# Similarity solution

In study of partial differential equations, particularly fluid dynamics, a similarity solution is a form of solution in which at least one co-ordinate lacks a distinguished origin; more physically, it describes a flow which 'looks the same' either at all times, or at all length scales. These include, for example, the Blasius boundary layer or the Sedov-Taylor shell.[1]

## 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. If we have catalogued these effects we will occasionally find that the system has not fixed a natural lengthscale (timescale), but that 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 ${\displaystyle \nu }$. These constructs are not 'guessed' but are derived immediately from the scaling of the governing equations.

## Example - The impulsively started plate

Consider a semi-infinite domain bounded by a rigid wall and filled with viscous fluid.[2] At time ${\displaystyle t=0}$ the wall is made to move with constant speed ${\displaystyle U}$ in a fixed direction (for definiteness, say the ${\displaystyle x}$ direction and consider only the ${\displaystyle x-y}$ plane). We can see that there is no distinguished length scale given in the problem, and we have the boundary conditions of no slip

${\displaystyle u=U}$ on ${\displaystyle y=0}$

and that the plate has no effect on the fluid at infinity

${\displaystyle u\rightarrow 0}$ as ${\displaystyle y\rightarrow \infty }$.

Now, if we examine the Navier-Stokes equations

${\displaystyle \rho \left({\dfrac {\partial {\vec {u}}}{\partial t}}+{\vec {u}}\cdot \nabla {\vec {u}}\right)=-\nabla p+\mu \nabla ^{2}{\vec {u}}}$

we can observe that this flow will be rectilinear, with gradients in the ${\displaystyle y}$ direction and flow in the ${\displaystyle x}$ direction, and that the pressure term will have no tangential component so that ${\displaystyle {\dfrac {\partial p}{\partial y}}=0}$. The ${\displaystyle x}$ component of the Navier-Stokes equations then becomes

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

and we may apply scaling arguments to show that

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

which gives us the scaling of the ${\displaystyle y}$ co-ordinate as

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

This allows us to pose a self-similar ansatz such that, with ${\displaystyle f}$ and ${\displaystyle \eta }$ dimensionless,

${\displaystyle u=Uf\left(\eta \equiv {\dfrac {y}{(\nu t)^{1/2}}}\right)}$

We have now extracted all of the relevant physics and need only solve the equations; for many cases this will need to be done numerically. This equation is

${\displaystyle -\eta f'/2=f''}$

with solution satisfying the boundary conditions that

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

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

## References

1. ^ Pringle and King, 2007, Astrophysical Flows, p54
2. ^ Batchelor (2006 edition), An Introduction to Fluid Dynamics, p189