Kerr–Dold vortex

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The stagnation point flow, which is already an exact solution of the Navier–Stokes equation is given by ${\displaystyle \mathbf {U} =(0,-Ay,Az)}$, where ${\displaystyle A}$ is the strain rate. To this flow, an additional periodic disturbance can be added such that the new velocity field can be written as

${\displaystyle \mathbf {u} ={\begin{bmatrix}0\\-Ay\\Az\end{bmatrix}}+{\begin{bmatrix}u(x,y)\\v(x,y)\\0\end{bmatrix}}}$

where the disturbance ${\displaystyle u(x,y)}$ and ${\displaystyle v(x,y)}$ are assumed to be periodic in the ${\displaystyle x}$ direction with a fundamental wavenumber ${\displaystyle k}$. Kerr and Dold showed that such disturbances exist with finite amplitude, thus making the solution an exact to Navier–Stokes equations. Introducing a stream function ${\displaystyle \psi }$ for the disturbance velocity components, the equations for disturbances in vorticity-streamfunction formulation can be shown to reduce to

${\displaystyle {\begin{aligned}\omega &=-\left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}\right)\psi \\[6pt]{\frac {\partial \psi }{\partial y}}{\frac {\partial \omega }{\partial x}}-{\frac {\partial \psi }{\partial x}}{\frac {\partial \omega }{\partial y}}&-Ay{\frac {\partial \omega }{\partial y}}-A\omega =\nu \left({\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}\right)\omega \end{aligned}}}$

where ${\displaystyle \omega }$ is the disturbance vorticity. A single parameter

${\displaystyle \lambda ={\frac {A}{\nu k^{2}}}}$

can be obtained upon non-dimensionalization, which measures the strength of the converging flow to viscous dissipation. The solution will be assumed to be

${\displaystyle \psi =\sum _{k=-\infty }^{\infty }[a_{k}(y)+ib_{k}(y)]e^{-ikx}.}$

Since ${\displaystyle \psi }$ is real, it is easy to verify that ${\displaystyle a_{k}=a_{-k},\,b_{k}=-b_{-k},\,b_{0}=0.}$ Since the expected vortex structure has the symmetry ${\displaystyle \psi (x,y)=\psi (-x,-y),\,\psi (x,y)=-\psi (\pi -x,y)}$, we have ${\displaystyle a_{0}=b_{1}=0}$. Upon substitution, an infinite sequence of differential equation will be obtained which are coupled non-linearly. To derive the following equations, Cauchy product rule will be used. The equations are^[6][7]

${\displaystyle {\begin{aligned}&a_{k}''''+Aya_{k}'''+(A-2k^{2})a_{k}''-k^{2}Aya_{k}'-k^{2}Aa_{k}+k^{4}a_{k}\\[6pt]&{}+i\left[b_{k}''''+Ayb_{k}'''+(A-2k^{2})b_{k}''-k^{2}Ayb_{k}'-k^{2}Ab_{k}+k^{4}b_{k}\right]\\[6pt]={}&i\sum _{\ell =-\infty }^{\infty }\left\{\left(a_{k-\ell }'+ib_{k-\ell }'\right)\left[\ell a_{\ell }''-\ell ^{3}a_{\ell }+i(\ell b_{\ell }''-\ell ^{3}b_{\ell })\right]-(k-\ell )\left(a_{k-\ell }+ib_{k-\ell }\right)\left[a_{\ell }'''-\ell ^{2}a_{\ell }'+i(b_{\ell }'''-\ell ^{2}b_{\ell }')\right]\right\}.\end{aligned}}}$

The boundary conditions

${\displaystyle a_{k}'(0)=b_{k}(0)=a_{k}(\infty )=b_{k}(\infty )=0}$

and the corresponding symmetry condition is enough to solve the problem. It can be shown that non-trivial solution exist only when ${\displaystyle \lambda >1.}$ On solving this equation numerically, it is verified that keeping first 7 to 8 terms suffice to produce accurate results.^[8] The solution when ${\displaystyle \lambda =1}$ is ${\displaystyle \psi =\cos x}$ was already discovered by Craik and Criminale in 1986.^[9]

• Sullivan vortex