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

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

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

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

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,

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

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

$f = 1 - \operatorname{erf} (\eta / 2)$ 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

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