Self-similar solution

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

The normal self-similar solution is also referred to as self-similar solution of the first kind since an another type of self-similar exists for finite-sized problems, which cannot be derived from dimensional analysis, known as self-similar solution of the second kind. The discovery of solution of the second kind was due to Yakov Borisovich Zel'dovich, who also named it as second kind in 1956.^[3] A complete description was made in 1972 by Grigory Barenblatt and Yakov Borisovich Zel'dovich.^[4]

Concept

A powerful tool in physics is the concept of dimensional analysis and scaling laws; by looking at 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 lengthscale (timescale) while the solution depends on space (time). It is then necessary to construct a lengthscale (timescale) using space (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.

Example - Rayleigh problem

A simple example is a semi-infinite domain bounded by a rigid wall and filled with viscous fluid.^[5] 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

$u=U$ on $y=0$

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

$u\rightarrow 0$ as $y\rightarrow \infty$ .

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 we 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

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

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

References

↑ Gratton, J. (1991). Similarity and self similarity in fluid dynamics. Fundamentals of Cosmic Physics, 15, 1-106.
↑ Barenblatt, Grigory Isaakovich. Scaling, self-similarity, and intermediate asymptotics: dimensional analysis and intermediate asymptotics. Vol. 14. Cambridge University Press, 1996.
↑ 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.
↑ Batchelor (2006 edition), An Introduction to Fluid Dynamics, p189

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[1] Gratton, J. (1991). Similarity and self similarity in fluid dynamics. Fundamentals of Cosmic Physics, 15, 1-106.

[2] Barenblatt, Grigory Isaakovich. Scaling, self-similarity, and intermediate asymptotics: dimensional analysis and intermediate asymptotics. Vol. 14. Cambridge University Press, 1996.

[3] Zeldovich, Y. B. (1956). The motion of a gas under the action of a short term pressure shock. Akust. zh, 2(1), 28-38.

[4] Barenblatt, G. I., & Zel'Dovich, Y. B. (1972). Self-similar solutions as intermediate asymptotics. Annual Review of Fluid Mechanics, 4(1), 285-312.

[5] Batchelor (2006 edition), An Introduction to Fluid Dynamics, p189