives the scaling of the co-ordinate as

This allows one to pose a self-similar ansatz such that, with and dimensionless,

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

with solution satisfying the boundary conditions that
which is a self-similar solution of the first kind.


  1. ^ Gratton, J. (1991). Similarity and self similarity in fluid dynamics. Fundamentals of Cosmic Physics. Vol. 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. Bibcode:1972AnRFM...4..285B. 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. S2CID 126488392.
  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. S2CID 123824364.
  10. ^ Batchelor (2000) [1967]. An Introduction to Fluid Dynamics. p. 189. ISBN 9780521663960.