{{Short description|Concept in partial differential equations}} 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.<ref>{{cite book |last=Gratton |first=J. |year=1991 |title=Similarity and self similarity in fluid dynamics |series=Fundamentals of Cosmic Physics |volume=15 |pages=1–106 |location=New York |publisher=Gordon and Breach |oclc=35504041 }}</ref><ref>{{cite book |last=Barenblatt |first=Grigory Isaakovich |title=Scaling, self-similarity, and intermediate asymptotics: dimensional analysis and intermediate asymptotics |volume=14 |publisher=Cambridge University Press |year=1996 |isbn=0-521-43522-6 }}</ref>
==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 <math>\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 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 (Guderley–Landau–Stanyukovich problem), analyzed by G. Guderley (1942) and Lev Landau and K. P. Stanyukovich (1944),<ref>Stanyukovich, K. P. (2016). Unsteady motion of continuous media. Elsevier. Page 521</ref> and propagation of shock waves by a short impulse, analysed by Carl Friedrich von Weizsäcker<ref>Weizsäcker, CF (1954). Approximate representation of strong unsteady shock waves through homology solutions. Zeitschrift für Naturforschung A, 9 (4), 269-275.</ref> and Yakov Borisovich Zel'dovich (1956), who also classified it as the second kind for the first time.<ref>{{cite journal |last=Zeldovich |first=Y. B. |year=1956 |title=The motion of a gas under the action of a short term pressure shock |journal=Akust. Zh |volume=2 |issue=1 |pages=28–38 }}</ref> An independent study about the same field was published by Leonid Ivanovich Sedov in 1959.<ref>{{Cite book |last=Sedov |first=L.I. |title=Similarity and Dimensional Methods in Mechanics |publisher=Elsevier Inc |year=2014 |orig-date=1959 |isbn=978-1-4832-0088-0}}</ref> A complete description was made in 1972 by Grigory Barenblatt and Yakov Borisovich Zel'dovich.<ref>{{cite journal |last1=Barenblatt |first1=G. I. |last2=Zel'dovich |first2=Y. B. |year=1972 |title=Self-similar solutions as intermediate asymptotics |journal=Annual Review of Fluid Mechanics |volume=4 |issue=1 |pages=285–312 |doi=10.1146/annurev.fl.04.010172.001441 |bibcode=1972AnRFM...4..285B }}</ref> The self-similar solution of the second kind also appears in different contexts such as in boundary-layer problems subjected to small perturbations,<ref>{{cite journal |last1=Coenen |first1=W. |last2=Rajamanickam |first2=P. |last3=Weiss |first3=A. D. |last4=Sánchez |first4=A. L. |last5=Williams |first5=F. A. |year=2019 |title=Swirling flow induced by jets and plumes |journal=Acta Mechanica |volume=230 |issue=6 |pages=2221–2231 |doi=10.1007/s00707-019-02382-2 |s2cid=126488392 }}</ref> as was identified by Keith Stewartson,<ref>{{cite journal |last=Stewartson |first=K. |year=1957 |title=On asymptotic expansions in the theory of boundary layers |journal=Journal of Mathematics and Physics |volume=36 |issue=1–4 |pages=173–191 |doi=10.1002/sapm1957361173 }}</ref> Paul A. Libby and Herbert Fox.<ref>{{cite journal |last1=Libby |first1=P. A. |last2=Fox |first2=H. |year=1963 |title=Some perturbation solutions in laminar boundary-layer theory |journal=Journal of Fluid Mechanics |volume=17 |issue=3 |pages=433–449 |doi=10.1017/S0022112063001439 |s2cid=123824364 }}</ref> Moffatt eddies are also a self-similar solution of the second kind.
