Moffatt eddies

Near the corner, the flow can be assumed to be Stokes flow. Describing the two-dimensional planar problem by the cylindrical coordinates $(r,\backslash theta)$ with velocity components $(u\_r,u\_\backslash theta)$ defined by a stream function such that

:$u\_r\; =\; \backslash frac\{1\}\{r\}\backslash frac\{\backslash partial\backslash psi\}\{\backslash partial\backslash theta\},\; \backslash quad\; u\_\backslash theta=-\backslash frac\{\backslash partial\backslash psi\}\{\backslash partial\; r\}$

the governing equation can be shown to be simply the biharmonic equation $\backslash nabla^4\backslash psi=0$. The equation has to be solved with homogeneous boundary conditions (conditions taken for two walls separated by angle $2\backslash alpha$)

:$$

\begin{align}

r>0,\ \theta =-\alpha: &\quad u_r = 0, \ u_\theta= 0\\

r>0,\ \theta =\alpha: &\quad u_r = 0, \ u_\theta= 0.

\end{align}

The Taylor scraping flow is similar to this problem but driven inhomogeneous boundary condition. The solution is obtained by the eigenfunction expansion,Shankar, P. N. (2007). Slow Viscous Flows: Qualitative Features and Quantitative Analysis Using Complex Eigenfunction Expansions (With CD-ROM). World Scientific.

:$\backslash psi\; =\; \backslash sum\_\{n=1\}^\backslash infty\; A\_n\; r^\{\backslash lambda\_n\}\; f\_\{\backslash lambda\_n\}(\backslash theta)$

where $A\_n$ are constants and the real part of the eigenvalues are always greater than unity. The eigenvalues $\backslash lambda\_n$ will be function of the angle $\backslash alpha$, but regardless eigenfunctions can be written down for any $\backslash lambda$,

:$$

\begin{align}

f_0 &= A + B\theta + C\theta^2 + D\theta^3,\\

f_1 &= A\cos\theta + B \sin\theta + C\theta\cos\theta + D\theta\sin\theta,\\

f_2 &= A\cos 2\theta + B \sin 2\theta + C\theta + D,\\

f_\lambda &= A\cos\lambda\theta + B\sin\lambda\theta + C\cos(\lambda-2)\theta + D\sin(\lambda-2)\theta, \quad \lambda\geq 2.

\end{align}

For antisymmetrical solution, the eigenfunction is even and hence $B=D=0$ and the boundary conditions demand $\backslash sin\; 2(\backslash lambda-1)\backslash alpha\; =\; -(\backslash lambda-1)\; \backslash sin\; 2\backslash alpha$. The equations admits no real root when °. These complex eigenvalues indeed correspond to the moffatt eddies. The complex eigenvalue if given by $\backslash lambda\_n\; =\; 1+(2\backslash alpha)^\{-1\}(\backslash xi\_n+i\backslash eta\_n)$ where

:$$

\begin{align}

\sin\xi \cosh\eta &= - k\xi,\\

\cos\xi \sinh \xi &= - k\eta.

\end{align}

Here $k=\backslash sin\; 2\backslash alpha/2\backslash alpha$.

• Taylor scraping flow

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Category:Fluid dynamics

Category:Flow regimes