(5) We can solve this equation with the appropriate boundary conditions in several interesting cases

This note briefly summarises problems which concern flows confined to a two-dimensional space in the presence of a corner. This might also be useful for a 3D system that is translationally invariant, so effectively a 2D description may be adopted.

TWO-DIMENSIONAL FLOW DESCRIPTION

We have shown that in a Stokes flow vorticity is a harmonic function, so

∇²ω = 0. (1)

If the flow is additionally 2D, we can write it using the scalar Stokes stream function ψ as

u = ∇ × (ψez), (2)

since then the components are given by

u = ∂ψ

∂y, −∂ψ

∂x, 0



, (3)

and the flow incompressibility condition ∇ · u = 0 is satisfied automatically. We also have

ω = (0, 0, −∇²ψ), (4)

which means that the stream function is biharmonic

∇²∇²ψ = 0. (5)

We can solve this equation with the appropriate boundary conditions in several interesting cases. A particular class of these involves boundaries forming a corner, near which the flow structure can be resolved analytically.

CORNER FLOWS

There are numerous examples of flows with the properties mentioned above. Here, we give a couple of interesting ones.

Taylor’s scraping flow

Consider a corner formed by two planar surfaces in a setting when one of the walls is sliding over the other with constant velocity. This has been named a Taylor’s scraping flow after G. I. Taylor, a Cambridge fluid dynamicist. It is also called a paintbrush or knife problem:

0

Figure 1. Stokes flow streamlines in the Taylor’s scaper problem. The scale bar shows the velocity field magnitude relative to the surface velocity.

If we assume in polar coordinates the walls to be at θ = 0 and θ = α, the problem is formulated as

∇⁴ψ = 0 with ur= 1 3

∂ψ

∂θ, u_θ= −∂ψ

∂r, (6)

with the boundary conditions

r > 0, θ = 0 :ur= −U, u_θ = 0, r > 0, θ = α : ur= 0, u_θ= 0. (7) Note that the problem is singular at the origin, where we have a mismatch of the boundary conditions and therefore we expect the stresses to be infinite. We attempt a solution in the separable form

ψ = U rf (θ). (8)

This leads to the equation

f⁰⁰⁰⁰+ 2f⁰⁰+ f = 0, (9)

with the boundary conditions f (0) = 0, f⁰(0) = −1, f (α) = 0, f⁰(α) = 0. It is a simple excercise to see (if you have some excess time, it is left as an exercise to the Reader ;) that the solution is

f (θ) = 1

α²− sin²α[θ sin α sin(α − θ) − α(α − θ) sin θ] , (10) from which the velocity field can be found by differentiation. The streamlines are shown in Fig. 1. It is interesting to see the distribution of stresses which justifies the different choices of the inclination

Figure 2. Stokes flow streamlines in the hinged plates problem. The plates are opening slowly with a constant angular velocity Ω causing the fluid to flow in, as the visible streamlines demonstrate.

angle e.g. when a painter wants to scrape paint off a pallette and when a platsterer wants to squeeze paint into hollows on a surface. Wikipedia offers a detailed discussion of the solution [1]. Notably, this problem raises the questions on how far from the corner is the flow still a Stokes flow? The Reynolds number in this case can be estimated as Re = U r/ν (note that there is no length scale in this problem!), and so the requirement Re  1 determines the region of applicability of the analysis above.

Hinged plates separating

Consider a fluid containing a region −α < θ < α between two hinged plates. Thus the velocity components are

ur= 0, uθ= ±Ωr on θ = ±α, (11)

and the system is illustrated in Fig. 2. The resulting Stokes flow problem may then be solved analyt- ically, as discussed in Exercise 7.5 and 7.6 in Acheson’s book [2].

Moffatt eddies

Another interesting example is a flow that is induced in a narrow corner not by the boundary conditions but by a flow far away from the corner itself. The problem was first considered by Keith Moffatt in 1964 [3, 4] (both files on the course Dropbox, I recommend the JFM paper as a nice summary). See also Ch. 7.3 in Acheson [2]. The flow external to the corner causes a disturbance but the no-slip boundary condition ensures the velocity to vanish on the walls.

The biharmonic equation becomes

 ∂²

∂r² +1 r

∂

∂r + 1 r²

∂²

∂θ²

2

ψ = 0. (13)

We now postulate the solution to be of the form

ψ(r, θ) = r^λf (θ), (14)

and find the general solution (for λ 6= 0 and λ 6= 2) to be

ψ = r^λ[A cos λθ + B sin λθ + C cos(λ − 2)θ + D sin(λ − 2)θ] . (15) The boundary conditions f (±α) = f⁰(±α) = 0 create an eigenvalue problem for λ of the form [2]

λ tan λα = (λ − 2) tan(λ − 2)α, (16)

which may be solved numerically, e.g. using Matlab. The solution is an infinite sequence of roots, most of them complex. We are, however, looking for those that have the smallest real part. It turns our that the roots of this equation are purely real if 2α > 146.3^◦, and for corner angles less than this λ is necessarily complex. What does it mean? Take λ = a + ib. Then

r^λ = r^ae^{ib ln r}, (17)

and thus

ψ(r, θ) = r^a[cos(b ln r)Re(f (θ)) − sin(b ln r)Im(f (θ))] , (18) so ψ oscillates indefinitely near r → 0, so there is a sequence of eddies (vortices) created of ever decreasing size, called the Moffatt’s eddies

and the magnitude of the eddies decreases fast too. [See also the discussion in Acheson]

Interestingly, these vortices have been seen experimentally. See the paper by Taneda [5] (also on the Dropbox) for beautiful demonstrations of eddies in corner flows in a wedge geometry (as Taneda’s Fig 17 is copied below together with a sketch from Acheson), as well as in other geometries in which eddies play a role in very viscous flows

[1] https://en.wikipedia.org/wiki/Taylor_scraping_flow.

[2] D. J. Acheson, Elementary Fluid Dynamics (Oxford Univ. Press, 1990).

[3] H. K. Moffatt, Arch. Mech. Stosowanej 2, 365 (1964).

[4] H. K. Moffatt,J. Fluid Mech. 18, 1 (1964).

[5] S. Taneda,Journal of the Physical Society of Japan 46, 1935 (1979),https://doi.org/10.1143/JPSJ.46.1935.