Self-similar_solution Knowpia

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 edit

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 $\nu$ . These constructs are not 'guessed' but are derived immediately from the scaling of the governing equations.

Classification edit

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 $t=0$ the wall is made to move with constant speed $U$ in a fixed direction (for definiteness, say the $x$ direction and consider only the $x-y$ 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

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

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

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

Now, from the Navier-Stokes equations

\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

y

direction and flow in the

x

direction, and that the pressure term will have no tangential component so that

{\dfrac {\partial p}{\partial y}}=0

. The

x

component of the Navier-Stokes equations then becomes

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

and the scaling arguments can be applied to show that

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

which gives the scaling of the

y

co-ordinate as

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

.

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

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

-\eta f'/2=f''

with solution satisfying the boundary conditions that

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 edit

^ 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.
^ 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.
^ Stanyukovich, K. P. (2016). Unsteady motion of continuous media. Elsevier. Page 521
^ Weizsäcker, CF (1954). Approximate representation of strong unsteady shock waves through homology solutions. Zeitschrift für Naturforschung A, 9 (4), 269-275.
^ Zeldovich, Y. B. (1956). "The motion of a gas under the action of a short term pressure shock". Akust. Zh. 2 (1): 28–38.
^ 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.
^ 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.
^ 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.
^ 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.
^ Batchelor (2000) [1967]. An Introduction to Fluid Dynamics. p. 189. ISBN 9780521663960.

[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.

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Summary

Concept edit

Classification edit

Self-similar solution of the second kind edit

Example: Rayleigh problem edit

References edit