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 $\nu$. 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 $t=0$ 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). We can see that there is no distinguished length scale given in the problem, and we have the boundary conditions of no slip

$u=U$ on $y=0$

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

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

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 $y$ 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 may apply scaling arguments to show that

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

which gives us the scaling of the $y$ co-ordinate as

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

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

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

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

$-\eta f'/2=f''$

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