==Examples ==
=== Rayleigh problem === A simple example is a semi-infinite domain bounded by a rigid wall and filled with viscous fluid.<ref>{{cite book |last=Batchelor |year=2000 |orig-year=1967 |url=https://books.google.com/books?id=Rla7OihRvUgC |title=An Introduction to Fluid Dynamics |page=189 |publisher=Cambridge University Press |isbn=9780521663960 }}</ref> At time <math>t=0</math> the wall is made to move with constant speed <math>U</math> in a fixed direction (for definiteness, say the <math>x</math> direction and consider only the <math>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 display="block">u{(y\!=\!0)} = U</math>
Also, the condition that the plate has no effect on the fluid at infinity is enforced as <math display="block">u{(y\!\to\!\infty)} = 0.</math>
Now, from the Navier–Stokes equations <math display="block">\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>y</math> direction and flow in the <math>x</math> direction, and that the pressure term will have no tangential component so that <math>\dfrac{\partial p}{\partial y} = 0</math>. The <math>x</math> component of the Navier–Stokes equations then becomes <math display="block">\dfrac{\partial \vec{u}}{\partial t} = \nu \frac{\partial^2 \vec{u}}{\partial y^2}</math> and the scaling arguments can be applied to show that <math display="block"> \frac{U}{t} \sim \nu \frac{U}{y^{2}}</math> which gives the scaling of the <math>y</math> co-ordinate as <math display="block">y \sim (\nu t)^{1/2}.</math>
This allows one to pose a self-similar ansatz such that, with <math>f</math> and <math>\eta</math> dimensionless, <math display="block">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 display="block">- \eta f'/2 = f''</math> with solution satisfying the boundary conditions that <math display="block">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)</math> which is a self-similar solution of the first kind.
=== Semi-infinite solid approximation === In transient heat transfer applications, such as impingement heating on a ship deck during missile launches and the sizing of thermal protection systems, self-similar solutions can be found for semi-infinite solids.<ref>{{Cite journal |last=Chang |first=Lang-Mann |date=1986 |title=Transient Heat Conduction in Semi-Infinite Solids with Temperature Dependent Properties |url=https://apps.dtic.mil/sti/tr/pdf/ADA166613.pdf |journal=Technical Report BRL-TR-2720 |volume=86 |page=30105 |publisher=US Army Ballistic Research Laboratory|bibcode=1986STIN...8630105C }}</ref><ref name=":0">{{Cite journal |last=Dec |first=John |title=Lecture #1: Stagnation Point Heating |url=https://tfaws.nasa.gov/TFAWS12/Proceedings/Aerothermodynamics%20Course.pdf |journal=Aerothermodynamics Course |publisher=NASA |pages=105–106}}</ref> The governing equation when heat conduction is the primary heat transfer mechanism is the one-dimensional energy equation:<math display="block">\rho c_{p} \frac{\partial T}{\partial t} = \frac{\partial}{\partial x}\left( k \frac{\partial T}{\partial x} \right)</math>where <math>\rho</math> is the material's density, <math>c_{p}</math> is the material's specific heat capacity, <math>k</math> is the material's thermal conductivity. In the case when the material is assumed to be homogeneous and its properties constant, the energy equation is reduced to the heat equation:<math display="block">\frac{\partial T}{\partial t} = \alpha \frac{\partial^{2}T}{\partial x^{2}}, \quad \alpha = \frac{k}{\rho c_{p}}</math>with <math>\alpha</math> being the thermal diffusivity. By introducing the similarity variable <math>\eta = x/\sqrt{t}</math> and assuming that <math>T(t,x) = f(\eta)</math>, the PDE can be transformed into the ODE:<math display="block">f''(\eta) + \frac{1}{2\alpha}\eta f'(\eta) = 0</math>If a simple model of thermal protection system sizing is assumed, where decomposition, pyrolysis gas flow, and surface recession are ignored, with the initial temperature <math>T(0,x) = f(\infty) = T_{i}</math> and a constant surface temperature <math>T(t,0) = f(0) = T_{s}</math>, then the ODE can be solved for the temperature at a depth <math>x</math> and time <math>t</math>:<ref name=":0" /><math display="block">T(t,x) = \text{erf}\left( \frac{x}{2\sqrt{\alpha t}} \right) \left( T_{i} - T_{s} \right) + T_{s}</math>where <math>\text{erf}(\cdot)</math> is the error function.
==References== {{reflist|30em}}
Category:Fluid dynamics Category:Partial differential equations