Similarity solution

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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[edit]

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 . These constructs are not 'guessed' but are derived immediately from the scaling of the governing equations.

Example - The impulsively started plate[edit]

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

on

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

as .

Now, if we examine the Navier-Stokes equations

we can observe that this flow will be rectilinear, with gradients in the direction and flow in the direction, and that the pressure term will have no tangential component so that . The component of the Navier-Stokes equations then becomes

and we may apply scaling arguments to show that

which gives us the scaling of the co-ordinate as

.

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

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

with solution satisfying the boundary conditions that

or

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

References[edit]

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