Physics: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]
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 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 [math]\displaystyle{ \nu }[/math]. 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 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 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] The self-similar solution of the second kind also appears in different contexts[5] such as in boundary-layer problems subjected to small perturbations, as was identified by Keith Stewartson[6], Paul A. Libby and Herbert Fox[7]. 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.[8] At time [math]\displaystyle{ t=0 }[/math] the wall is made to move with constant speed [math]\displaystyle{ U }[/math] in a fixed direction (for definiteness, say the [math]\displaystyle{ x }[/math] direction and consider only the [math]\displaystyle{ x-y }[/math] 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
[math]\displaystyle{ u = U }[/math] on [math]\displaystyle{ y = 0 }[/math]
Also, the condition that the plate has no effect on the fluid at infinity is enforced as
[math]\displaystyle{ u \rightarrow 0 }[/math] as [math]\displaystyle{ y \rightarrow \infty }[/math].
Now, from the Navier-Stokes equations
[math]\displaystyle{ \rho \left( \dfrac{\partial \vec{u}}{\partial t} + \vec{u} \cdot \nabla \vec{u} \right) =- \nabla p + \mu \nabla^{2} \vec{u} }[/math]
one can observe that this flow will be rectilinear, with gradients in the [math]\displaystyle{ y }[/math] direction and flow in the [math]\displaystyle{ x }[/math] direction, and that the pressure term will have no tangential component so that [math]\displaystyle{ \dfrac{\partial p}{\partial y} = 0 }[/math]. The [math]\displaystyle{ x }[/math] component of the Navier-Stokes equations then becomes
[math]\displaystyle{ \dfrac{\partial \vec{u}}{\partial t} = \nu \partial^{2}_{y} \vec{u} }[/math]
and the scaling arguments can be applied to show that
[math]\displaystyle{ \frac{U}{t} \sim \nu \frac{U}{y^{2}} }[/math]
which gives the scaling of the [math]\displaystyle{ y }[/math] co-ordinate as
[math]\displaystyle{ y \sim (\nu t)^{1/2} }[/math].
This allows one to pose a self-similar ansatz such that, with [math]\displaystyle{ f }[/math] and [math]\displaystyle{ \eta }[/math] dimensionless,
[math]\displaystyle{ u = U f \left( \eta \equiv \dfrac{y}{(\nu t)^{1/2}} \right) }[/math]
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
[math]\displaystyle{ - \eta f'/2 = f'' }[/math]
with solution satisfying the boundary conditions that
[math]\displaystyle{ f = 1 - \operatorname{erf} (\eta / 2) }[/math] or [math]\displaystyle{ u = U \left(1 - \operatorname{erf} \left(- y / (4 \nu t)^{1/2} \right)\right) }[/math]
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.
- ↑ 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.
- ↑ Stewartson, K. (1957). On asymptotic expansions in the theory of boundary layers. Journal of Mathematics and Physics, 36(1-4), 173-191.
- ↑ Libby, P. A., & Fox, H. (1963). Some perturbation solutions in laminar boundary-layer theory. Journal of Fluid Mechanics, 17(3), 433-449.
- ↑ Batchelor (2006 edition), An Introduction to Fluid Dynamics, p189
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