Similarity solution

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 ν . 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. 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 → 0 as y → ∞ .

Now, if we examine the Navier-Stokes equations

ρ ( ∂ u → ∂ t + u → ⋅ ∇ u → ) = − ∇ p + μ ∇ 2 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 ∂ p ∂ y = 0 . The x component of the Navier-Stokes equations then becomes

∂ u → ∂ t = ν ∂ y 2 u →

and we may apply scaling arguments to show that

U t ∼ ν U y 2

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

y ∼ ( ν t ) 1 / 2 .

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

u = U f ( η ≡ y ( ν t ) 1 / 2 )

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

− η f ′ / 2 = f ″

with solution satisfying the boundary conditions that

f = 1 − erf ⁡ ( η / 2 ) or u = U ( 1 − erf ⁡ ( − y / ( 4 ν t ) 1 / 2 ) )

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