Lecture 11

Question 1

Why do eddies appear in low Reynolds number corner flows despite the absence of inertial effects?

Answer 1

Eddies emerge due to the geometry of the flow near corners, especially when the opening angle is sharp enough. The streamfunction solution can have complex exponents (imaginary parts in λ), leading to oscillatory behavior in the velocity field—a hallmark of eddies. These are Moffatt vortices.

Question 2

In the solution form 𝜓(𝑟,𝜃)=𝑟^𝜆𝑓(𝜃), what determines the value of 𝜆, and why is its nature (real vs. imaginary) significant?

Answer 2

𝜆 is determined by satisfying boundary conditions, typically through solving an eigenvalue problem. If 𝜆 is imaginary (which occurs when corner angle 𝛼<73∘), the solution becomes oscillatory in ln𝑟, leading to the formation of nested eddies (Moffatt vortices).

Question 3

What physical quantity diverges as 𝑟→0 in G.I. Taylor’s scraping flow, and why is this a problem?

Answer 3

The shear stress 𝜏𝑟_𝜃 diverges as 𝑟→0, since it scales like 1/𝑟.
This indicates an unphysical singularity—real materials would yield or break, showing the breakdown of the model near corners.

Question 4

Describe the general form of the streamfunction solution for a corner flow when no physical length scale is present.

Answer 4

The streamfunction satisfies ∇^4𝜓=0 and is assumed in the form 𝜓(𝑟,𝜃)=𝑟^𝜆𝑓(𝜃). This leads to a 4th order ODE in 𝑓(𝜃), with λ emerging as part of solving this eigenvalue problem under given boundary conditions.

Question 5

What happens to the solution near the corner if multiple 𝜆𝑖 values are possible?

Answer 5

The term with the smallest real part of 𝜆𝑖 dominates near the corner, since it decays slowest as 𝑟→0. If this smallest real part is imaginary, it results in oscillatory flow and vortex structures.

Question 6

In Taylor’s scraping flow, what boundary conditions are imposed at 𝜃=0 and 𝜃=𝛼, and how do they affect the streamfunction?

Answer 6

At 𝜃=0, 𝑢𝜃=−𝑈, and at 𝜃=𝛼, 𝑢𝑟=𝑢_𝜃=0. These conditions constrain the form of 𝑓(𝜃) in the solution 𝜓(𝑟,𝜃)=𝑟^𝜆𝑓(𝜃), leading to a unique λ and specific constants in the general solution.

Question 7

In the anti-symmetric corner eddy configuration (e.g. Moffatt vortices), what symmetry is imposed and how does it simplify the streamfunction?

Answer 7

The symmetry condition 𝜓(𝑟,𝜃)=−𝜓(𝑟,−𝜃) implies that 𝜓 is an odd function in θ, eliminating even cosine terms in the solution and reducing the number of coefficients to determine.

Question 8

Why does the streamfunction solution for corner flows take the form
𝜓(𝑟,𝜃)=𝑟^𝜆𝑓(𝜃)?

Answer 8

This form arises because there is no physical length scale in the local corner geometry. Seeking separable solutions with self-similar scaling leads to a power-law dependence in 𝑟, where 𝜆 is determined by solving the associated angular eigenvalue problem.

Question 9

What determines whether eddies will form in a corner flow?

Answer 9

Eddies form if the smallest eigenvalue 𝜆 has an imaginary part. This occurs when the corner angle 𝛼<73∘, resulting in complex λ and oscillatory flow patterns—i.e., Moffatt vortices.

Question 10

How does the singularity in stress appear in corner flows, and what does it imply physically?

Answer 10

The tangential stress 𝜏𝑟𝜃 scales like 1/𝑟 near the corner, diverging as 𝑟→0. Physically, this means the model predicts infinite stress, which is unrealistic—real fluids or materials would deform or fail before this point.

Question 11

What is the form of the full general solution for 𝜓(𝑟,𝜃) near a corner, and how is it simplified?

Answer 11

The full solution is a sum of modes:
𝜓(𝑟,𝜃)=∑𝐴_𝑖𝑟^𝜆_𝑖 𝑓_𝑖(𝜃)
But near the corner (𝑟→0r→0), the term with the smallest real part of 𝜆𝑖 dominates, simplifying the analysis and governing local behavior.